I wrote about motivic cell structures and the Białynicki-Birula decomposition before. Here I'll explain how to compute a motivic cell structure out of the BB-decomposition explicitly and how to get an explicit BB-decomposition for any smooth complete toric variety. Then I'll do the examples.

Explicit Motivic Cell Structure

As described in the last post about Białynicki-Birula's Annals paper from 1972/73, for a $\mathbb{G}_m$-action on a smooth complete variety $X$ over a (possibly non-closed) field $k$ (of arbitrary characteristics), there is the plus-decomposition, giving us for each fixed point $a \in X^{\mathbb{G}_m}$ a subscheme $X_a^+ \subset X$ which is isomorphic to an affine space $X_a^+ \simeq \mathbb{A}^{n_a}$. The number $n_a$ is the dimension of the positive weight part (with respect to the induced $\mathbb{G}_m$-action) of the (co)tangent space of $X$ at $a$.

There is another paper of Białynicki-Birula, published in 1976 in the Bulletin de l'académie polonaise des sciences, Série des sciences math., where it is proved that the BB-decomposition of smooth projective varieties is filtrable, which means that one can choose an order on the fixed points $X^{\mathbb{G}_m} = \{a_0,\dots,a_m\}$ and a partition of $m$ into $d$ blocks (where $d$ is the dimension of $X$) such that for $X_i$ the union of cells of each block, the $X_i$ are closed subschemes of $X$ that form a finite decreasing sequence

$$X = X_d \supset X_{d-1} \supset \cdots \supset X_{0} \supset X_{-1} = \emptyset.$$

Now we build a motivic cell structure out of this. This is slightly unusual, as the attaching maps arise "the other way around" than one would expect. If it confuses you, the examples below might provide illumination.
The open subscheme $X \setminus X_i \to X \setminus X_{i-1}$ has complement isomorphic to a disjoint union of some $X^+_{a_i}$.
Look at the closed immersion of smooth schemes $\iota : X_{a_i}^+ \to X \setminus X_{i-1}$ and its normal bundle $N_{\iota}$. By the homotopy purity theorem of Morel and Voevodsky we have

$$N_\iota / (N_\iota \setminus X^+_{a_i}) =: Th(N_{\iota}) \simeq (X \setminus X_{i-1}) / ((X \setminus X_{i-1})\setminus \iota(X_{a_i}^+)$$

$$= (X \setminus X_{i-1}) / (X \setminus X_i) \simeq S^{2n_i,n_i}.$$

That toric varieties have a motivic cell structure (without referring to the BB-decomposition) is already contained in the paper of Dugger and Isaksen Motivic Cell Structures.

Motives

From a homotopy cofiber sequence $X \to Y \to Z \to \cdots$ we get a distinguished triangle in the category of motives $DM_-$ of the reduced motives $\tilde{h}(X) \to \tilde{h}(Y) \to \tilde{h}(Z) \to[1] \cdots$. Like with reduced and unreduces cohomology theories, $\tilde{h}(Y) \oplus \mathbb{Z} = h(Y)$, so we can compute motives of varieties from homotopy cofiber sequences.

Concrete $\mathbb{G}_m$-actions with isolated fixed points

It is long known for toric varieties, that any torus cocharacter in general position (in the cocharacter lattice) has the same fixed points as the torus. To describe explicit cell decompositions, one needs to know which cocharacter is "in general position" and which cocharacter has a larger fixed point set. A cocharacter $\alpha$ fixes an orbit $O_\tau$ if $\alpha$ is inside the linear subspace generated by $\tau$. The full torus has as fixed points those $O_\tau$ with $\tau$ of maximal dimension. To pick a good cocharacter, one just has to avoid the hyperplanes spanned by the codimension 1 cones in the fan.

The Projective Line

The fan of the projective line consists of the cone $\{0\}$ and the cones generated by $1$ and $-1$ respectively. The affine variety corresponding to the $1$-cone is $\mathbb{A}^1_0 := \mathbb{P}^1 \setminus \{\infty\} \subset \mathbb{P}^1$ and the affine variety corresponding to the $-1$-cone is $\mathbb{A}^1_\infty := \mathbb{P}^1 \setminus \{0\} \subset \mathbb{P}^1$. Their intersection is the affine variety corresponding to the cone $\{0\}$, which is the torus $\mathbb{G}_m = \mathbb{P}^1 \setminus \{0,\infty\}$.

The torus $\mathbb{G}_m$ acts on itself via the group multiplication, and it acts on $\{0\}$ and $\{\infty\}$ trivially, i.e. these are the fixed points of the action. If you prefer homogeneous coordinates, $\lambda \in \mathbb{G}_m(k)$ acts on $[x:y] \in \mathbb{P}^1(k)$ as $\lambda.[x:y] = [\lambda x : y]$, so we have $\lambda.[0:1] = [0:1]$ and $\lambda.[1:0] = [\lambda:0] = [1:0]$.

The Kähler differentials are $\Omega^1_{\mathbb{P}^1/k,0} \simeq \Omega^1_{\mathbb{A}^1_0/k,0} = \langle dX \rangle_k$ and $\Omega^1_{\mathbb{P}^1/k,\infty} \simeq \Omega^1_{\mathbb{A}^1_\infty/k,0} = \langle dX^{-1} \rangle_k$. The induced $\mathbb{G}_m(k)$-action is $\lambda.dX = \lambda dX$ and $\lambda.dX^{-1} = \lambda^{-1} dX^{-1}$, respectively. We see that the positive weight part at $0$ is everything, while at $\infty$ it is nothing. Consequently, the orthogonal at $0$ is nothing and at $\infty$ it is everything. Under $\mathfrak{m} \to \mathfrak{m}/\mathfrak{m}^2$ we get an isomorphic preimage of this orthogonal and take the ideal generated by it. This gives us ideals $\mathfrak{n}_0 = (0)$ and $\mathfrak{n}_\infty = (X^{-1})$. They correspond to the cells $X_0^+ = V(\mathfrak{n}_0) = \mathbb{A}^1_0$ and $X_\infty^+ = V(\mathfrak{n}_\infty) = \{\infty\}$.

This is already the BB-decomposition: $\mathbb{P}^1 = \mathbb{A}^1 \cup \{\infty\}$.
The BB-filtration $X = X_d \supset X_{d-1} \supset \cdots \supset X_{0} \supset X_{-1} = \emptyset.$ of $X = \mathbb{P}^1$ is (with $d=1$) just $\mathbb{P}^1 \supset \{\infty\}$.
To get a motivic cell structure, we need attaching maps for the cell $Th(N_\iota) \simeq X/X_0 = \mathbb{P}^1 \simeq S^{2,1}$ to the set of $0$-cells $X_0 = \{\infty\}$. The stable attaching map is the cofibration $S^{2,1} \to \Sigma (\mathbb{P}^1 \setminus \{\infty\})$, which one can see as the homotopy cofiber of $\mathbb{P}^1 \to \mathbb{P}^1$, i.e. the gluing of $\mathbb{P}^1$ along $\infty$ to a point $\infty$.

The homotopy cofiber sequence yields a distinguished triangle

$$\tilde{h}(\mathbb{P}^1) \to \tilde{h}(S^{2,1}) \to \tilde{h}(\Sigma (\mathbb{P}^1 \setminus \{\infty\})) \to \tilde{h}(\mathbb{P}^1)[1] \to \cdots$$

$$\tilde{h}(\mathbb{P}^1) \to \mathbb{Z}(1)[2] \to 0 \to \cdots.$$

Okay, that was kind of stupid, given that we already knew that $\mathbb{P}^1$ is a $(2,1)$-cell. It was also kind of stupid that we have computed a stable cell structure, while it is also quite easy to describe an unstable cell structure of projective spaces.

We can also take a different $\mathbb{G}_m$-action on $\mathbb{P}^1$, by taking any cocharacter (i.e. group homomorphism) $\mathbb{G}_m \to \mathbb{G}_m$. These are all of the form $\lambda \mapsto \lambda^n$ for some $n \in \mathbb{Z}$. If $n > 0$, we get the same weight decomposition of (co)tangent spaces (Kähler differentials), hence the same BB-decomposition. If $n = 0$, we get no decomposition because the fixed points are not isolated (they are everything). If $n < 0$, we get the decomposition $\mathbb{P}^1 = \{0\} \cup \mathbb{A}^1_\infty$, which one might also call the minus-decomposition w.r.t. the first action considered.

A product of two lines

Here we have a torus $\mathbb{G}_m \times \mathbb{G}_m$ acting and it becomes a slightly more interesting question which cocharacter $\mathbb{G}_m \to \mathbb{G}_m \times \mathbb{G}_m$ gives which BB-decomposition.

The fixed points of the torus are the orbit closures corresponding to the maximal cones, which are $(0,0),(0,\infty),(\infty,\infty),(\infty,0)$.

Take the diagonal $\lambda \mapsto (\lambda,\lambda)$ corresponding to the weight $(1,1)$ in the weight lattice. It doesn't hit any linear subspace generated by codimension one cones, so it has the same isolated fixed points, as the original torus. (In contrast, e.g. $\lambda \mapsto (\lambda,1)$ fixes a whole $\{0\} \times \mathbb{P}^1$).

From analyzing the positive weight subspaces of the cotangent spaces of $\mathbb{P}^1 \times \mathbb{P}^1$ at these fixed points, we get
$(\Omega^1_{(0,0)})^+ = \langle dX,dY\rangle_k$
$(\Omega^1_{(0,\infty)})^+ = \langle dX\rangle_k$
$(\Omega^1_{(\infty,\infty)})^+ = 0$
$(\Omega^1_{(\infty,0)})^+ = \langle dY\rangle_k$
and from this the ideals defining the cells
$\mathfrak{n}_{(0,0)} = (0) \leq k[X,Y]$
$\mathfrak{n}_{(0,\infty)} = (Y^{-1}) \leq k[X,Y^{-1}]$
$\mathfrak{n}_{(\infty,\infty)} = (X^{-1},Y^{-1}) \leq k[X^{-1},Y^{-1}]$
$\mathfrak{n}_{(\infty,0)} = (X^{-1}) \leq k[X^{-1},Y]$
so the cells are
$X^+_{(0,0)} = \mathbb{A}^1_0 \times \mathbb{A}^1_0$
$X^+_{(0,\infty)} = \mathbb{A}^1_0 \times \{\infty\}$
$X^+_{(\infty,\infty)} = \{(\infty,\infty)\}$
$X^+_{(\infty,0)} = \{\infty\} \times \mathbb{A}^1_0$

It is pretty obvious now how much influence the choice of a cocharacter has on the cell decomposition. There are only four different cell decompositions, corresponding to the four maximal-dimensional cones in the fan.

The BB-filtration is $\mathbb{P}^1 \times \mathbb{P}^1 \supset \mathbb{P}^1 \times \{\infty\} \cup \{\infty\} \times \mathbb{P}^1 \supset \{(\infty,\infty)\} \supset \emptyset$. The motivic cell structure is built inductively, we start with the cellular space $\{(\infty,\infty)\} \simeq \mathbb{A}^1 \times \mathbb{A}^1$ and attach the cellular space $Th(N_1) \simeq S^{2,1} \vee S^{2,1}$ to it (to obtain $\mathbb{P}^1\times\mathbb{P}^1 \setminus \{(\infty,\infty)\}$) via the homotopy cofiber sequence

$$\mathbb{A}^1 \times \mathbb{A}^1 \to \mathbb{P}^1\times\mathbb{P}^1 \setminus \{(\infty,\infty)\} \to Th(N_1) \to \Sigma (\mathbb{A}^1 \times \mathbb{A}^1).$$

$$\mathbb{P}^1\times\mathbb{P}^1 \setminus \{(\infty,\infty)\} \to \mathbb{P}^1 \times \mathbb{P}^1 \to Th(N_0) \to \Sigma(\mathbb{P}^1\times\mathbb{P}^1 \setminus \{(\infty,\infty)\}).$$

This is also the motivic cell structure you would get as product cell structure from the previously considered cell structure for $\mathbb{P}^1$, and it is all parallel to classical topology, up to homotopy (though classically the gluing maps don't look that strange).

For the sake of completeness, let's compute the motive from this (i.e. let's look at the distinguished triangles):

$$0 \to \tilde{h}(\mathbb{P}^1\times\mathbb{P}^1 \setminus \{(\infty,\infty)\})) \to \mathbb{Z}(1)[2] \oplus \mathbb{Z}(1)[2] \to 0 \to \cdots$$

$$\mathbb{Z}(1)[2] \oplus \mathbb{Z}(1)[2] \to \tilde{h}(\mathbb{P}^1 \times \mathbb{P}^1) \to \mathbb{Z}(2)[4] \to \mathbb{Z}(1)[3] \oplus \mathbb{Z}(1)[3] \to \cdots$$

The Projective Plane

The fixed points of the torus $\mathbb{G}_m \times \mathbb{G}_m$ are $[0:0:1]$, $[1:0:0]$ and $[0:1:0]$ (corresponding to the maximal dimensional cones $1,2,3$ in the fan, in counter-clockwise order). Denote by $U_i$ the affine toric variety corresponding to the maximal dimensional cone $i$.

The cotangent spaces at the fixed points are
$\Omega_{\mathbb{P}^2,[0:0:1]} = \Omega_{U_1,(0,0)} = \langle dX, dY \rangle_k$,
$\Omega_{\mathbb{P}^2,[1:0:0]} = \Omega_{U_2,(0,0)} = \langle dX^{-1}Y, dX^{-1} \rangle_k$,
$\Omega_{\mathbb{P}^2,[0:1:0]} = \Omega_{U_3,(0,0)} = \langle dXY^{-1}, dY^{-1} \rangle_k$.

The diagonal cocharacter $\lambda \mapsto (\lambda,\lambda)$ is no longer good, since $(1,1)$ lies in the linear subspace generated by a cone of the fan -- it fixes the projective line $\{[x:y:0] | [x:y] \in \mathbb{P}^1\}$.

We can choose the cocharacter $\lambda \mapsto (\lambda^{-1},\lambda)$, which acts with the same isolated fixed points as the whole torus on $\mathbb{P}^2$. For this one we get:
$(\Omega^1_{[0:0:1]})^+ = \langle dY \rangle_k$,
$(\Omega^1_{[1:0:0]})^+ = \langle dX^{-1}Y, dX^{-1} \rangle_k$,
$(\Omega^1_{[0:1:0]})^+ = 0$,
so that we have cells
$X^+_{[0:0:1]} = V(X) = \{[0:y:1] | y \in \mathbb{A}^1\} \subset U_1$,
$X^+_{[1:0:0]} = V(0) = U_2 \simeq \mathbb{A}^2$,
$X^+_{[0:1:0]} = \{[0:1:0]\}$.
This decomposition is one of the common decompositions of $\mathbb{P}^2$ into $\mathbb{A}^2$ and a $\mathbb{P}^1$ at infinity, which is decomposed into $\mathbb{A}^1$ and $\infty$.
The corresponding BB-filtration is just $\mathbb{P}^2 \supset \mathbb{P}^1 \supset \mathbb{P}^0$.

The homotopy cofiber sequences that give the motivic cell structure are

$$\mathbb{A}^2 \to \mathbb{P}^2 \setminus \mathbb{P}^0 \to Th(N_1) \simeq S^{2,1} \to \Sigma \mathbb{A}^2$$

$$\mathbb{P}^2 \setminus \mathbb{P}^0 \to \mathbb{P}^2 \to Th(N_0) \simeq S^{4,2} \to \Sigma(\mathbb{P}^2 \setminus \mathbb{P}^0).$$

If we pick another cocharacter $\lambda \mapsto (\lambda^{-2},\lambda^{-1})$, this is still inside the cone $2$, but it has a different scalar product with one of the rays, so the decomposition should be different. Indeed, from computations we find that the big cell now is part of $U_3$ and in $U_2$ we have only a $0$-cell.

Hirzebruch surfaces

The fan for a Hirzebruch surface $\mathbb{F}_a = \mathbb{P}(\mathcal{O}(a) \oplus \mathcal{O})$ looks similar to the fan of $\mathbb{P}^1\times\mathbb{P}^1 = \mathbb{P}(\mathcal{O} \oplus \mathcal{O})$:

The image shows the fan of $\mathbb{F}_2$.

We pick the torus cocharacter of weight $(1,1)$ again, since it works for all Hirzebruch surfaces.

The cone generated by $(0,1)$ and $(1,0)$ as well as the cone generated by $(1,0)$ and $(0,-1)$ are just as in the situation of $\mathbb{F}_0 = \mathbb{P}^1 \times \mathbb{P}^1$, and the corresponding affine toric varieties glue together to a $\mathbb{P}^1 \times \mathbb{A}^1$, which is visible in the cell structure. The big cell is $\mathbb{A}^2$ corresponding to the cone with faces $(0,1)$ and $(1,0)$.

The cone generated by $(-1,a)$ and $(0,-1)$ corresponds to a fixed point which is a BB-cell itself and the cone generated by $(0,1)$ and $(-1,a)$ corresponds to a fixed point with attached BB-cell of dimension $1$.

The BB-filtration is $\mathbb{F}_a = X_2 \supset X_1 \supset X_0 \supset \emptyset$ with $X_0$ the fixed point which already is a cell and $X_1$ everything except the big cell (big cell = unique open cell). The motivic cell structure is built from the homotopy cofiber sequences

$$\mathbb{A}^2 \to \mathbb{F}_a \setminus X_0 \to Th(N_1) \to \Sigma \mathbb{A}^2$$

$$\mathbb{F}_a \setminus X_0 \to \mathbb{F}_a \to Th(N_0) \to \Sigma(\mathbb{F}_a \setminus X_0).$$

The motive is just the same as the motive of $\mathbb{P}^1 \times \mathbb{P}^1$.

More complete nonsingular surfaces

It is a well-known fact (and not hard to prove) that all complete nonsingular toric surfaces (implicitly assuming normal) are either $\mathbb{P}^2$, a Hirzebruch surface $\mathbb{F}_a$ or a blow-up of one of these at torus fixed points, since one can describe such blow-ups with fans. One easily sees that blowing up a fixed point introduces an additional fixed point with "BB-cell" $\mathbb{A}^1$, therefore an additional $S^{2,1}$ to the motivic cell structure (the additional $\mathbb{P}^1$). This introduces an additional $\mathbb{Z}(1)[2]$ to the motive.

Białynicki-Birula's algebraic Morse theory

The paper is essentially about algebraic torus actions on varieties and relating the induced action on the tangent space of a fixed point to the variety itself. The most simple torus is just the multiplicative group $\mathbb{G}_m$ (think of $\mathbb{C}^\times$ or $\mathbb{R}^\times$). In classical Morse theory, one considers "Morse functions", which are a particular kind of function $X \to \mathbb{R}$, and their gradient flow, which is the flow associated to the gradient vector field. Such a flow is nothing but a $\mathbb{G}_m$-action! Where the Morse-theory people look at smooth manifolds and apply the exponential function from the tangent space (of a critical point of the Morse function, i.e. a fixed point of the flow) to the whole space $X$, an algebraic geometer has to do something else (as the exponential function is not algebraic). This something else is a gimmick invented by Białynicki-Birula. With this gimmick, a $\mathbb{G}_m$-action with isolated fixed points provides a cell decomposition, like the CW decomposition from classical Morse theory.

Proof outline

We work over an algebraically closed field $k$. Let $X$ be a quasi-affine algebraic scheme and $a \in X$ a nonsingular closed point. We denote by $G$ a reductive algebraic group, though in the end only the 1-dimensional torus $\mathbb{G}_m$ is relevant.

Given a $G$-action on a scheme $X$ with fixed point $a \in X$, the tangent space $T_a(X)$ gets a natural $G$-action. For any vector space $V$ with $\mathbb{G}_m$-action there is a decomposition $V = V^- \oplus V^0 \oplus V^+$ into the weight-graded pieces. I call the action definite if either the minus- or the plus-part vanishes, and fully definite if also the zero-part vanishes.

Theorem 2.1: Given a reductive group $G$ acting on $X$ with a closed irreducible $G$-invariant subscheme $X_0$ containing a closed fixed point $a$ nonsingular in $X_0$ and $X$, to any $G$-invariant subspace $U_1$ of the tangent space $T_a(X)$ that contains $T_a(X_0)$ one can find a closed irreducible $G$-invariant subscheme $X_1$ that contains $X_0$ and has the prescribed tangent space.
(This is what I consider a replacement for the exponential function).

Proof idea: The maximal ideal $\mathfrak{m} \leq k[X]$ corresponding to $a \in X^G$ maps $G$-equivariant surjective to $\mathfrak{m}_a/\mathfrak{m}_a^2 = T_a(X)^\vee$. Denote by $U_0 \subset T_a(X)$ the tangent space of $X_0$ and by $\mathfrak{n}_0 \leq \mathfrak{m}$ the ideal corresponding to $X_0$. Since $G$ is reductive, there exists a $G$-submodule $N_1 \subset \mathfrak{n}_0$ that maps isomorphically to $U_1^\perp \subset T_a(X)^\vee$. Then $\mathfrak{n}_1 := N_1k[X]$ is an ideal in $\mathfrak{n_0}$, so the corresponding closed subscheme of $X$ has an irreducible component $X_1$ containing $X_0$. By construction, $T_a(X_1) = U_1$.

Uniqueness of the subspace is also discussed in Theorem 2.2, in particular we have a Corollary to Theorem 2.2: Let $\mathbb{G}_m$ act on $X$ with fixed point $a$. If $U_1$ is either the positive, the negative, the non-negative, the non-positive or the zero-part of the graded vector space $T_a(X)$, then there exists exactly one closed, irreducible and reduced $\mathbb{G}_m$-invariant subscheme $X_1$ through $a$ such that $a \in X_1$ is non-singular and $T_a(X_1)=U_1$.

There is a morphism-version of Theorem 2.1, which is slightly weaker. Roughly, to a $G$-isomorphism of some tangent spaces of two $G$-schemes, you get a third scheme with étale maps to the two others, and if you already have a $G$-isomorphism on subschemes, this is taken into account. The precise statement is

Theorem 2.4: Given for $i=1,2$ sequences $\{a_i\} \to Y_i \to X_i$ of closed immersions of $G$-invariant subschemes of quasi-affine algebraic $G$-schemes and a $G$-isomorphism $\alpha : (Y_1,a_1) \to (Y_2,a_2)$, such that the $G$-modules $T_{a_1}(X_1)$ and $T_{a_2}(X_2)$ are isomorphic, there exists such a sequence $\{a_0\} \to Y_0 \to X_0$ and étale $G$-morphisms $\beta_i : (X_0,Y_0,a_0) \to (X_i,Y_i,a_i)$ that map $Y_0$ onto an open subscheme of $Y_i$.

Proof idea: Inside $X_1\times X_2$ embed $Y_1$ as $Y_0' := Y_1 \times \alpha Y_1$ and apply Theorem 2.1 to get a subscheme $X' \subset X_1\times X_2$ that contains $a_0 = (a_1,a_2)$ and $Y_0'$, with $T_{a_0}(X')=\Delta$. The projections to the factors $X_i$ are étale at $a_0$, hence over a smaller subscheme $X''$ that still contains $a_0$. Denote by $Y'$ the union of the preimages of the $Y_i$ in $X''$, then $Y_0 := Y_0' \cap Y'$ and $X_0 := X'' \setminus (Y' \setminus Y_0')$ do the job.

The local structure of affine "cells" comes from

Theorem 2.5: For any torus $G$ acting on $X$ such that $a$ is a fixed point and the induced action on $T_a(X)$ is definite, there exists a $G$-invariant open neighborhood $U$ of $a$ which is $G$-isomorphic to $(U\cap X^G) \times V$, with $V$ a finite-dimensional fully definite $G$-module.

Proof idea: The $V$ arises as the complement of $T_a(X^G) \subset T_a(X)$.
WLOG (as one can show) $X$ is reduced, irreducible and $G=\mathbb{G}_m$ acts effectively. Apply Theorem 2.4 to $X_1 := X$, $X_2 := X^G \times V$, $Y_1 := X^G$, $Y_2 := X^G \times 0$, $a_1 := a$, $a_2 := (a,0)$, then the resulting $\beta_i : X_0 \to X_i$ are not only étale, but also birational (as one can show), hence open immersions. Then $X_0$ contains an open subscheme $X_0'$ which is $G$-isomorphic to $U_0 \times V$ for $U_0$ some open beighbourhood of $a$ in $X^G$ and $U := \beta_1(X_0')$ gives the statement of the theorem.

In the full proof of Theorem 2.5, the notion of a universal domain $\Omega$ is frequently used. This is a device to handle generic points without talking about prime ideals, which I explained in this blog posts about points.

Given a $G$-rep $\alpha : G \to GL(V)$ one defines the notion of an $\alpha$-fibration $X \to Y$, which carries a $G\times Y$-action on $X$ and Zariski-locally on $Y_i \subset Y$ looks like $V \times Y_i \to Y_i$, with $G \times Y_i$-action induced by $\alpha$. We call $\dim V$ the dimension of the $\alpha$-fibration.
One should remark that an $\alpha$-fibration needn't be a vector bundle, since there might be more $G$-equivariant automorphisms of $V$ than the linear ones.

The following gives us a uniqueness property for $\alpha$-fibration-structures on maps $X \to X^G$.

Corollary to Proposition 3.1: For any torus $G$ acting on a nonsingular $X$, two $G$-representations $\alpha_i : G \to GL(V_i)$ for $i=1,2$ and $\alpha_i$-fibrations $\gamma_i : X \to X^G$ (respectively), then $\alpha_1$ is equivalent to $\alpha_2$. If furthermore $a$ is a $G$-fixed point and $T_a(X)$ is definite, then $\gamma_1 = \gamma_2$.

Proof idea: For any closed point $a \in X^G$, as $G$-modules, $V_i \cong T_a(X)/T_a(X^G)$, so the $\alpha_i$ are equivalent. Note that the $U$ in $U \oplus T_a(X^G) = T_a(X)$ is uniquely determined, since there is no nonzero $G$-homomorphism $T_a(X^G) \to T_a(X)/T_a(X^G)$. By (the corollary to) Theorem 2.2 there exists exactly one $G$-invariant subscheme $X_a$ with $a \in X_a$ nonsingular and $T_a(X_a)=U$, but $\gamma_1^{-1}(a)$ and $\gamma_2^{-1}(a)$ both fulfill these conditions, hence $\gamma_1^{-1}(a) = \gamma_2^{-1}(a)$. This shows $\gamma_1 = \gamma_2$.

We call a morphism $X \to Y$ with $G \times Y$-action on $X$ a $G$-fibration if it is Zariski-locally over $Y_i \subset Y$ an $\alpha_i$-fibration $X \times_Y Y_i \to Y_i$ for some $G$-reps $\alpha_i$. If the dimensions of the $\alpha_i$ all coincide, we call that number the dimension of the $G$-fibration.

Now, let $G=\mathbb{G}_m$ and $X$ any non-singular reduced algebraic $G$-scheme that can be covered by $G$-invariant quasi-affine open subschemes (for example any smooth projective $X$ will do, maybe normal quasiprojective suffices, by Sumihiro's equivariant compactification).

Theorem 4.1: Let $X^G = \bigcup (X^G)_i$ be the decomposition into connected components. For any $i$ there exists a unique locally closed $G$-invariant subscheme $X_i^+ \subset X$ and a unique morphism $\gamma_i^+ : X_i^+ \to (X^G)_i$ such that

1. $\gamma_i^+$ is a retraction, i.e. $(X^G)_i$ is a closed subscheme of $X_i^+$ and $\gamma_i^+|_{(X^G)_i}$ is the identity,
2. $\gamma_i^+$ is a $G$-fibration,
3. for any closed fixed point $a \in (X^G)_i$, the tangent space is $T_a(X_i^+) = T_a(X)^0 \oplus T_a(X)^+$ and the dimension of the $G$-fibration $\gamma_i^+$ is $dim T_a(X)^+$.

Proof idea: Let $a \in X^G$. By Theorem 2.1 there exists a closed $G$-invariant irreducible subscheme $Y_a' \subset X$ with $a \in Y_a'$ nonsingular and $T_a(Y_a')=T_a(X)^0 \oplus T_a(X)^+$. By Theorem 2.5, there is an open $G$-stable nonsingular subscheme $Y_a \subset Y_a'$ that still contains $a$ and $\gamma_a : Y_a \to Y_a \cap X^G$ is a trivial $G$-fibration. Using the Corollary to Theorem 2.2 and the Corollary to Proposition 3.1 (the uniqueness statements) we know for $a,b \in X^G$ that $\gamma_a|_{Y_a \cap Y_b} = \gamma_b|_{Y_a \cap Y_b}$ and for any third fixed point $c \in Y_a \cap Y_b \cap X^G$ we have $Y_a \cap Y_b \supset \gamma_c^{-1}(Y_a \cap Y_b \cap Y_c \cap X^G)$.
Since every $(X^G)_i$ is noetherian, we find $\{a_1,\dots,a_n\} \subset (X^G)_i$ such that $(X^G)_i = \bigcup (Y_{a_i} \cap (X^G)_i)$, so $X_i^+ := \bigcup Y_{a_i}$ is a $G$-invariant, locally closed subscheme of $X$ and a $G$-fibration $\gamma_i^+ : X_i^+ \to (X^G)_i$ can be uniquely glued together from the $\gamma_{a_i}$.

Actually, there is also a minus-decomposition, where you use $T_a(X)^-$ instead. The interplay of these two decompositions for the same $\mathbb{G}_m$-action is explained in Theorem 4.2: Let $G=\mathbb{G}_m$ act on a quasi-affine $X$. For a rational point $t \in X(k)$, the orbit closure $\overline{G(k)t}$ intersects a connected component $(X^G)_i$ in a non-empty set iff $t \in X_i^+$ or $t \in X_i^-$. Moreover, $X_i^+ \cap X_i^- = (X^G)_i$ for all connected components.

Proof idea: The direction $t \in X_i^+ \cup X_i^- \Rightarrow \overline{G(k)t} \cap (X^G)_i \neq \emptyset$ is clear. For the other direction apply Theorem 2.4 to $X_1 := X$, $X_2 := (X^G)_i \times (T_a(X)^+ \oplus T_a(X)^-)$, $Y_1 := (X^G)_i$, $Y_2 := (X^G)_i \times 0$, $a_1 := a$, $a_2 := (a,0)$. For $\beta_1^{-1}(t) = \{t_1,\dots,t_s\}$ we have $\beta_1^{-1}(\overline{G(k)t}) = \bigcup \overline{G(k)t_i}$. Only one $\overline{G(k)t_i}$ contains $a_0$ and one can show (using again Theorem 2.1 and 2.2) that actually $a_0 \in G(k)t_i$ and $t \in X_i^+ \cup X_i^-$.
Moreover, from $((X^G)_i \times T_a(X)^+) \cap ((X^G)_i \times T_a(X)^-) = (X^G)_i \times 0$ we get $X_i^+ \cap X_i^- = (X^G)_i$.

Theorem 4.3: Let $G=\mathbb{G}_m$ act on a complete nonsingular algebraic $k$-scheme $X$, with $X^G = \bigcup (X^G)_i$ the decomposition of the fixed points into connected components. Then there exists a unique locally closed $G$-invariant decomposition $X = \bigcup X_i$ and $G$-fibrations $\gamma_i : X_i \to (X^G)_i$ such that $(X_i)^G = (X^G)_i$ and for any closed fixed point $a \in (X^G)_i$, $T_a(X_i) = T_a((X^G)_i) \oplus T_a(X)^+$.

Proof idea: Take the plus-decomposition from Theorem 4.1, then what's missing for the statement ($X_i \cap X_j = \emptyset$ for $i \neq j$ and $(X_i)^G = (X^G)_i$) follows from analyzing orbit closures (that is actually Theorem 4.2 together with Lemma 4.1 which I didn't include in this summary).

From this follows Theorem 4.4: If the $\mathbb{G}_m$-action in Theorem 4.3 has isolated fixed points, then any $X_i$ is isomorphic to an affine space $\mathbb{A}^{n_i}_k$.

(This looks like a cell structure!)

The proofs and results have been improved a little bit (Hesselink removed the assumption that $k$ is algebraically closed), so that the current level of generality provides the following
Theorem:
Let $X$ be a smooth projective $\mathbb{G}_m$-variety over a field $k$. Then
1) $X^{\mathbb{G}_m}$ is a smooth closed subscheme of $X$ (Iversen),
2) Given the connected components $X^{\mathbb{G}_m} = \bigcup_{i=1}^n Z_i$, there is a filtration $X = X_n \supset X_{n-1} \supset \cdots \supset X_0 \supset X_{-1} = \emptyset$ and affine fibrations $\phi_i : X_i \setminus X_{i-1} \to Z_i$,
3) The relative dimension of $\phi_i$ is the dimension of $T_a(X)^+$ for any $a \in Z_i$.

I want to remark that any generalization of this theorem to quasiprojective or singular situations would be a very impressive result.

The only generalizations I know of are papers of Skowera and Choudhury on Deligne-Mumford stacks and papers of Carrell and Sommese on the Kähler analogue.

Karpenko's, Chernousov-Gille-Merkurjev's and Brosnan's motivic decompositions

Theorem (Karpenko):
Let $X$ be a smooth projective variety over a field $k$, equipped with a filtration $X = X_n \supset X_{n-1} \supset \cdots \supset X_0 \supset X_{-1} = \emptyset$ where the $X_i$ are closed subvarieties, and affine fibrations $\phi_i : X_i \setminus X_{i-1} \to Z_i$ of relative dimension $n_i$. Then the Chow motive decomposes $h(X) = \bigoplus_{i=0}^n h(Z_i)(n_i)$.

Corollary (Brosnan):
Let $X$ be a smooth projective $\mathbb{G}_m$-variety over a field $k$. Then $h(X) = \bigoplus_{i=0}^n h(Z_i)(n_i)$, where the $Z_i$ are the connected components of the fixed point locus $X^{\mathbb{G}_m}$.

From this, Brosnan re-proved
Theorem (Chernousov-Gille-Merkurjev):
For $X$ a projective homogeneous variety (for a reductive group) over a field $k$, the kernel of the map $End(M(X)) \to End(M(X_{\overline{k}}))$ consists only of nilpotent elements.

Brosnan proved more, in particular how the motive of $G/P$ decomposes, using this method: $M(G/P) = \bigoplus_{w \in E} M(Z_w)(\ell(w))$, where $\ell(w)$ is length and $E$ is the set of minimal length coset representatives of $W_I\backslash W/W_J$, with $J$ the set of roots corresponding to $P$ and $I$ the set of roots that are killed by a non-central cocharacter of the maximal torus (taking care of the possible non-splitness of the maximal torus).

Wendt's cellular decomposition of the stable motivic homotopy type

Using the BB-decomposition, one gets a decomposition of the motive. Actually, one gets a bit more, namely a decomposition in the stable A¹-homotopy category. This is even more analogous to CW decompositions coming from Morse theory.

What you need for a cellular decomposition (in the homotopy-theoretic sense), but what's missing in a direct sum decomposition of the motive, are the gluing maps. One has to extract these gluing maps from the BB-decomposition. This was done by Wendt, who used this approach to show stable cellularity of connected split reductive groups and their classifying spaces, as well as stable cellularity of smooth projective spherical varieties under connected split reductive groups.
As this post is already too long, I might explain the motivic cell structures in another post. Actually, you can just take a look at the preprint.

It remains to see how these cell structures look like explicitly!

A list of some things that have a fundamental group flavor:

• The usual fundamental group $\pi_1(M,m_0)$ of a topological space $M$ with basepoint $m_0 \in M$.
• The simplicial fundamental group $\pi_1(N,n_0)$ of a simplicial set $N$ with basepoint $n_0 \in N_0$.
• The fundamental group $\pi_1(A,a_0)$ of an object $A$ with basepoint $a_0 \in A$ in a Quillen model category.
• The absolute Galois group $Gal(k^{sep}/k)$ of a field $k$ with separable hull $k^{sep}$.
• The fundamental group of a linear algebraic group: root lattice modulo weight lattice.
• The étale fundamental group $\pi_1^{et}(X)$ of a scheme $X$.
• The fundamental group scheme $\pi_1(X,x_0)$ of a scheme $X$ over a perfect field $k$ with rational basepoint $x_0 \in X(k)$.
• The motivic fundamental group $\pi_1(X,x_0)_M$ of a scheme $X$ over a perfect field $k$ with rational basepoint $x_0 \in X(k)$.
• The Tannakian fundamental group object $\pi_1(\omega)$ of a fiber functor $\omega$ of a Tannakian category.
• The polynomial fundamental group presheaf $\pi_1^{poly}(X,x_0)$ of a scheme $X$ over a field $k$, with rational basepoint $x_0 \in X(k)$.
• The A¹-fundamental group (pre)sheaf $\pi_1^{\mathbb{A}^1}(X,x_0)$ of a scheme $X$ over a field $k$, with rational basepoint $x_0 \in X(k)$.

Related concepts and observations:

• For an object $M$, the set of objects of some type over $M$, i.e. the set of certain maps to $M$, is often classified by something like a fundamental group. In the case of the topological fundamental group, the covering spaces are in 1:1 correspondence with subgroups of the fundamental group. In particular, there is one covering space which has a trivial fundamental group itself (it is then called simply connected) and the automorphisms of this covering map form precisely the fundamental group of the base space. A similar covering space theory holds for the absolute Galois group and field extensions instead of coverings.
• There are operations that convert one notion of fundamental group to another. For example the singular set of a topological space is a simplicial set that has as simplicial fundamental group exactly the topological fundamental group of the original space (and the geometric realization goes the other way around).
• For many fundamental-group concepts, there are higher homotopy groups $\pi_n$ and long exact sequences that connect the information of $\pi_1$ with the $\pi_n$. For some fundamental-group concepts, there are no "higher groups" (known so far).
• One may often neglect basepoints, if the resulting groups are isomorphic anyway (under some connectedness assumption). This, however, can be dangerous, as one gets an action of the fundamental group at one basepoint on each other homotopy group, and this action might be non-trivial. Another danger comes from fundamental groups that are not just groups, but also carry some other structure (that of an algebraic variety, or a mixed Hodge structure, for example) -- these additional structures may then depend on the basepoint. If one wants to forget about basepoints, there is the safe way of working with fundamental groupoids instead.
• There are more delicate relations between the various fundamental groups. For a connected projective algebraic variety $X$ over $\mathbb{C}$ one may use an embedding $X \to \mathbb{P}^N$ to put the submanifold topology of the complex manifold $\mathbb{P}^N(\mathbb{C})$ on $X(\mathbb{C})$, call this $X(\mathbb{C})^{an}$ and compare $\pi_1^{et}(X)$ with $\pi_1(X(\mathbb{C})^{an})$. It turns out that the étale fundamental group is the profinite completion of the topological fundamental group. This may be explained by the fact that the étale topology only allows finite covers, so for example $exp : \mathbb{R} \to S^1$ is a topological covering that doesn't come from any étale covering of $\mathbb{C}^\times$.
• Some of the fundamental groups on the list are special cases of others. For example, both the topological and the simplicial fundamental group can be defined in the context of Quillen model categories. This also clarifies the relation between them, since the singular set functor and the geometric realization form what is called a "Quillen adjunction", which axiomatizes this relationship. On the other hand, the Galois group of a field is just the étale fundamental group of the spectrum of that field -- which has no direct connection to model categories at all.
• Well, actually it is a bit more complicated: the precise relation between étale fundamental groups and Galois groups with respect to basepoints is described in this MO answer by Minhyong Kim.
• Some of these fundamental groups are really not very fundamental-group-like: The algebraic fundamental group of a reductive linear algebraic group, as explained on MO by Brian Conrad here, is not the étale fundamental group and its relation to it (and other fundamental groups) can be very loose.

Systematically:

I claim that there are two or three major approaches to fundamental groups. The first is via Quillen model categories, which captures the intution of loops at some basepoint modulo basepointed homotopy, and which directly gives us higher homotopy groups and long exact sequences. The second is via Tannakian categories, which captures the intuition of being an automorphism group, and which may directly give us some extra structure on the fundamental group.
Grothendieck's Galois Theory of Topoi captures the intuition of monodromy, but essentially is a variation of the Tannakian point of view (in my eyes).

Fundamental groups definable in terms of Quillen model categories

The brief definition of the fundamental group of an object $X$ in a (base-pointed) Quillen model category: Take a fibrant replacement $\tilde{X} \to X$ and then $\pi_1(X) := \pi_1(\tilde{X}) := [S^1,\tilde{X}]$ where the brackets mean base-pointed homotopy, i.e. $Map(S^1,\tilde{X})/\sim$ with $\sim$ the (left)homotopy relation. For that to make sense, you obviously need an object $S^1$ in the model category; this can be constructed once you have some kind of interval, for example if the model category is in fact a simplicial model category. Alternatively, without an interval you can still define the loop space via the path space fibration $\Omega X \to P X \to X$ and then take $\pi_1(X) := \pi_0(\Omega X) := \Omega X / \sim$ where $\sim$ is the (left)homotopy relation.

• The usual fundamental group of a topological space comes from the model structure with homotopy equivalences as weak equivalences and Hurewicz fibrations as fibrations (or any other Quillen equivalent model structure).
• The simplicial fundamental group comes from the Kan model structure, with Kan fibrations as fibrations (and the weak equivalences can be defined without reference to fundamental groups).
• The polynomial fundamental group presheaf is a presheaf of simplicial fundamental groups of singular resolutions.
• The A¹-fundamental group (pre)sheaf is the group (pre)sheaf coming out of the Morel-Voevodsky model category.
• Étale fundamental groups may be defined in terms of an étale homotopy category as well, though this is not widely used.

Fundamental groups definable in terms of Tannakian categories

The brief definition of the fundamental group of a fiber functor $\omega$ of a Tannakian category over a field $k$ is just $\underline{\mathcal{A}ut}^{\otimes}(\omega)$, the $\otimes$-automorphisms of the functor (which are invertible natural transformations from $\omega$ to itself). This is, in general, a pro-algebraic group scheme over $k$. If the Tannakian category is $\otimes$-generated by a single object $X$, then $\underline{\mathcal{A}ut}^{\otimes}(\omega) \subset \mathcal{A}ut(\omega(X))$.

• The topological fundamental group of a space $X$ with basepoint $x_0 \in X$ is the Automorphism group of the fiber functor from the covering category. The covering category is the Tannakian category of covering spaces $Y \to X$ with lift $y_0$ of the basepoint, and the fiber functor forgets $X$. If there is a single generator of the Tannakian category, i.e. a universal covering, then the Automorphism group of the fiber functor is just the automorphism group of this single covering (where we're talking about automorphisms of the morphism, not just of the space).
• The absolute Galois group of a field $k$ comes out of the category of all separable algebraic field extensions $L/k$, with the fiber functor $\omega : L/k \mapsto L$, since $\mathcal{A}ut(\omega(k^{sep})) = Gal(k^{sep}/k)$.
• The motivic fundamental group of Tate motives over a scheme $S$ is given by the category of mixed Tate motives $MTM(S)$ and the Betti realization. Here, $MTM(k)$ can be constructed as subcategory of $DM_-$ cut out by a t-structure if $k$ is a number field. For rings of integers of number fields, one can give an ad-hoc modification of $MTM(k)$ that gives the right answer. There are also models from rational homotopy theory (see Deligne-Goncharov, Esnault-Levine).

One might even ask, as Harry Gindi did on MO, whether one could define higher homotopy groups in the Tannakian setting. Obviously, one would need higher Tannakian categories.

Grothendieck's Galois Theory

Given a Topos $Sh(C)$ of sheaves on some category C, and a point p, we can look at the functor that assigns to each sheaf its stalk at that point. If we take only locally finite, locally constant sheaves (aka local systems), we get a functor to finite sets
$\rho_p : Loc(C) \to FinSet$
which results in an equivalence of $Loc(C)$ to a category of modules under a group, the corresponding Galois group.

See this MO question of Lars Kindler for the relation to the Tannakian POV.

Two more fundamental groups:

• The geometric fundamental group of a scheme $X \to k$ over a field $k$ is given as the kernel of the morphism $\pi_1^{et}(X) \to \pi_1^{et}(k) = Gal(k)$. This morphism can be seen to come from a morphism $Sch/X \to Sch/k$. If $k$ is a number field, the geometric fundamental group is the profinite completion of the usual topological fundamental group of the associated complex space.
• The motivic fundamental group of a scheme $X \to k$ is similarly given as the kernel of the morphism $\pi_1(MTM(X)) \to \pi_1(MTM(k))$. There has been some recent research around this object, especially the special case of motivic $\pi_1$ of $\mathbb{P}^1 \setminus \{0,1,\infty\}$. The reason is a deep connection with multiple Zeta values.

Once you have any definition of a fundamental group, it is interesting whether you can find a sheaf topos such that the local systems correspond 1:1 to the representations of this fundamental group. That would be in the spirit of Caramello's unification of mathematics via topos theory.

A PDF/A file is a document that probably ends in .pdf, complies to the PDF 1.4 standard (not more or less), has ORCed text in the background layer to allow for full-text search, has valid metadata in XMP format (yay!), and the compression is Mixed Raster Compression (MRC) which allows quite small documents (though DJVU is still slightly smaller in my experience). Actually that was more-or-less PDF/A-1b, the basic version. Now there is also PDF/A-2, where you can use better compression (JPEG2000), transparencies and layers, since it's based on PDF 1.7. The "A" in PDF/A stands for "archive-able".

Quick solution

In Debian or Ubuntu GNU/Linux, if you like graphical user interfaces:
sudo apt-get install scantailor
will bring you all you need. Under the hood works a command-line tool:
sudo apt-get install unpaper
and you can get ScanTailor and unpaper also from their websites.

Long Story

Step 1: produce high-quality input data

To find good hardware for scanning that is linux-compatible, just compare technical specifications and prices as usual, and once you have a list of 1--5 devices you might buy, check against the list of linux-supported devices here.
An alternative to the usual flatbed-scanner setup is to construct something yourself, like an open-source book scanner, another open-source book scanner, or a slide-scanner made from a camera.

I use a recent Canon model (LiDe 210) that works without quirks in Ubuntu Linux 12.10. I use scanimage on the command-line and the GUI of XSane (though it looks a bit old-fashioned) so let me tell you about the available options on XSane. Using other scanning software on linux most probably means using another UI to the Sane library, so the options are the same.

For OCR, the best mode is "Gray" or "Color", but not lineart. The resolution should be 300 or 600 DPI, more is usually not necessary and slows down the post-processing. If you're low on memory, high DPI values might even make the post-processing impossible. There is a green-blueish button for automatic gamma, brightness and contrast, which makes sense after acquiring a preview of your scan; I recommend the default enhancement values (1,0,0) since we can post-process later in the proper tools. Some post-processing tools have problems with 16-bit images, so I recommend to use 8-bit (in the "Bit depth"/"Standard Options" window of XSane). For most post-processing tools, it is convenient to have the scans in TIFF or PNG format. With TIFF you have to make sure that lossless compression is activated in the Sane configuration (see also these scanning tips).

Speckles and black borders on a document can make it really hard for OCR software, so you should try to get your scan as clean as possible. It may help to acquire a preview and crop manually.

To make sure the end-result contains all available relevant metadata, I recommend taking as much information as possible into your filename already, like some date attached to the scanned piece (if it is a letter or a photo) and some context. This will later make it easy to move this information into the PDF, especially if you intend to scan many pieces at once.

If you want to generate PDF/A compliant PDFs, one solution is to use LaTeX, where you just insert your scan(s) as embedded images, and the metadata where it belongs. There is a tutorial for PDF/A compliant PDFs out of LaTeX, though it doesn't touch the issue of embedding scanned images or OCRed text.

Step 2 (optional): use unpaper to remove artifacts

UnPaper is a very useful software to remove any paper artifacts from you scans. In principle, this enables you to get printouts of your scan that look like actual re-prints, not photocopies. This is especially useful for the purpose of OCR.

The standard interface for UnPaper is the command-line, but there are also GUIs available. Some of them are still at an early stage of development, like GScan2PDF, others seem to be discontinued, like OCRFeeder, so I recommend using ScanTailor (download!).

ScanTailor has the assumption of scanning books in mind, so it is optimized to scan two pieces at once and later splitting them into two separate pages. This was useful to me when I wanted to scan large amount of photos, 4 at a time, to split them later in ScanTailor.

Warning: with high resolution come large files, so the post-processing that happens in ScanTailor can be slow. If you have a whole book to scan, I would recommend finding out the right parameters by hand and using the command-line UnPaper instead of ScanTailor.

UnPaper and ScanTailor take image files like TIFF or PNG and give back TIFF.

Step 3a: compress into a djvu

DjVu files are known for their incredible compression. However, the magic ingredient for that is "Mixed Raster Compression" (MRC), which you can also use in PDFs. Since PDF/A is the archive standard, not DJVU/A, and future tools enable MRC in PDF, DjVu will become even less important.

There is already a wonderfully detailed tutorial online on how to digitize books to DjVu, even with a section covering OCR.

As far as I know, this must be done on the command-line, since no free GUI is available.

Step 3b: compress into a pdf

To convert a bunch of TIFFs to PDF, there is tiff2pdf. You can supply some metadata on the command-line, to be included in the PDF.

Example usage:
tiff2pdf -o outputfile.pdf -z -u m -p "A4" -F inputfile.tif

The switch "-z" enables lossless compression, instead you could use "-j -q 95" for 95% quality JPEG compression. The switch "-p "A4"" specifies the paper size, which could also be "letter". The switch "-F" causes the TIFF to fit the entire PDF page, to avoid borders.

Another example:
tiff2pdf -o outputfile.pdf -z -u m -p "A4" -F -c "tiff2pdf" -a "Author Name" -t "Document Title" -s "Document Subject" -k "keyword1,keyword2,keyword3"
−e 20130324103000 inputfile.tif

This line will include the given metadata into the resulting PDF.

Step 3c: OCR

Between post-processing the scans and compressing them into a PDF, we might want to run OCR on them. I still use tesseract/hocr2pdf to do that, since the Tesseract engine tends to give me the best results, and hocr2pdf is the only solution I know of that can "hide" the scanned text in a layer behind the scanned image, to give you true full-text search without damaging the scan quality at all.

With whatever input data you have, I recommend the following:
convert -normalize -density 300 -depth 8 "inputfile.ext" "normalized-input.png"
since tesseract really works best with normalized images at density 300 and bit-depth 8, in PNG format.

Tesseract is language-sensitive. If you do
tesseract -l deu -psm 1 "normalized-input.png" "output.pdf" hocr
it will assume german text (deu=deutsch=german), but the switch "-l eng" will change that to english language. There are many other languages available (see "man tesseract"), and you can build your own.

To merge back the hocr data into the PDF, you need to convert the PNG to JPEG and run hocr2pdf:
convert "normalized-input.png" "normalized-input.jpg"
hocr2pdf -i "normalized-input.jpg" -s -o "output.pdf" < "output.pdf.html"

To get the metadata right, you might want to use PDFTk and its dump_data,update_info commands. Take a look at the final shell script below for this.

Step 4 (optional): validate

Standards are only good as long as you can validate them. This is possible for PDF/A with JHOVE, the JSTOR/Harvard Object Validation Environment (pronounced "jove"). Though it still has some bugs, it is the only viable free alternative to Adobe's Windows-only Preflight mode (which is still better, I admit).

After extracting the JHOVE files to some directory "jhove", you have to edit the file "jhove/conf/jhove.conf" and change something in "something" to the actual directory (ending in "/jhove").

After you got that right, run
java -jar jhove/bin/JhoveView.jar
to get the interactive program. You can change the configuration there as well. Once I had the strange issue that I had to change the directory from the UI tool to make the CLI tool work...

If you prefer to stay on the command-line, to automate your workflow, try
java -jar jhove/bin/JhoveApp.jar -m PDF-hul "filename.pdf"
and watch out for the lines beginning with "Status" and "ErrorMessage".

You'll notice that most documents have some errors, but these don't affect reading the documents. It is actually quite hard to get a PDF/A-conforming document!

I did a little survey on my own archive of PDFs, mostly from the arXiv and mathematical journals, in total about 500 PDFs. The errors (also happening in files that seem to be generated from a TeX source and files from JSTOR or Journal homepages) were:

• InfoMessage: Too many fonts to report; some fonts omitted.: Total fonts = ...
• InfoMessage: Outlines contain recursive references.
• ErrorMessage: Improperly formed date
• ErrorMessage: Lexical error
• InfoMessage: File header gives version as 1.4, but catalog dictionary gives version as 1.6
• ErrorMessage: Invalid page dictionary object
• ErrorMessage: Invalid outline dictionary item
• ErrorMessage: Invalid object number in cross-reference stream
• ErrorMessage: Invalid destination object
• ErrorMessage: Invalid Resources Entry in document
• ErrorMessage: Malformed dictionary
• ErrorMessage: Malformed filter
• ErrorMessage: No PDF header
• ErrorMessage: No PDF trailer
• ErrorMessage: Unexpected error in findFonts: java.lang.ClassCastException: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject cannot be cast to edu.harvard.hul.ois.jhove.module.pdf.PdfDictionary
• ErrorMessage: Unexpected error in findFonts: java.lang.ClassCastException: edu.harvard.hul.ois.jhove.module.pdf.PdfStream cannot be cast to edu.harvard.hul.ois.jhove.module.pdf.PdfDictionary

The last two are obviously bugs in JHOVE. The "too many fonts to report" info message came about 100 times. About 100 files (not the same, but with some overlap) out of the total 500 were invalid PDF/A. Nevertheless, all these files are perfectly readable. It is not clear, if they would be readable on other devices, like a Kindle or Android. I also encountered printing errors with malformed PDFs in the past, so I recommend getting rid of these errors at least in the files you produce after scanning.

One Shell Script to Rule Them All

This is a script to call from the command-line, to scan and OCR directly to PDF/A.

usage:
./scan-archive.sh filename.pdf title subject keywords

example usage:
konrad@sagebird:~/Documents/scans$ ./scan-archive.sh Letter-20130324-Bankaccount-closing.pdf "Letter from the bank" finances bank,account,closing

full script (also available on pastebin):
#!/usr/bin/env bash
echo "usage: ./scan-archive.sh filename.pdf title subject keywords"
echo "scanning \"$2\" on \"$3\" about \"$4\"... ($1)"
scanimage --mode Color --depth 8 --resolution 600 --format pnm > out.pnm
echo "processing... ($1)"
scantailor-cli --color-mode=black_and_white --despeckle=normal out.pnm ./
rm -rf cache out.pnm
tiff2pdf -o "$1" -z -u m -p "A4" -F -c "scanimage+unpaper+tiff2pdf+pdftk+imagemagick+tesseract+exactimage" -a "Author Name" -t "$2" -s "$3" -k "$4" out.tif
rm -f out.tif
echo "converting to PDF 1.4 ($1)..."
mv "$1" "$1.bak"
pdftk "$1.bak" dump_data > data_dump.info
pdftk "$1.bak" cat output "$1.bk2" flatten
echo "OCR in lang deu... ($1)"
convert -normalize -density 300 -depth 8 "$1.bk2" "$1.png"
tesseract -l deu -psm 1 "$1.png" "$1" hocr
convert "$1.png" "$1.jpg"
hocr2pdf -i "$1.jpg" -s -o "$1.bk2" < "$1.html"
echo "Inserting metadata... ($1)"
pdftk "$1.bk2" update_info data_dump.info output "$1"
rm -f "$1.bak" "$1.bk2" data_dump.info
rm -f "$1.png" "$1.jpg" "$1.html" "$1.pdf"
echo "done. wrote file. ($1)"
echo "validating... ($1)"
java -jar jhove/bin/JhoveApp.jar -m PDF-hul "$1" |egrep "Status|Message"


You should obviously customize "Author Name", and you might want to skip the validation step in the end. In other environments, "A4" might be better replaced with "Letter" or "A3", depending on your scan format. Purists might want to skipt the conversion to JPEG, which I used to get smaller files. In JPEG2000, the same compression technique that powers DjVu (MRC) is possible.

Maybe one should try the suggestions here for other Tesseract UIs, but I'll stick to the command-line for now. Any other suggestions?

Measuring the size of an object is done by comparing it to some other object of known size. The other object of known size is used to define a unit of size (or length, if you prefer). Measuring mass of an object is done by comparing it to some other object of known mass, which are both measured indirectly by measuring the gravitational force influenced by the earth. With very precise instruments or very heavy objects, one could also measure the force without thinking about a third object. In any case, one would actually measure the work over some known length.

How do we measure time? We can compare with some known time-frame, like the duration of one full trip of the earth around the sun (also known as "year"), but only if we're ignoring that this duration changes over time. We can take a look at our watches; There, small Quartz crystals vibrate such that the frequency doesn't change over time, with high precision. To get higher precision, one can only take oscillating signals that are even more stable, like atomic clocks that use the frequency of (microwave) photons that are emmited in electron energy-level transitions. These are the best clocks we have on earth and this level of precision is not reached out of sheer scientific curiosity, but it's necessary for GPS to work.

What exactly does it mean to compare a time-frame with some frequency of an oscillation? In the easy example of a year, it means that we take a look at the position of the earth around the sun (for example, it could be summer solstice) when we start the measurement, and we observe how this position changes during measurement. When the measurement is done we integrate ("sum") over all the position changes and get the total length of the path the earth has travelled around the sun. This means that a unit of time is essentially specified by an oscillation in space. While this is not directly true for the atomic clocks, it is true indirectly for two reasons: we observe the electron energy-level transitions ultimately as oscillation of the position of something and every oscillation in space specifies just a scalar mutliple of the time frame specified by the microwave oscillation.

So, to measure time, we have to observe how things move over some distances, and our units of time are intimately linked to the distance used in the observation - it is impossible to give a precise definition of "a second" without reference to any distances, lengths or sizes if we want to be able to measure "a second".

How did we measure distances? By comparing them with known objects, where "known" means that we ultimately use one thing as reference for all the others, somewhat arbitrarily. A century ago, people used the "Ur-meter", a certain object of 1 meter length, for comparisons. Since then, molecules of the Ur-meter were displaced and the length changed measurably, so 1983 the definition of 1 meter was changed to "the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second". We can safely forget about the archaic factor 299,792,458 and observe that length is measured via time now. Didn't we observe before that time is measured via length?

The definition of length units in terms of time the light takes has the advantage that light is the same everywhere and available in abundance, so we can compare measurements without having to move or make copies of any other reference object. The same is proposed for the kilogramm - it can be defined via the energy of a photon (by Einstein's famous formula).

Measuring time can be a very complicated issue, since we don't want to measure the time at one place, but usually at multiple places, to compare them. For example, our artificial satellites measure their position by measuring how long the light takes to go between a point on the earth and the satellite. For this, they need to have a clock on board. From the change of their position, they can calculate their speed. This is where things become interesting, since a change of speed alters the perception of time (general relativity). Taking general relativity into account, one can use insights from quantum physics to build high-precision atomic clocks and computers and launch them into space, which gives us GPS.

What do we mean by "observe the frequency of a photon"? In the end, it means we build a complicated apparatus that transfers information in the form of (other) photons to us, maybe via a computer screen that displays a number. To build that apparatus, we need some model of the reality, some units of length, time and mass. This means that the measurement is always bound by the theory that is used to make the measurement.

By the way: this article of Terence Tao about dimensional analysis inspired me to finish this article now, and I can only recommend it! While we're at it: you should also take a look at Longitude, a book about the importance of time measurement for sea navigation and a true story how we got the first usable clocks.

In short:
Is there a Homotopy Theory of Proofs?

Suppose you take two sets of statements A and B, for example A says that some set X and a product on X satisfies the axioms of a group, and B says something about X and the product which is satisfied for any group. There are many ways to prove $A \Rightarrow B$, though we might be inclined to see many of them (if not all) somewhat equivalent. Suppose two proofs are considered equivalent, can you image that there are two different ways to see that/how they are equivalent? This is, very roughly, what I think should a homotopy theory of proofs be.

Statement sets (or just statements?) would be points (0-cells), proofs the arrows/morphisms (1-cells) and equivalences of proofs (whatever that may be) the 2-morphisms (2-cells) and so on, with higher homotopies.

One way to get that rigorous might lie in the syntax: take as statements syntactic statements and as proofs full syntactic proofs. Then one can at least assign to two proofs $A \Rightarrow B$ the Hamming distance, which gives us the structure of a (pseudo)metric space.

My question is: is there anyone out there already working on such stuff? Is there any hope or can you give an argument why that is inherently nonsensical?

My next steps would be to seek more examples of different proofs for the same elementary implications to get some feeling for how the transformations between proofs might look like.

Another idea was to look at Turing machines instead, and take reductions as 2-morphisms. Can you think of anything that goes in that direction?

Why this question isn't suitable for MO:
I didn't even try to do an extensive search in the literature (as I don't really know where to start, not being a logician), I can give no good answer to the question "what would you expect from such a theory to prove/do?" and I don't know what would qualify as acceptable answer(s). Maybe in 1-2 years I will have thought enough about this topic to distill a MO-level question. Up to then, this will have to suffice.

I'll be happy with any input (and I know there are at least 2 people who also like this idea)!

[UPDATE 2013-03-29] Thank you for the comments so far, also on Google+!

A Feynman graph is just a (non-directed) graph with a finite number of vertices and a finite number of edges. Physicists are interested in computing certain integrals defined in terms of Feynman graphs, which they call amplitudes.

To a graph one can associate the graph polynomial which is defined as a sum over all spanning trees. A tree is a subgraph which is 1-connected (connected and without loops) and a spanning tree is a tree which meets every vertex. Consider the polynomial ring over the integers with one polynomial variable for each edge of the graph. Then, for a spanning tree, take the product over all variables that correspond to edged not in the tree. Sum these over all spanning trees and you get the graph polynomial. This is also called Kirchhoff polynomial, since it was discovered by Kirchhoff in work on electric circuits. Minor caveat: whereas the classical Kirchhoff polynomial of a disconnected graph should be 0, the graph polynomial should be defined as the product of the Kirchhoff polynomial of the connected components.

With a polynomial comes an affine variety, i.e. the zero-set of the polynomial. In general, this variety will be highly singular. Nevertheless, we can assign a motive (in Voevodsky's category DM) to this variety.

Now one can analyze whether these motives are of a special kind, which periods they give (the amplitudes mentioned above are actually periods) and so on. Since many small graphs give multiple zeta values as periods, there was the conjecture floating around (by Kontsevich) that Feynman motives (=motives coming from Feynman graphs) are Tate motives (because that would explain the MZVs appearing as periods). But that's not true in general, by a result of Belkale and Brosnan. What can be said about these motives and these periods? It seems to be unknown, right now.

My questions would be: How large is the tensor-category generated by Feynman motives (as subcategory of DM)? It certainly is larger than Tate motives (though I don't even know whether it contains all Tate motives, but other people probably know that). Can one still hope to find a t-structure that cuts out Feynman motives? Would that give a Tannakian category? Probably the graph motives are just not a very "motivic" subcategory, so there is no good answer to these questions. Since the periods appear to be rather special (and close to MZVs), these questions are still of interest.

Instead of telling you more about this story, I send you to these interesting papers (in which I hope to read more this week):

• Spencer Bloch, Hélène Esnault, Dirk Kreimer: On motives associated to graph polynomials, the paper that started the story.
• Christian Bogner, Stefan Weinzierl: Feynman graph polynomials gives an easy introduction to graph polynomials, spanning trees and matroids.
• Dominique Manchon: Hopf algebras, from basics to applications to renormalization tells you how these Feynman graphs form a Hopf algebra (and what a Hopf algebra is).
• Prakash Belkale, Patrick Brosnan: Matroids, motives and conjecture of Kontsevich shows that Kontsevich's conjecture was wrong, by demonstrating that graph polynomials generate the Grothendieck ring of counting functions.
• Stefan Müller-Stach, Benjamin Westrich: Motives of graph hypersurfaces with torus operations analyzes what one can say about Feynman motives which admit a torus operation.

Adobe released two fonts under a open license (free to use, open source: SIL OpenFont OFL 1.1), in the Open@Adobe project (hosted at SourceForge). I give you some more links to follow, then I tell you how to employ them.

Getting the Fonts

Source Sans Pro is a sans-serif font suitable for screen reading and printing. If you use a modern browser, you might read it right now.

• Look at Source Sans Pro @ Google, also look at pairings
• Integrate Source Sans Pro @ Google Web Fonts into your website
• Original Download Page for Source Sans Pro, most likely you need only the latest FontsOnly package.

Source Code Pro is a monospace font suitable for coding (and any other kind of text editing, if you ask me) and terminal emulators.

• Look at Source Code Pro @ Google, also look at pairings
• Integrate Source Code Pro @ Google Web Fonts into your website
• Original Download Page for Source Code Pro, most likely you need only the latest FontsOnly package.

So many other fonts are available, too. Not all of them are as high-quality as Adobe's, not all of them are free, but take a look yourself, at Google Web Fonts, for example. Other popular fonts on the web are the Droid fonts from Android or Neue Frutiger or Yanone Kaffeesatz, etc.

Using the Fonts

To use these fonts on your Ubuntu Linux (and probably many other Linux Distros), download the FontsOnly files from Adobe, unzip them, copy the OTF files to ~/.fonts/ and run fc-cache -f -v. The process is also explained and packaged into a ready-to-use script at this answer from AskUbuntu, where you have to make the obvious changes for Source Sans Pro instead of Source Code Pro (and similarly for other fonts).

Once you have the fonts installed, you can tell some applications to use them, for example the gnome-terminal has a profile-system, which you can access by the menu entry Edit->Profiles, then select your profile (most likely "Default", the only one) and Edit again, where you can now choose to either use the system fixed width font or some other font. Try the newly installed Source Code Pro!
In Emacs, the menu entry Options directly offers "Use System Font" versus "Set Default Font" (at least emacs24, I don't know about xemacs and the like).

If you want to change your system font easily under Ubuntu, you can install gnome-tweak-tool, which allows you to set default font, document font, monospace font and window title font.

To use these fonts (or many others) on your website or blog or wiki, you just need to follow the steps at "Google Web Fonts". For WordPress instances, there is a plug-in to use Google Web Fonts which I haven't tried, since it is really easy to customize your CSS. I also customized the wiki I use for personal knowledge management, which is an Instiki instance; here one has to go to the main page and "Edit Web" to enter some custom CSS.

To use these fonts on Windows or Mac OSX, download the FontsOnly file and read the installation instructions from Adobe (I didn't).

Final Remarks

After using Ubuntu's default font settings (and the horrible Arial in the original theme of my blog) for years, I was surprised how beautiful new fonts can be, and how much more readable it feels. I urge you to at least try something different, since it's quite easy nowadays.

Nota bene: I read somewhere that someone found out that serif fonts are not more readable than sans-serif fonts. That's why you don't see so much serif around these days. You should try it, sans-serif looks just "clean".

I have skipped the aspect of using shiny new fonts in LaTeX, but then I'm not sure whether it's a good idea to use a text font that doesn't match the math symbol font... this will have to wait for a more thorough investigation at some other time.

Happy font switching!

In this post I look at some computations around projective bundle formulae for the Chow ring, the algebraic K-Theory and the (Chow) motive of some spaces, in particular flag varieties. We recover some results from the previous posts on cohomology, cycles & bundles and motive of projective space.

Motivation and History

In the first section, I want to talk about (nice) topological spaces $X$ and their real vector bundles.

Classical topology

The idea of a projective bundle formula comes from classical topology.
Theorem (Projective Bundle formula for singular cohomology):
Given a vector bundle $\mathcal{E} \to X$ of rank $r+1$ the singular cohomology of its projectivization $\mathbb{P}(\mathcal{E}) \to X$ is a module over the singular cohomology of the base $X$, and there is a module homomorphism $H^\bullet(\mathbb{P}(\mathcal{E}),\mathbb{Z}) \simeq H^\bullet(X)[H]/(H^{n+1})$ where the $H$ stands for "hyperplane" and is in degree $2$.

A special case of the projective bundle formula is projective space $\mathbb{P}^n$ itself, the projectivization of a vector space $V = k^{n+1}$, considered as a vector bundle over a point. Since the cohomology of a point is concentrated in degree $0$ and there just $\mathbb{Z}$, we get $H^\bullet(\mathbb{P}^n)=\mathbb{Z}[H]/(H^{n+1})$.

One should pay attention to the fact that not every bundle with fibers projective spaces are projectivizations of vector bundles. The obstruction to this is a class in $H^2$ with values in $GL_1$, as one can see explicitly by a Cech resolution.

The projective bundle formula for singular cohomology can be seen as a special case of the Leray-Hirsch theorem, which states that a fiber bundle $F \to E \to B$ which has the property that $F \to E$ induces a surjection $H^\bullet(E) \to H^\bullet(F)$ has cohomology $H^\bullet(E) \simeq H^\bullet(B) \otimes H^\bullet(F)$, where the isomorphism is an isomorphism of $H^\bullet(B)$-modules and the tensor product is taken in the graded sense.
If the basis $B$ is a point (or just contractible, i.e. a point from the homotopy point of view) the theorem is trivial. If you take $E \to B$ to be a projective bundle of rank $r$, the fiber over any point is $F = \mathbb{P}^r$ and one can show that the classes $H^k$ that generate the cohomology of $\mathbb{P}^r$ are in the image of $H^\bullet(E) \to H^\bullet(F)$, hence Leray-Hirsch can be applied.
Leray-Hirsch follows from the Leray-Serre spectral sequence (which is, of course, just a special case of the Grothendieck spectral sequence for the composition of two functors and derivation), which is $H^p(B,\mathcal{H}^q(F)) \Rightarrow H^{p+q}(E)$.

Singular cohomology also satisfies a Künneth formula, which is $H^\bullet(X,\mathbb{Q}) \otimes H^\bullet(Y,\mathbb{Q}) \simeq H^\bullet(X \times Y,\mathbb{Q})$. If we try to do this with integral coefficients, there's not an isomorphism but a short exact sequence with a Tor-term which doesn't vanish in general. A Künneth formula would also give us a projective bundle formula for trivial projective bundles $\mathbb{P}^n \times X \to X$. In some sense, I like to think of bundle formulas as generalizations or versions of the Künneth formula (since bundles are locally products).

The Formulas in Algebraic Geometry

Now we change our focus and switch to algebraic geometry. I've written about definitions and relations between Chow groups and algebraic K-Theory before.

Chow ring

Let $X$ be an algebraic scheme over a field for the rest of the article.

Chow groups don't satisfy a Künneth formula! They do satisfy a projective bundle formula. I want to start with a baby version, which I'd like to call projective Künneth formula:
Theorem: $CH^\bullet(X \times \mathbb{P}^n) \simeq CH^\bullet(X)[H]/(H^{n+1})$, where $H$ is a hyperplane class in degree $1$.

This formula, together with the Yoneda lemma (in a variant known as Manin Identity Principle), already gives a decomposition of the Chow motive of projective space into irreducibles.

The full projective bundle theorem can be deduced from a localization sequence for higher Chow groups.

Theorem (localization sequence):
For $Z \to X$ a closed immersion of pure codimension $c$ with open complement $U$ there is a long exact sequence
$\cdots \to CH^{i-c}(Z,n) \to CH^{i}(X,n) \to CH^{i}(U,n) \to CH^{i-c}(Z,n-1) \to \cdots$

Theorem (projective bundle formula):
Let $\mathcal{E} \to X$ be a vector bundle of rank $r+1$ and $\mathbb{P}(\mathcal{E}) \to X$ its projectivization. Then $CH^\bullet(\mathbb{P}(\mathcal{E}),\bullet)$ is a $CH^\bullet(X,\bullet)$-module isomorphic to $CH^\bullet(X,\bullet)[H]/(H^{n+1})$, with $H$ in degree $1$.

To prove this by induction on the dimension on $X$, we can just take an open subset $U \subset X$ such that $\mathcal{E}$ is trivial over $U$, then for $U$ we have a projective Künneth formula and for $Z$ we use the induction hypothesis.

If you take only the $n=0$ part of the localization sequence (which I think was known for a longer time), you can prove the projective bundle formula for ordinary Chow groups with this method.

Algebraic K-Theory

We start with a projective bundle formula for $K_0$, i.e. the K-Theory of coherent sheaves on an algebraic scheme.
Theorem
For a vector bundle $E \to X$ of rank $n+1$, $K_0(\mathbb{P}(E))$ is a $K_0(X)$-module which is module-isomorphic to $K_0(X)[H]/(H^{n+1})$.

For trivial bundles, this is quite easy to see:
By a Theorem of Serre, any coherent sheaf on $\mathbb{P}^n$ is a quotient of some $\mathcal{O}(k)^{\oplus m}$ (with $k,m \geq 0$), hence the $\mathcal{O}(k)$ generate $K_0(\mathbb{P}^n)$.
The Koszul complex on $\mathbb{A}^{n+1}$ is a $n+2$-term exact sequence which gives a relation in $K_0$ between any $\mathcal{O}(k+1),\dots,\mathcal{O}(k+n+2)$ by applying a shift and taking the coherent sheaf on $\mathbb{P}^n$ associated to a graded module. Therefore we see that $H := [\mathcal{O}(-1)]$ and $H^2,\dots,H^n$ and $H^0 = [\mathcal{O}]$ together form a generating set. There are no more relations between these generators.

The proof of the non-trivial case is given by Grothendieck and Berthelot in SGA6 Exposé VI. Such a bundle formula also holds for higher algebraic K-Theory (due to Quillen):

Theorem (projective bundle formula):
For a vector bundle $E \to X$ of rank $n+1$, $K_\bullet(\mathbb{P}(E))$ is a $K_0(X)$-module which is module-isomorphic to $K_\bullet(X)[H]/(H^{n+1})$, with $H$ in degree $1$.

Chow Motive

If we're talking about bundles over a smooth projective (or smooth complete) base, we can talk about the Chow motive of the total space in relation to the base.

Theorem (projective bundle formula):
For a vector bundle $E \to X$ of rank $n+1$, $h(\mathbb{P}(E)) = \bigoplus_{s=0}^n h(X)(s)[2s]$ ($=h(X)[H]/(H^{n+1})$, with $H = \mathbb{Z}(1)[2]$).

Manin's Identity Principle implies that a morphism of Chow motives $M\to N$ is an isomorphism iff the morphism of associated functors $\omega_M \to \omega_{N}$ is an isomorphism, where
$\omega_M : \mathcal{P}(k) \ni Y \mapsto M_\sim(h(Y),M(\ast)) := \bigoplus_{r \in \mathbb{Z}} M_\sim(h(Y),M(r))$

We guess the motive of a projective bundle $\mathbb{P}(E)$ of rank $n$ over a base $X$ to be $M := \bigoplus_{s=0}^n h(X)(s)[2s]$, then from the projective bundle formula for Chow groups we see that $\omega_M(\mathbb{P}(E)) \simeq \omega_{\mathbb{P}(E)}(\mathbb{P}(E))$, so the identity morphism on the right hand side yields a morphism $h(\mathbb{P}(E)) \to M$ which induces an isomorphism on the functors $\omega$, thus is an isomorphism.

Voevodsky Motive

If we look at an arbitrary smooth base (no properness assumption) we need Voevodsky's triangulated motives. I assume that the base is a $k$-scheme for $k$ a perfect field.

Theorem (projective bundle formula):
For a vector bundle $E \to X$ of rank $n+1$, the canonical morphism $\bigoplus_{s=0}^n \mathbb{Z}_{tr}(X)(s)[2s] \to \mathbb{Z}_{tr}(\mathbb{P}(E))$ is an isomorphism.

One can take a local trivialization of $E$ over a cover $\mathcal{U}$ of $X$, which allows (by Mayer-Vietoris triangles) to compute the motive of $\mathbb{P}(E)$ in terms of $\mathbb{P}(E)|_U$ for $U \in \mathcal{U}$. Thus one can look at the trivial case, where $\mathbb{P}(E) \simeq \mathbb{P}^n_X$ and the formula holds.

From this projective bundle formula one can deduce a Gysin triangle:
Gysin/localization triangle:
Let $X$ be a smooth scheme, $Z$ a smooth closed subscheme of codimension $c$. Then
$C_\ast \mathbb{Z}_{tr}(X \setminus Z) \to C_\ast \mathbb{Z}_{tr}(X) \to C_\ast \mathbb{Z}_{tr}(Z)(c)[2c] \to [1]$
is a distinguished triangle.

This can be proved by looking at the situation étale-locally, where $X= Y \times \mathbb{A}^c$ and $Z = Y \times 0$. Then the morphism $C_\ast \mathbb{Z}_{tr}(X) \to C_\ast \mathbb{Z}_{tr}(Z)(c)[2c]$ is just $C_\ast(\mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^n\setminus 0)) \to \mathbb{Z}(n)[2n]$ from the projective bundle formula, tensored with $C_\ast\mathbb{Z}_{tr}(Y)$.

More general bundle formulas

One could ask what happens for a general $G$-bundle $E \to X$, let's say for $G$ a (split semi-simple) reductive linear algebraic group, if we take $E/P \to X$ for any subgroup $P$ of $G$. Maybe in the case of a parabolic $P$ we can describe the motive of $E/P$ in terms of $X$. At least in the case of a certain maximal parabolic of $G=SL_{n+1}$ this yields $G/P = \mathbb{P}^n$, so the analogy should be clear.

Using localization techniques becomes more interesting, since a $G$-bundle which is étale-locally trivial needn't be Zariski- (or even Nisnevich-) locally trivial (though this is the case for $G=GL_n$ and $G=SL_n$). Of course, one can restrict attention to Zariski-locally trivial bundles first.

Köck has further developed the techniques used to prove the projective bundle formula for Chow motives, and computed the motive of $G/P$ itself.
Theorem:
Let $G$ be a split reductive group over a field $k$ and $P$ a parabolic subgroup, with $Y:=G/P$ the quotient homogeneous space. Denote by $Y_w := BwB/P$ the Bruhat cells in $Y = \bigcup_{w \in W} Y_w$. Then $h(Y) \simeq \bigoplus_{w \in W} \mathbb{Z}(-dim(Y_w))$.

Habibi and Rad have recently proved for a connected reductive group $G$ over a characteristics $0$ field $k$ that the motive of a Zariski-locally trivial $G$-bundle over an irreducible base with mixed Tate motive is itself a mixed Tate motive.

Is there more known about $G/P$-bundles? I would love to see a formula that takes the type of the parabolic (i.e. the corresponding subset of roots) and spits out a motivic decomposition. I have the impression that one should be able to prove this along the lines of the projective bundle formula for Voevodsky motives. Locally, one has just $G/P$ and there Köck described the motive. Alternatively, one could use the Gysin sequence.

Example

To compute the Voevodsky motive of $\mathbb{P}^n$ from the bundle formula above would be cheating, since I did the proof by reduction to the motive of $\mathbb{P}^n$ (as it is done in the book of Mazza-Voevodsky-Weibel). I wrote about the Motive of $\mathbb{P}^n$ without a bundle formula before.

Flag varieties

One can see $\mathbb{P}^n$ as a special kind of partial flag variety which parametrizes flags of the form $0 = V_0 \subset V_1 \subset V_n = V$ with $V_1$ one-dimensional and $V$ an $n$-dimensional vector space (affine space).

Every partial flag variety $X$ parametrizing flags of the form $0 = V_0 \subset V_{i_1} \subset \cdots \subset V_{i_k} \subset V_n = V$ with $0 < i_1 < \cdots < i_k < n$ can be written as homogeneous space $X = GL_n/P(i_1,\dots,i_k)$, where $P(i_1,\dots,i_k)$ denotes a standard parabolic in $GL_n$.

The $1$-flags in a vector space $V$ (of dimension $n+1$) form a variety $Fl_1$ which maps to a point, which we can consider as the space of $0$-flags $Fl_0$. This map $Fl_1 \to Fl_0$ is obviously just $\mathbb{P}^{n} \to \mathbb{P}^0$, hence we can compute the motive of $Fl_1$. Now $Fl_{1,2}$, the space of $1,2$-flags, fibers over $Fl_1$, since over each $1$-flag there is a projective space of $2$-flags containing this $1$-flag. This means $Fl_{1,2} \to Fl_1$ is a projective bundle, and we can compute the motive. And so on.

Explicitly, this gives us $h(Fl_{1,2}) = \bigoplus_{s_2=0}^n \left(\bigoplus_{s_1=0}^n \mathbb{Z}(-s_1)[-2s_1] \right)(-s_2)[-2s_2]$ $= \bigoplus_{s_1,s_2=0}^n \mathbb{Z}(-s_1-s_2)[-2s_1-2s_2]$ and in general we have $h(Fl_{1,\dots,n}) = \bigoplus_{s_1,\dots,s_n=0}^n \mathbb{Z}(-(\sum s_i))[-2\sum(s_i)]$ reflecting the structure of the Bruhat cells of $Fl_{1,\dots,n}(V) \simeq GL(V)/B$ ($B$ a Borel subgroup).

It would be nice if one could use a projective bundle formula to compute the motive of any homogeneous space $G/P$ for $P$ a parabolic, but if one tries to do that, the bundles one encounters are no longer projective.

Do you know of other neat applications of the projective bundle formula?

Motives

Chow motive of projective space

One can compute the Chow motive of projective space by guessing it and using Manin's identity principle, a variant of the Yoneda lemma. To do such a "calculation" in general, the guessing part will be a problem. We can try to do systematic guessing. In fact, the previous two posts on projective space have prepared this. We expect, from the Weil cohomology computations, to have a motive $h(\mathbb{P}^n) = \bigoplus_{s=0}^n 1(-s)[-2s]$ (we can forget about the grading and the $[-2s]$ for now).

Manin's identity principle states that the functor that maps a motive $M$ to the functor
$\omega_M : \mathcal{P}(k) \ni Y \mapsto M_\sim(h(Y),M(\ast)) := \bigoplus_{r \in \mathbb{Z}} M_\sim(h(Y),M(r))$
is fully faithful (where the grading on the right hand side is taken to be the grading in intersection groups). If you know the Yoneda lemma, this is an easy consequence. If you don't know the Yoneda lemma, you should be sitting at your desk, trying to prove it!

We compare the motives $h(\mathbb{P}^n)$ and $\bigoplus_{s=0}^n 1(-s)$ by looking at their corresponding functors $\omega_M$. The only input we need from intersection theory is a projective bundle formula
$A^\bullet(X \times \mathbb{P}^n) \simeq A^\bullet(X)_F[H]/(H^{n+1}) = \bigoplus_{s=0}^n A^\bullet(X) \cdot H^s.$
We compute for any $Y \in \mathcal{P}(k)$:
$M_\sim(h(Y),h(\mathbb{P}^n)(r)) = Z_\sim^{d_Y+r}(Y \times \mathbb{P}^n) \simeq \bigoplus_{s=0}^n Z_\sim^{d_Y+r-s}(Y),$
$M_\sim(h(Y),\left(\bigoplus_{s=0}^n 1(-s)\right)(r)) = \bigoplus_{s=0}^n Z_\sim^{d_Y+r-s}(Y).$
Using the first line with $Y=\mathbb{P}^n$ and $r=0$ we can take the identity on $\mathbb{P}^n$ to yield a canonical morphism $h(\mathbb{P}^n) \to \bigoplus_{s=0}^n 1(-s)$, which is an isomorphism, since it induces an isomorphism of corresponding functors.

We can use this now to compute realizations of the Chow motive. As we have seen, we just need to know how the Lefschetz motive $1(-1)[-2]$ realizes, i.e. what a Weil cohomology does on $\mathbb{P}^1$. For example, Betti realization $\omega_B$ gives us a one-dimensional vector space $\omega_B(1(-1)) = V \simeq \mathbb{Q}$ with Hodge decomposition $V = V^{1,1}$. The $\ell$-adic realization gives us a one-dimensional vector space $\omega_\ell(1(-1)) = \mathbb{Q}_\ell(1) = T_\ell(\mu) \otimes \mathbb{Q}_\ell$, where $T_\ell(\mu)$ is the $\ell$-adic Tate module of roots of unity $\mu$, which has a natural Galois action (which is part of the realization). Consequently, $H_B^{2k}(\mathbb{P}^n) = \left(H_B^{1}(\mathbb{P}^1)\right)^{\otimes k} = V^{\otimes k}$ with Hodge decomposition $V^{\otimes k} = (V^{\otimes k})^{(k,k)}$ and $H_\ell^{k}(\mathbb{P}^n) = \left(H_\ell^{1}(\mathbb{P}^1)\right)^{\otimes k} = \mathbb{Q}_\ell(k) = T_\ell(\mu)^{\otimes k}$.

We could have skipped the Weil cohomology computations for $\mathbb{P}^n$, it would have sufficed to compute the motive and the realizations of $\mathbb{P}^1$. But how would we have guessed the motive then? Well, using geometry. I will come to that later in this article.

Voevodsky motives

Now I want to describe the motive of projective space (and its decomposition) in Voevodsky's framework of the derived category of mixed motives (which isn't constructed as the derived category of an abelian category, however it looks like that). By a general theorem (which is not too hard), the motive of a smooth projective variety is in the image of a (contravariant!) functor from Chow motives, so we don't need to work any longer for projective space. The following is for educational purposes only. I want to consider only perfect fields $k$, since I don't know what's possible for non-perfect fields.

The triangulated category of effective geometrical motives over $k$, denoted $DM_{gm}^{eff}(k)$ is defined as the pseudo-abelian envelope of a localization (at the minimal thick subcategory containing $X\times \mathbb{A}^1 \to X$ and Mayer-Vietoris sequences) of the homotopy category of bounded complexes over $SmCor(k)$, the category of finite correspondences of smooth schemes over $k$. We denote the image of a smooth scheme $X$ in this category by $M_{gm}(X)$ (following Voevodsky).

A few easy calculations:
Since we have the pseudo-abelian property, we can do at least the usual splitting $M_{gm}(\mathbb{P}^1) = \mathbb{Z} \oplus \mathbb{L}$, where $\mathbb{Z} := M_{gm}(Spec k)$ is the unit object for the tensor structure and $\mathbb{L}$ is the reduced motive of $\mathbb{P}^1$. The Tate object is defined as $\mathbb{Z}(1) := \mathbb{L}[-2]$ (warning: Voevodsky motives are covariant, while Chow motives are contravariant, hence some formula look different; this is such a formula).
For Mayer-Vietoris, we can take the usual two charts $U,V : \mathbb{A}^1 \to \mathbb{P}^1$ which are a Zariski open covering of $X := \mathbb{P}^1$, so there is a distinguished triangle
$M_{gm}(U\cap V) \to M_{gm}(U) \oplus M_{gm}(V) \to M_{gm}(X) \to M_{gm}(U \cap V)[1]$
which in our case looks like
$M_{gm}(\mathbb{G}_m) \to 0 \oplus 0 \to M_{gm}(\mathbb{P}^1) \to M_{gm}(\mathbb{G}_m)[1]$
so that $M_{gm}(\mathbb{P}^1) \to M_{gm}(\mathbb{G}_m)[1]$ is an isomorphism.

To work with these motives, it is better to look at another category, which is denoted by $DM_{-}^{eff}(k)$. A presheaf with transfers on $Sm_k$ is an additive contravariant functor from $SmCor(k)$ to abelian groups. Such a presheaf with transfers can be seen as a presheaf on $Sm_k$ with additional restriction maps for each correspondence which isn't the graph of a morphism; in particular for the transposed graphs, which give restriction maps "in the other direction", hence the name "with transfers". A presheaf with transfers on $Sm_k$ is called Nisnevich sheaf if the corresponding presheaf of abelian groups on $Sm_k$ is a Nisnevich sheaf. A (pre)sheaf with transfers is called homotopy invariant if every projection map $X \times \mathbb{A}^1 \to X$ induces an isomorphism of sections. Now $DM_{-}^{eff}(k)$ is the full subcategory of $D^{-1}(Shv_{Nis}(SmCor(k)))$ of complexes with homotopy invariant cohomology sheaves. This category is a triangulated pseudo-abelian category. One can show that $DM_{gm}^{eff}(k)$ admits a full embedding (as tensor triangulated category) into $DM_{-}^{eff}(k)$.

Both categories of effective motives yield larger categories of motives by inverting the Tate twist operation $\otimes 1(1)$, thus one has $DM_{gm}(k)$ a full tensor triangulated subcategory of $DM_{-}(k)$. Now one could write down a Gysin sequence and a projective bundle theorem which can be used to compute the motive of $\mathbb{P}^n$, entirely in terms of $DM_{gm}^{eff}(k)$; the problem is that the proof I know of goes through the computation of the motive of $\mathbb{P}^n$, in terms of $DM_{-}(k)$.

Denote by $\mathbb{Z}_{tr}(X) := Cor(-,X)$ the presheaf with transfers associated to a smooth scheme $X$ and by $C_\ast\mathbb{Z}_{tr}(X)$ the complex obtained from the simplicial object $Cor(- \times \Delta^\bullet,X)$.

Voevodsky motive of projective space

We use $\mathbb{Z}(q) := C_\ast\mathbb{Z}_{tr}(\mathbb{G}_m^{\wedge q})[-q]$, which is called a motivic complex. We have already discussed $M_{gm}(\mathbb{P}^1) = M_{gm}(\mathbb{G}_m)[1]$. In a similar spirit, we have $C_\ast\mathbb{Z}_{tr}(\mathbb{P}^1) = \mathbb{Z}(1)[2]$, using the following lemma: for $\mathcal{U}$ a Zariski covering of $X$ the Cech resolution $Tot C_\ast \mathbb{Z}_{tr}(\mathcal{U}) \to C_\ast \mathbb{Z}_{tr}(X)$ is a quasi-isomorphism in the Zariski topology.

Let's denote $0 := [1:0:\cdots:0] \in \mathbb{P}^n$ and look at the map $f : \mathbb{P}^n \setminus 0 \to \mathbb{P}^{n-1}$ given by $[x_0:\cdots:x_n] \mapsto [x_1:\cdots:x_n]$. The fibers of this map are just $\mathbb{A}^1$. There is an $\mathbb{A}^1$-homotopy inverse $g : \mathbb{P}^{n-1} \to \mathbb{P}^n \setminus 0$ given by the section $[x_1:\cdots:x_n] \mapsto [0:x_1:\cdots:x_n]$, and the $\mathbb{A}^1$-homotopy of $g \circ f$ to $id$ is given by multiplication of $x_0$ with $\lambda \in \mathbb{A}^1$. Such a pair of $\mathbb{A}^1$-homotopy inverse maps yield a chain homotopy equivalence $C_\ast \mathbb{Z}_{tr}(\mathbb{P}^n \setminus 0) \to C_\ast \mathbb{Z}_{tr}(\mathbb{P}^{n-1})$.

Theorem: For each $n$, there are quasi-isomorphisms of Zariski sheaves:
$C_\ast\left( \mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^{n-1}) \right) \simeq C_\ast \mathbb{Z}_{tr}(\mathbb{G}_m^{\wedge n})[n] = \mathbb{Z}(n)[2n]$.

The proof of this theorem (which I got from the Mazza-Voevodsky-Weibel book, Chapter 15) uses a few facts about presheaves with transfers, for example the Nisnevich sheafification $F_{Nis}$ of a homotopy invariant presheaf with transfers $F$ is again homotopy invariant; and more is true: all the presheaves $H^n_{Nis}(-,F_{Nis})$ are homotopy invariant. Using this, one can show that a presheaf with transfers $F$ that satisfies $F_{Nis} =0$ also satisfies $(C_\ast F)_{Nis} \simeq 0$ and $(C_\ast F)_{Zar} \simeq 0$.

Proof sketch: Let $\mathcal{U}$ be the usual cover of $\mathbb{P}^n$ by $n+1$ charts $\mathbb{A}^n$ and let $\mathcal{V}$ be the cover by $n$ charts of $\mathbb{P}^{n} \setminus 0$. Intersecting $i+1$ charts gives $\mathbb{A}^{n-i} \times (\mathbb{A}^1 \setminus 0)^i$. There are quasi-isomorphisms (of complexes of Nisnevich sheaves with transfers) $\mathbb{Z}_{tr}(\mathcal{U}) \to \mathbb{Z}_{tr}(\mathbb{P}^n)$ and $\mathbb{Z}_{tr}(\mathcal{V}) \to \mathbb{Z}_{tr}(\mathbb{P}^n\setminus 0)$, so $Q_\ast := \mathbb{Z}_{tr}(\mathcal{U})/\mathbb{Z}_{tr}(\mathcal{V})$ is a resolution of $\mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^n \setminus 0)$ (as Nisnevich sheaf). Now $Tot C_\ast Q_\ast$ is quasi-isomorphic to $C_\ast\left(\mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^n \setminus 0)\right)$, hence to $C_\ast\left(\mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^{n-1})\right)$ for the Zariski topology.
One can write down a resolution $R_\ast$ of $\mathbb{Z}_{tr}(\mathbb{G}_m^{\wedge n})[n]$ such that one gets a map $Q_\ast \to R_\ast$ whose terms are direct sums of $\mathbb{A}^1$-homotopy equivalences, so $C_\ast Q_\ast \to C_\ast R_\ast$ is a quasi-isomorphism. Applying $Tot$ gives us
$C_\ast\left(\mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^{n-1})\right) \simeq Tot C_\ast Q_\ast \simeq Tot C_\ast R_\ast \simeq C_\ast \mathbb{Z}_{tr}(\mathbb{G}_m^{\wedge n})[n]$.

One can show that the isomorphism in this theorem factors through every inclusion $C_\ast \mathbb{Z}_{tr}(\mathbb{P}^i) \to C_\ast \mathbb{Z}_{tr}(\mathbb{P}^n)$ with $n > i$.

Corollary: There is a quasi-isomorphism $M(\mathbb{P}^n)=C_\ast\mathbb{Z}_{tr}(\mathbb{P}^n) \to \bigoplus_{s=0}^n \mathbb{Z}(s)[2s]$.

Proof: by induction, where the case $n=1$ is already done. The map $\mathbb{Z}_{tr}(\mathbb{P}^{n-1}) \to \mathbb{Z}_{tr}(\mathbb{P}^{n})$ is split injective in $DM_{-}^{eff}(k)$, since the quasi-isomorphism $\mathbb{Z}_{tr}(\mathbb{P}^{n-1}) \to \bigoplus_{s=0}^{n-1} \mathbb{Z}(s)[2s]$ (from the induction hypothesis) factors through it. Hence the distinguished triangle
$C_\ast \mathbb{Z}_{tr}(\mathbb{P}^{n-1}) \to C_\ast \mathbb{Z}_{tr}(\mathbb{P}^{n}) \to \mathbb{Z}(n)[n] \to [1]$
splits.

Computations from the motive: motivic cohomology

Now that we have the motive, we can (in principle) compute motivic cohomology
$H^{p,q}_{mot}(\mathbb{P}^n,\mathbb{Z}) := H^p_{Zar}(\mathbb{P}^n,\mathbb{Z}(q)) = Ext^p(\mathbb{Z}_{tr}(X),\mathbb{Z}(q))$
where the Ext is in the category of Nisnevich sheaves with transfer.
So we have to understand $Hom_{DM_{-}}(\mathbb{Z}(s)[2s],\mathbb{Z}(q)[p])$. This can be simplified to $Hom(\mathbb{Z},\mathbb{Z}(q-s)[p-2s]) = H^{p-2s,q-s}_{mot}(Spec(k))$, so we are reduced to understand the motivic cohomology of a point.

From the motivic cycle class isomorphism $CH^q(X,2q-p) \simeq H^{p,q}(X)$ one can recover the Chow groups of $\mathbb{P}^n$. The motivic Chern character yields an isomorphism computing the algebraic K-Theory of $\mathbb{P}^n$ (see this article about the divisorial jungle, where I discuss this briefly).

A¹-homotopy type and motivic cell structure

We can take one step backwards and look at $\mathbb{P}^n$ not with (co)homological eyes, but with homotopical ones. The functor $M : Sm_k \to DM_{-}(k)$ factors through a model category $\mathcal{M}$ of simplicial Nisnevich sheaves on $Sm_k$, where one can do homotopy theory.

In this model category $\mathcal{M}$ one can write down $X := \bigvee_{s=0}^n \mathbb{G}_m^{\wedge s} \wedge S^s_s$, a space (=simplicial Nisnevich sheaf) which has the same motive as $\mathbb{P}^n$, since wedging with the $s$-dimensional simplicial sphere $S_s^s$ induces a shift by $s$, hence $M(\mathbb{G}_m^{\wedge s} \wedge S^s_s) = \mathbb{Z}(s)[2s]$. One could now try to write down (or prove existence of) an $\mathbb{A}^1$-homotopy equivalence of $X$ with $\mathbb{P}^n$.

This turns out to be impossible in general, since the homotopy types are different (but they become isomorphic over a quadratically closed base field). We can already see that in the real realization, which is a functor that assigns to a homotopy type a topological space which acts like the $\mathbb{R}$-points. The projective space has non-orientable real realization, while $X$ is orientable. This is like the difference between $\mathbb{RP}^2$ and $S^2$.

One can write down a motivic cell structure for $\mathbb{P}^n$, where the attaching maps split over a quadratically closed field. One can say that motives don't distinguish between $X$ and $\mathbb{P}^n$, but the homotopy type does (even the stable homotopy type).

Such a motivic cell structure can be constructed like in topology: start with a point $\mathbb{P}^0$ and attach a 1-cell $\mathbb{A}^1$ along the attaching map $\eta_1 : \mathbb{A}^1 \setminus 0 \to \mathbb{P}^0$ which is the quotient map after the $\mathbb{G}_m$-action. You get a $\mathbb{P}^1$. Then attach a 2-cell $\mathbb{A}^2$ along $\eta_2 : \mathbb{A}^2 \setminus 0 \to \mathbb{P}^1$, you get $\mathbb{P}^n$ and so on.

We get the motive out of a motivic cell structure, since a cofiber sequence $\mathbb{A}^n \setminus 0 \to \mathbb{P}^{n-1} \to \mathbb{P}^n$ yields a distinguished triangle and $\mathbb{A}^n \setminus 0 \simeq S^{2n-1,n}$ has the motive $\mathbb{Z}(n)[2n-1]$, so we can write the distinguished triangle as
$M(\mathbb{P}^{n-1}) \to M(\mathbb{P}^n) \to \mathbb{Z}(n)[2n] \to$
which splits, i.e. $M(\mathbb{P}^n) = M(\mathbb{P}^{n-1}) \oplus \mathbb{Z}(n)[2n]$, since the morphism $\mathbb{Z}(n)[2n-1] \to M(\mathbb{P}^{n-1})$ is trivial.

This is the calculation of the motive, thus of higher Chow groups, algebraic K-Theory and all Weil cohomology theories, that I like most.

The invariants discussed in this article are closely related and I wrote an article about the divisorial jungle before.

Divisor class group, Picard group

Since projective space is an integral scheme of finite type, smooth over the base, we have four different isomorphic characterizations of the Picard group. It is the group of invertible sheaves (lines bundles) with tensor product, up to isomorphism $Pic(\mathbb{P}^n)$. It is isomorphic to the Cartier class group $CaCl(\mathbb{P}^n)$ of Cartier divisors modulo principal divisors, which is isomorphic to the sheaf cohomology group $H^1(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n}^\times)$. It is the Class group $Cl(\mathbb{P}^n)$ of Weil divisors modulo principal divisors.

Any hypersurface $D$ of degree $d$ in projective $n$-space is linearly equivalent to the Weil divisor $d H$, where $H = \{[x_0:\cdots:x_n] \in \mathbb{P}^n : x_0=0\}$ is a (generic) hyperplane. In particular, all hyperplanes are linearly equivalent to each other. The degree of $H$ is $1$, so the degree homomorphism $Cl(\mathbb{P}^n) \to \mathbb{Z}$ is an isomorphism. In particular, $Cl(\mathbb{P}^1)$ is non-trivial. One can show $D \sim d H$ by first proving $deg(f)=0$ for $(f)$ the divisor of a non-zero rational function $f \in \mathcal{K}_{\mathbb{P}^n}^\times$ and then writing $D = (g)-(h)$ for $g,h$ some rational functions, so $D-dH = (g/x_0^d h)$.

The line bundles on $\mathbb{P}^n_k$, for $k$ a field, are generated (up to isomorphism) by the tautological bundle $\mathcal{O}(-1)$, its dual $\mathcal{O}(1)$ and their tensor powers $\mathcal{O}(n)$, which are all called twisting sheaves. The usual isomorphism $Cl(X) \to Pic(\mathbb{P}^n)$ identifies $H$ with $\mathcal{O}(1)$.

Chow ring

The Chow ring $A^\bullet(\mathbb{P}^n)$ of $\mathbb{P}^n$ is the graded ring whose $k$-th graded component is the Chow group $A^k(\mathbb{P}^n)$ of codimension $k$ cycles up to rational equivalence, with ring structure coming from the intersection product. Here, we consider Chow groups with integer coefficients.

Let us first look at $A^\bullet(\mathbb{P}^1)$. Can we come up with any cycles, hopefully non-trivial ones? In dimension $0$, that is codimension $1$, we have the points $p \in \mathbb{P}^1$ which are all rationally equivalent, since any two points $p,q$ in $\mathbb{P}^1$ can be joined by a $\mathbb{P}^1$ via the map $\phi : [x_0:x_1] \mapsto [x_0p_0+x_1q_0,x_0p_1+x_1q_1]$ (check $\phi([1:0]) = p, \phi([0:1]) = q$). In dimension $1$, that is codimension $0$, there is $\mathbb{P}^1$ and that's the only closed subvariety, since $\mathbb{P}^1$ is irreducible. In higher or lower codimensions, there is nothing.

The trick that shows that two points on $\mathbb{P}^1$ are rationally equivalent actually works for hyperplanes in $\mathbb{P}^n$, and we have seen (in the section above) that one can show (using linear equivalence of divisors) that these are all classes of cycles. Since rational equivalence of codimension 1 cycles coincides with linear equivalence of divisors, this shows that $A^\bullet(\mathbb{P}^1) = \mathbb{Z}[H]/(H^2)$, for $H$ the class of a hyperplane $\{[x_0:x_1] : x_0=0\}$ (in fact, a point).

The structure of $A^k(\mathbb{P}^n)$ looks less clear at first, but one can show that it is generated by any codimension $k$ linear subspace via the following "excision lemma":

Lemma: Let $i : Y \to X$ be a closed immersion, $j : U := X \setminus Y \to X$ the open complement, then the sequence $A_k(Y) \to A_k(X) \to A_k(U) \to 0$ is exact for all $k$ (where the arrows are pushforward by $i$ and $j$, so we need to take the grading by dimension).

We apply this with $k=n-1$, $X = \mathbb{P}^n$, $Y = H$ a hyperplane, and then $U = \mathbb{A}^n$. Then we use $A_k(\mathbb{A}^n)=0$ for $k < n$ to get that $i_\ast : A_{n-1}(H) \to A_{n-1}(\mathbb{P}^n)$ is a surjection. In fact, also $A_k(H) \to A_k(\mathbb{P}^n)$ is a surjection for $k < n$, and because of $H \simeq \mathbb{P}^{n-1}$ we can use induction.

A small remark: the excision lemma can be improved to a long exact localization/Gysin sequence for higher Chow groups, and the next term on the left would be $A_k(U,1)$, which gives a much smoother proof. The localization sequence, on the other hand, is very very hard to prove (at least that's what Voevodsky writes, so I believe it).

To see that there are no relations in $A^k(\mathbb{P}^n)$ (for $0 \leq k \leq n$), i.e. $A^k(\mathbb{P}^n) = \mathbb{Z} [H^k]$ with $H^k$ the class of a codimension $k$ linear subspace, we look at the cases $k > 1$ (since the case $k=0$ is clear and we have seen the case $k=1$ above, that was the divisor-setting). Any relation would be of the form $d H^k = \sum n_i [div(r_i)]$ for $r_i \in \mathcal{K}^\times(V_i)$ some rational functions. Let $Z := \bigcup V_i$ and $f : Z \to \mathbb{P}^{n-k+1}$ the projection to a linear $k-2$-dimensional subspace disjoint from $Z$. Then $f$ is proper, so there is proper pushforward $f_\ast : A_\bullet(Z) \to A_\bullet(\mathbb{P}^{n-k+1})$ which can not annull $d H^k$ because of compatibility with the degree morphism $deg : A_\bullet(Z) \to \mathbb{Z}$ (and $d H^k$ has degree $d$ over $\mathbb{P}^{n-k+1}$), but has to annul $\sum n_i [div(r_i)]$.

We see that two hyperplanes in $\mathbb{P}^n$ in general position intersect to a linear subspace of codimension $2$, and in general $H^i \cdot H^j = H^{i+j} \in A^\bullet(\mathbb{P}^n)$, so we have
$A^\bullet(\mathbb{P}^n) = \mathbb{Z}[H]/(H^{n+1})$.

Algebraic K-Theory

From the general theory, we can use smoothness of $\mathbb{P}^n_k$ over a perfect field $k$ to deduce that the Chern character with rational coefficients $K^\bullet(\mathbb{P}^n_k)_{\mathbb{Q}} \to A^\bullet(\mathbb{P}^n_k)_{\mathbb{Q}}$ is an isomorphism. I think this also works for $\mathbb{P}^n$ over $\mathbb{Z}$, since only regularity is used. However, the rational coefficients are quite unsatisfying. At least from $K^\bullet(\mathbb{P}^n_k)_{\mathbb{Q}} \simeq \mathbb{Q}[H]/(H^{n+1})$ we can guess how $K^\bullet(\mathbb{P}^n)$ might look like. Or can we? Actually it is way more complicated than the rationalized picture, and I think the general answer is still unknown!

Serre proved a theorem, that every coherent sheaf on $\mathbb{P}^n$ admits a surjective map of some $\mathcal{O}(k)^m$ for positive $k,m$. This shows that $K_0(\mathbb{P}^n)$ is generated by the set of all $\mathcal{O}(k)$. Furthermore, one can show that there is a relation in $K_0$ between any set of $n+2$ coherent sheaves on $\mathbb{P}^n$, by analyzing the Koszul complex on $\mathbb{A}^{n+1}$.

For $\mathcal{E} \to X$ a rank $k+1$ vector bundle on a quasiprojective variety $X$, one can look at the projective bundle $\mathbb{P}(\mathcal{E}) \to X$ which is fiber-wise a projective space of the fiber. On the total space $\mathbb{P}(\mathcal{E})$ one has canonically the tautological bundle and its tensor powers $\mathcal{O}(-i)$, which generate (for $i=0,...,k$) $K_0(\mathbb{P}(\mathcal{E})$. One can take a vector bundle $V \to X$, pull it back to $\mathbb{P}(\mathcal{E})$ and twist it (i.e. tensor it with some $\mathcal{O}(-i)$). This gives functors $u_i : VB(X) \to VB(\mathbb{P}(\mathcal{E}))$.

The projective bundle theorem for algebraic K-Theory says that these functors $u_i$ induce an equivalence $K(X)^{r+1} \simeq K(\mathbb{P}(\mathcal{E}))$ and $K_\bullet(X) \otimes_{K_0(X)} K_0(\mathbb{P}(\mathcal{E})) \to K_\bullet(\mathbb{P}(\mathcal{E}))$ is a ring isomorphism.

If we apply this to a trivial rank $k+1$ vector bundle $\mathcal{E}$, where $\mathbb{P}(\mathcal{E}) = \mathbb{P}^k_X$, we see that $K_\bullet(\mathbb{P}^k_X) \simeq K_\bullet(X)[z]/[z^{k+1}]$.

In particular, $K_\bullet(\mathbb{P}^k) \simeq K_\bullet(\mathbb{Z})[z]/[z^{k+1}]$. But the higher algebraic K-groups of the integers are not known! One might naively guess that it becomes better by taking a field as base, but fields (even algebraically closed ones) have a very rich higher K-theory, too.

Still, the projective bundle theorem shows us that the K-Theory of projective space isn't really complicated, since all the complexity lies in the K-Theory of the base.

Some afterthoughts

By the way, I realise during writing this that I would love to hear something about other isomorphism invariants I'm missing, like the Kodaira dimension, existence of a spin structure on the associated complex manifold, volume of the Fubini-Study metric, geodesic completeness, Lusternik-Schnirelman category, whatever... I fell in love with the idea of computing everything of $\mathbb{P}^n$. Please tell me your favourite isomorphism invariant of $\mathbb{P}^n$ or $\mathbb{CP}^n$ or $\mathbb{RP}^n$ in the comments!

In this part 1, I discuss only the cohomology of $\mathbb{P}^n$. Part 2 contains a discussion of the intersection theory and bundles and part 3 contains the motivic stuff. I intentionally left out usage of projective bundle formulas, as I will discuss them separately.

CW structure on the associated analytic space

We can compute the Betti cohomology, i.e. the singular cohomology of the associated analytic space, by showing that the manifold $\mathbb{CP}^n := (\mathbb{P}^n(\mathbb{C}))^{an}$ admits a CW-structure and computing the cellular cohomology (which also satisfies the Eilenberg-Mac Lane axioms, hence is isomorphic).
For $n=0$ this is easy, as $\mathbb{CP}^0$ is just a single point. Suppose now we already have a CW-structure on $\mathbb{CP}^n$, then we will construct $\mathbb{CP}^{n+1}$ by attaching one $2n+2$-cell $\mathbb{C}^{n+1}$ by the gluing map $\eta : \mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{CP}^n$ given by quotienting out the $\mathbb{C}^\times$-action, i.e. I claim $\mathbb{CP}^{n+1} \simeq \mathbb{CP}^{n} \cup_{\eta} \mathbb{C}^{n+1}$. The map $\eta$ is known under the name "Hopf map", and it is a surjective map.
The CW structure of $\mathbb{CP}^n$, as we have just seen, is such that there is a single $2k$-dimensional cell for each $k = 0,\dots,n$. The chain complex computing cellular cohomology thus has no differentials, so we can quickly see that

$$H^i_{cell}(\mathbb{CP}^n,\mathbb{Z}) = \begin{cases} \mathbb{Z}, & \text{ for } i=2k, k\leq n\\ 0 & \text{else.} \end{cases}$$

Smooth de Rham theory and ring structure

We can also look at the deRham cohomology of $\mathbb{CP}^n$ as a smooth $2n$-dimensional real manifold. The advantage is that we would see the cup product structure (coming from the wedge of differential forms). From what we already know, there should be a closed smooth differential $2k$-form $\omega_k$ which isn't exact for each $k = 0,\dots,n$. We know that $\omega_n$ will be a volume form and similarly $\omega_k$ can be chosen to be a volume form of $\mathbb{CP}^k$ when restricted to that subspace. Then we have $\omega_k \wedge \omega_j$ homologous to $\omega_{k+j}$ for $k+j \leq n$, so that the multiplicative structure in the cohomology can be described as $[\omega_k] = [\omega_1]^k$ and by writing $x = [\omega_k]$ (of degree $2$) we have the identity of graded rings

$$H^\bullet_{dR}(\mathbb{CP}^n,\mathbb{R}) = \mathbb{R}[x]/(x^{n+1}).$$

Some people like to describe the cohomology of projective space by choosing an arbitrary hyperplane $H \subset \mathbb{CP}^n$ (so the complement is isomorphic to $\mathbb{CP}^{n-1}$) and then $H^\bullet_{sing}(\mathbb{CP}^n,\mathbb{Z}) = \mathbb{Z}[H]/(H^{n+1}),$ with $H$ in degree $2$. This description arises from the Poincaré duality map taking singular $k$-cycles in $\mathbb{CP}^n$ to singular $2n-k$-cocycles, which takes any complex hyperplane $H \subset \mathbb{CP}^n$ to the same cohomology class of degree $2$ (since the real codimension of a complex hyperplane is $2$).

Cup product with integer coefficients

Of course, it is unsatisfying to have the multiplicative structure only with real coefficients, so we can work a little bit more (or differently) and compute the cup product via Yoneda products in sheaf cohomology of the constant sheaf $\underline{\mathbb{Z}}$. I'm too lazy now, but in the end you get

$$H^\bullet_{Betti}(\mathbb{P}^n,\mathbb{Z}) = \mathbb{H}^\bullet(\mathbb{CP}^n, \underline{\mathbb{Z}}_{\mathbb{CP}^n}) = \mathbb{Z}[x]/(x^{n+1}).$$

Yet another way to get the cup product is the most "canonical" way with cellular cohomology (in my eyes), where we use only the diagonal morphism and the Künneth formula. I'm going to do that:
The diagonal $\mathbb{P}^n \to \mathbb{P}^n \times \mathbb{P}^n$ induces a homomorphism $H^\bullet_{cell}(\mathbb{P}^n \times \mathbb{P}^n) \to H^\bullet_{cell}(\mathbb{P}^n)$ and the Künneth formula tells us $H^\bullet_{cell}(\mathbb{P}^n \times \mathbb{P}^n) \simeq H^\bullet_{cell}(\mathbb{P}^n \otimes H^\bullet_{cell}(\mathbb{P}^n))$, so we can take classes $\alpha,\beta \in H^\bullet_{cell}(\mathbb{P}^n)$, form $\alpha \otimes \beta$, move this through the Künneth isomorphism and the morphism induced by the diagonal and end up in $H^\bullet_{cell}(\mathbb{P}^n)$ again, where we call the result $\alpha \cup \beta$. Since Künneth is a graded isomorphism, this map is graded and thus we have all properties of a cup product. All other constructions (via Yoneda Ext products or wedge products of differential forms) yield the same product (otherwise we wouldn't call it cup product).

Hodge theory

There is still something left out in this discussion, namely Hodge theory. Betti cohomology of a smooth projective variety carries a Hodge structure. For projective space, there isn't much to discuss, as the Hodge structure is trivial in the sense that every class in $H^{2k}_{B}(\mathbb{P}^n)$ is of Hodge type $(k,k)$, i.e. the Hodge diamond of $\mathbb{P}^n$ has only the Hodge numbers $h^{k,k}=1$ and all other Hodge numbers vanish. The computation can be found in Voisin's book, section 7.2 (page 167 of book one of the english edition).

The structure of a homogeneous space

Now, a little bit of general theory not necessary to proceed:
The $n$-dimensional projective space parametrizes 1-dimensional linear subspaces $L$ of affine $n+1$-space, which we can consider as partial flags of linear subspaces $0 \leq L \leq \mathbb{A}^{n+1}$. We have a natural $GL_{n+1}$-action on $\mathbb{A}^{n+1}$ (hence on flags). Partial flags of linear subspaces in $\mathbb{A}^{n+1}$ are stabilized by certain parabolic subgroups $P$ of $GL_{n+1}$, for example full flags are stabilized by a Borel subgroup. The quotient $GL_{n+1}/B$ thus parametrizes precisely full flags in $\mathbb{A}^{n+1}$. The quotients $GL_{n+1}/P$ that parametrize partial flags of a certain shape (determined by $P$) are called generalized Grassmannians.
I included this material here to give an outlook on how to proceed past projective space later on, in further calculations.

More concretely:
In $GL_{n+1}$ look at the subgroup $P$ given by block matrices with a block of size $n \times n$ just all of $GL_n$ (in the upper left corner) and a block of size $1 \times 1$ (in the lower right corner) just $GL_1$, i.e. $\mathbb{G}_m$, and a block of size $n \times 1$ just $\mathbb{A}^n$ (in the upper right corner), with a block of size $1 \times n$ just zeroes (in the lower left corner). If we look at $GL_{n+1}/P$, we see that, as a variety, the big $n \times n$ block in the upper left corner of $GL_{n+1}$ is killed and what remains is $\mathbb{A}^{n+1} \setminus \{0\}$ from the right column of $GL_{n+1}$, modulo the group action from $\mathbb{G}_m$ from the lower right corner of $P$. So we really have

$$GL_{n+1}/P = (\mathbb{A}^{n+1} \setminus \{0\})/\mathbb{G}_m = \mathbb{P}^n.$$

The good thing about this description is that $P$ contains a maximal torus (the diagonal matrices) and then there is a nice general theory of Schubert calculus to be applied. Topologically, this also gives us a CW structure, isomorphic to the structure we built "manually" before.

Algebraic de Rham cohomology over the rationals

By definition, $H^\bullet_{dR}(\mathbb{P}^n) = \mathbb{H}^\bullet(\mathbb{P}^n, \Omega^\bullet_{\mathbb{P}^n/\mathbb{Q}})$, a sheaf cohomology group. A cup product structure comes from the wedge product on the de Rham complex $\Omega^\bullet_{\mathbb{P}^n/\mathbb{Q}}$. One can compute algebraic de Rham cohomology by the Hodge-to-de Rham spectral sequence, which is

$$E^{p,q}_1 := \mathbb{H}^q(\mathbb{P}^n,\Omega^p_{\mathbb{P}^n/\mathbb{Q}}) \Rightarrow H^{p+q}_{dR}(\mathbb{P}^n).$$

In particular, we have $H^\bullet_{dR}(\mathbb{P}^1) \simeq \mathbb{Q}[x]/(x^2)$. With the Künneth formula we compute for an $n$-fold product $H^\bullet_{dR}(\mathbb{P}^1\times\cdots\times\mathbb{P}^1) \simeq \mathbb{Q}[x_1,\dots,x_n]/(x_i^2)$. The symmetric group $S_n$ acts on this polynomial ring as well as on the $n$-fold product $(\mathbb{P}^1)^{\times n}$. We let the symmetric group act trivially on $\mathbb{P}^n$ and the map $(\mathbb{P}^1)^{\times n} \to \mathbb{P}^n$ becomes $S_n$-equivariant. It induces an $S_n$-equivariant injection $H^\bullet_{dR}(\mathbb{P}^n) \to H^\bullet_{dR}((\mathbb{P}^1)^{\times n})$, so after taking invariants we obtain

$$H^\bullet_{dR}(\mathbb{P}^n) \simeq \mathbb{Q}[x]/(x^{n+1})$$

l-adic cohomology

I don't want to compute l-adic cohomology here, since even the definitions are a bit lengthy.

For example, $H^k_\ell(\mathbb{P}^n) \simeq 0$ for odd $k$ and $H^k_\ell(\mathbb{P}^n) \simeq \mathbb{Z}_\ell$ for even $k$. There is a Galois action on $H^k_\ell(\mathbb{P}^n)$ such that $H^2_\ell(\mathbb{P}^n)$ is equivariantly isomophic to $T_\ell(\mu)$, the limit over the roots of unity $\mu_{\ell^n} \subseteq \mathbb{C}$. Also, $H^0_\ell(\mathbb{P}^n) \simeq \mathbb{Z}_\ell$, where we equip $\mathbb{Z}_\ell$ with a trivial Galois action. Note that there are many non-equivariant isomorphisms $\mathbb{Z}_\ell \to T_\ell(\mu)$.

Similar in spirit, there is crystalline cohomology (or rigid cohomology) but I don't know enough about the subject to compute anything, so I have to leave that out.

Counting points over finite fields & Zeta function

I have written about this before (although with a different aim), so I don't want to repeat it.

The Zeta function of projective space is $Z(\mathbb{P}^n,s) = \prod_{k=0}^n{(1-p^kt)^{-1}}$,
as you can compute by hand from the decomposition of $\mathbb{F}_q$-rational points $\mathbb{P}^n(\mathbb{F}_q) = \mathbb{A}^n(\mathbb{F}_q) \cup \mathbb{A}^{n-1}(\mathbb{F}_q) \cup \cdots \cup \mathbb{A}^0(\mathbb{F}_q)$.

So far this is what I had to tell about the cohomology of projective space.