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Section 1.9 Polarizations and Étale cohomology (Alex)

Plan: polarizations, a little cohomological warmup and a cool finiteness result. Étale cohomology.

Subsection 1.9.1 Polarizations

Definition 1.9.1. Polarizations.

A polarization of an abelian variety \(A/k\) is an isogeny

\begin{equation*} \lambda \colon A \to \hat A \end{equation*}

such that

\begin{equation*} \lambda \simeq_{\overline k} \lambda_{\sheaf{L}} : a\mapsto t_a^*\sheaf L \otimes \sheaf L^{-1} \end{equation*}

for an ample invertible sheaf \(\sheaf L\) on \(A_{\overline k}\text{.}\)

We then have a notion of degree, polarizations of degree 1 (i.e. isomorphisms \(A\to \hat A\)) are called principal polarizations.

Natural questions: what does the line bundle \(\sheaf L\) tell us about the polarization? Can we tell principality?

To answer this we must (rapidly) recall (Zariski) sheaf cohomology. But this will help us in the next section too.

A line bundle (or indeed any sheaf) defines for us for any open subset \(U \hookrightarrow X\) an abelian group of sections \(\sheaf L(U)\text{.}\)

However taking (global) sections doesn't play well with exact sequences!

Example 1.9.3. Classic example.

Let \(X = \CC^*\) and consider

\begin{equation*} 0 \to \ZZ \hookrightarrow \sheaf O_X \xrightarrow{e^{2\pi i -}} \sheaf O_X^* \to 0 \end{equation*}

but

\begin{equation*} 0 \to \ZZ \to \sheaf O_X(X) \to \sheaf O_X^*(X) \end{equation*}

is not surjective on the right, for example \(f(z) = z\) is a nowhere vanishing meromorphic function on \(X\) but its not \(\exp\) of anything. Upshot: maps of sheaves can be surjective (by being so locally) but not globally.

To understand/control this phenomenon we introduce \(H^1(X, \sheaf F)\) fitting into the above and so on.

Explicitly: for a sheaf \(\sheaf F\) we fix an injective resolution

\begin{equation*} 0\to \sheaf F \to \sheaf I_0 \to \sheaf I_1 \to \cdots \end{equation*}

which we then take global sections of to get a chain complex

\begin{equation*} 0\to \Gamma(X,\sheaf F) \to \Gamma(X,\sheaf I_0) \to \Gamma(X,\sheaf I_1) \to \cdots \end{equation*}

and we truncate and take cohomology of this to measure “failure of exactness”

\begin{equation*} H^0(X, \sheaf F) , H^1(X, \sheaf F) , H^2(X, \sheaf F) , \ldots\text{.} \end{equation*}
Definition 1.9.4. Euler-Poincaré characteristic.

Define the Euler-Poincaré characteristic of a line bundle \(\sheaf L\) to be

\begin{equation*} \chi(\sheaf L) = \sum (-1)^i \dim_k H^i(A,\sheaf L)\text{.} \end{equation*}

Recall Subsection 1.6.3: So for ample \(\sheaf L\) we have \(K(\sheaf L)\) finite, so the vanishing theorem applies. Additionally for very ample \(\sheaf L\) we know \(H^0(A,\sheaf L) \ne 0\) so in this case we get vanishing of higher cohomology.

(Super sketch)

Over a finite field implies there is an ample \(\sheaf L\) with \(\lambda_{\sheaf L}\) a polarization of degree \(d^2\text{,}\) then using above \(\chi(\sheaf L^3) = 3^g d\) and \(\sheaf L^3\) is very ample hence \(\dim H^0(A, \sheaf L^3) = 3^g d\) so we get an embedding into \(\PP^{3^g d - 1}\text{.}\)

The degree of \(A\) in \(\PP^{3^g d - 1}\) is \(((3D)^g) = 3^g d(g!)\) . It is determined by its Chow form, which by these formulae has some (large) bounded degree, as we are over a finite field however there are only finitely many such.

Subsection 1.9.2 Étale Cohomology of Abelian Varieties

See [77] or [96].

Recall for abelian varieties over \(A/\CC\) we considered singular cohomology of the complex points \(A(\CC)\text{.}\) Indeed this theory was strongly connected to the lattice \(\Lambda\) defining \(A(\CC)\text{.}\)

We saw that in fact \(\pi_1(A,0) = \pi^{-1} (0) = \Lambda \subseteq V\) which was the universal covering space of \(A(\CC)\text{.}\) We want to emulate this over a general field.

We want to allow multiplication by \(n\) to define finite covers for our abelian varieties as they did before.

Problem: Zariski topology is too coarse: we can't find an open \(U\) set around \(0 \in A\) such that \(\lb 2\rb \colon U \to A\) is an isomorphism onto its image. Isogenies are not local isomorphisms for the Zariski topology.

How on earth do we “allow” maps which are clearly not local isomorphisms to become such? First what do we mean by local isomorphism?

\begin{equation*} \xymatrix{ f^{-1}(U)\ar[d] \ar[r]^{\sim} & U\ar@{^(->}[d]^i\\ X \ar[r]_f & Y }\text{.} \end{equation*}

There exists an open subset \(U\) such that the base change \(X \times_Y U\) is isomorphic with \(\coprod U\) of several copies of \(U\) in a compatible way with the map to \(U\text{.}\)

So let's cheat, the best isomorphism is the identity map

\begin{equation*} \xymatrix{ X\ar[d] \ar[r]^{\sim} & X\ar[d]^f\\ X \ar[r]_f & Y } \end{equation*}

if we define an “open set” \(U\) to be a morphism \(X \to Y\) with the properties we want, then all such become local isomorphisms.

By taking our topology to be given by some maps we decide are decent covering maps we can circumvent these difficulties.

What is the correct class of morphisms to take here, we feel like our \(\lb n\rb\) maps should count. Taking inspiration from differential geometry perhaps, we are led to the notion of a local diffeomorphism, an étale map.

Definition 1.9.8.

Let \(X,Y\) be nonsingular varieties over \(k = \overline k\text{.}\) Then \(f\colon X\to Y\) is étale at a point \(P\in X\) if

\begin{equation*} \diff f\colon\Tgt_{P}(X) \to \Tgt_{f(P)} (Y) \end{equation*}

is an isomorphism.

Example 1.9.10. A non-étale map.

Consider the map

\begin{align*} \aff^2 \amp\to \aff^2\\ (x,y) \amp\mapsto (x^3, x^2 + y) \end{align*}

we can see that the image of \(y= 0\) is the nodal cubic (\(Y^3 = X^2\)), which is messed up (singular) at \((0,0)\text{.}\) The jacobian is

\begin{equation*} \begin{pmatrix} 3x^2 \amp 0 \\ 2x \amp 1\end{pmatrix} \end{equation*}

so this matrix is singular exactly when \(x= 0\) (unless characteristic 3). So the map is not étale at these points.

Key point \(\diff (\alpha+ \beta)_0 = (\diff \alpha )_0 + (\diff \beta)_0\text{.}\) So the map on tangent spaces is simply multiplication by \(n\text{.}\)

Definition 1.9.12. Étale morphisms.

A morphism \(f\colon X\to Y\) of schemes is étale if it is flat and unramified.

Flatness for finite morphisms of varieties is equivalent to each fibre \(f^{-1}(t)\) being of equal cardinality, counting multiplicities.

All isogenies are finite and flat.

Definition 1.9.13.

Let \(\mathrm{FEt}/X\) be the category of finite étale maps \(\pi\colon Y \to X\) (i.e. finite étale coverings of \(X\)).

Then after picking a basepoint \(x\in X\) we can map

\begin{equation*} F\colon \mathrm{FEt}/X\to \mathrm{Set} \end{equation*}
\begin{equation*} \pi\mapsto \Hom_X(x,Y) \approx\pi^{-1}(x)\text{.} \end{equation*}

This is in fact pro-representable, i.e. there exists a system

\begin{equation*} \tilde X = (X_i)_{i\in I} \end{equation*}

with

\begin{equation*} F(Y) = \Hom(\tilde X ,Y) = \varinjlim_i \Hom(X_i, Y)\text{.} \end{equation*}

We then define

\begin{equation*} \pi_1(X,x) = \Aut_X(\tilde X) = \varprojlim_i \Aut_X(X_i)\text{.} \end{equation*}

So we need to understand étale covers of abelian varieties. Following [47]:

(Sketch)

Let

\begin{equation*} \tau\colon X\times X \to X\times X \end{equation*}
\begin{equation*} \tau(x,y) = (xy,y) \end{equation*}

so \(\tau^{-1} (e,e) = (e,e)\text{.}\) Some exercise in Hartshorne implies \(\im \tau\) has dimension \(2\dim X\text{.}\)

Reduce to algebraically closed case.

Let

\begin{equation*} \tau^{-1}(\{e\} \times X) = \{(x,y) : xy = e\} = \Gamma \subseteq X\times X \end{equation*}

as \(\tau \) is surjective we get \(p_2 \colon \Gamma \to X\) is also so pick an irreducible \(\Gamma_1 \subseteq \Gamma\) with \(p_2(\Gamma_1) = X\text{.}\) This also implies \(p_1(\Gamma_1) = X\text{.}\)

Let

\begin{equation*} f\colon \Gamma_1 \times X\times X \to X \end{equation*}
\begin{equation*} f((x,y),z,w) = x((yz)w) \end{equation*}

then

\begin{equation*} f(\Gamma_1 \times\{e\}\times\{e\}) = \{eee\} = \{e\} \end{equation*}

so a version of rigidity 1.2.11 gives

\begin{equation*} x((yz)w) = zw\ \forall (x,y) \in \Gamma_1,\,z,w\in X \end{equation*}

So letting \(w = e\) we get

\begin{equation*} x(yz) = z\text{.} \end{equation*}

Fix \(y \in X(k)\text{,}\) and then by surjectivity we can find \(x,z \in X(k)\) with \((x,y)\in \Gamma_1 \ni (y,z)\text{.}\) So we get

\begin{equation*} x = x(yz) = ze = z \end{equation*}

and so \(y\) has both a left and right inverse. We then multiply above by \(y\) to get

\begin{equation*} y(zw) = y(x((yz)w)) = (yz)w \end{equation*}

so \(X(k)\) is associative.

Must construct a group law on \(Y\text{:}\)

Take the graph of \(m\colon X\times X \to X\)

\begin{equation*} \Gamma_X \subseteq X\times X\times X \end{equation*}

and pullback along \(f\times f\times f\) to

\begin{equation*} \Gamma'_Y \subseteq Y\times Y\times Y \end{equation*}

fix the connected component \(\Gamma_Y\) containing \((e_Y,e_Y,e_Y)\text{.}\)

Call the projections from \(\Gamma_Y\) \(q_I\text{.}\) Now we must show that \(q_{12}\colon \Gamma_Y \to Y\times Y\) is an isomorphism, then \(m_Y\colon Y\times Y \to Y\) can be defined as \(q_3 \circ q_{12}^{-1}\text{.}\) \(q_{12}\) has sections \(s_1,s_2\) over \(\{e_Y\}\times Y\text{,}\) \(Y\times \{e_Y\}\) respectively given by \(s_1(e_Y,y) = (e_Y,y,y)\) and \(s_2(y,e_y,y) = (y,e_y,y)\text{.}\) So \(m_Y\) satisfies the conditions of the surprising proposition.

\begin{equation*} \xymatrix{ \Gamma_Y \ar[r]\ar[d]_{q_{12}} & \Gamma_X\ar[d]^{p_{12}} \\ Y\times Y \ar[r]_{f\times f} & X\times X } \end{equation*}

the horizontal maps are étale coverings and the rightmost an isomorphism so \(q_{12}\) is an étale covering. The projection \(p_2 \circ q_{12} = q_2 \colon \Gamma_Y\to Y\) is smooth proper. Fact: all fibres of \(q_2\) are irreducible. So \(Z = q_2^{-1} (e_Y) = q_{12}^{-1}(Y\times \{e_Y\})\) is irreducible. Moreover \(q_{12}\) restricts to an étale covering \(Z \to Y = Y\times\{e_Y\}\) of the same degree, but \(s_2\) is a section of this covering, hence it is an isomorphism. Hence \(q_{12}\) has degree 1 and is therefore an isomorphism as required.

So we have some control over the finite étale maps, what does the covering space look like? Last week we saw that for an isogeny \(\alpha\colon B \to A\) we could find \(\beta \colon A \to B\) with \(\beta\circ\alpha = \lb n\rb\colon A \to A\text{.}\) This means we can take our universal covering space to be

\begin{equation*} (A)_{i\in I} \end{equation*}

with multiplication by \(n\) maps.

So we find

\begin{equation*} \pi_1^\et(A,0) =\varprojlim_n \Aut_A(A\xrightarrow{[n]} A)= \varprojlim_n A[n]\text{.} \end{equation*}

Note that Milne gives a combined proof of the above two statements, this relies on some theorems on Hopf algebras such as [25, Theoreme 6.1].