Let ${X}$ be an abelian variety over an algebraically closed field ${k}$. If ${k = \mathbb{C}}$, then ${X}$ corresponds to a complex torus: that is, ${X}$ can be expressed complex analytically as ${V/\Lambda}$ where ${V}$ is a complex vector space of dimension ${\dim X}$ and ${\Lambda \subset V}$ is a lattice (i.e., a ${\mathbb{Z}}$-free, discrete submodule of rank ${2g}$). In this case, one can form the dual abelian variety

$\displaystyle X^{\vee} = \hom(X, S^1) = \hom_{\mathrm{cont}}(V/\Lambda, \mathbb{R}/\mathbb{Z}) \simeq \hom_{\mathbb{R}}(V, \mathbb{R})/2\pi i \hom(\Lambda, \mathbb{Z}).$

At least, ${X^{\vee}}$ as defined is a complex torus, but it turns out to admit the structure of an abelian variety.

The purpose of the next few posts is to describe an algebraic version of this duality: it turns out that ${X^{\vee}}$ can be constructed as a scheme, purely algebraically. I’d like to start with a couple of posts on Picard schemes. A useful reference here is this article of Kleiman.

1. The Picard scheme analytically

Let ${X }$ be a smooth projective variety over the complex numbers ${\mathbb{C}}$. The collection of line bundles ${\mathrm{Pic}(X)}$ is a very interesting invariant of ${X}$. Usually, it splits into two pieces: the “topological” piece and the “analytic” piece. For instance, there is a first Chern class map

$\displaystyle c_1: \mathrm{Pic}(X) \rightarrow H^2(X; \mathbb{Z}) ,$

which picks out the topological type of a line bundle. (Topologically, line bundles on a space are classified by their first Chern class.) The admissible topological types are precisely the classes in ${H^2(X; \mathbb{Z})}$ which project to ${(1,1)}$-classes in ${H^2(X; \mathbb{C})}$ under the Hodge decomposition. (more…)

Let ${X}$ be a variety over an algebraically closed field ${k}$. ${X}$ is said to be rational if ${X}$ is birational to ${\mathbb{P}_k^n}$. In general, it is difficult to determine when a variety in higher dimensions is rational, although there are numerical invariants in dimensions one and two.

• Let ${X}$ be a smooth projective curve. Then ${X}$ is rational if and only if its genus is zero.
• Let ${X}$ be a smooth projective surface. Then ${X}$ is rational if and only if there are no global 1-forms on ${X}$ (i.e., ${H^0(X, \Omega_{X/k}) = 0}$) and the second plurigenus ${H^0(X, \omega_{X/k}^{\otimes 2}) }$ vanishes. This is a statement about the negativity of the cotangent bundle (or, equivalently, of the positivity of the tangent bundle) which is a birational invariant and which holds for ${\mathbb{P}^2_k}$. The result is a criterion of Castelnuovo, extended by Zariski to characteristic $p$.

In higher dimensions, it is harder to tell when a variety is rational. An easier problem is to determine when a variety is unirational: that is, when there is a dominant rational map

$\displaystyle \mathbb{P}_k^n \dashrightarrow X;$

or, equivalently, when the function field ${k(X)}$ has a finite extension which is purely transcendental. In dimensions one and two (and in characteristic zero), the above invariants imply that a unirational variety is rational. In higher dimensions, there are many more unirational varieties: for example, a theorem of Harris, Mazur, and Pandharipande states that a degree ${d}$ hypersurface in ${\mathbb{P}^N}$, ${N \gg 0}$ is always unirational.

The purpose of this post is to describe a theorem of Serre that shows the difficulty of distinguishing rationality from unirationality. Let’s work over ${\mathbb{C}}$. The fundamental group of a smooth projective variety is a birational invariant, and so any rational variety has trivial ${\pi_1}$.

Theorem 1 (Serre) A unirational (smooth, projective) variety over ${\mathbb{C}}$ has trivial ${\pi_1}$.

The reference is Serre’s paper “On the fundamental group of a unirational variety,” in J. London Math Soc. 1959. (more…)

The purpose of this post and the next is to work through a basic example of intersection theory: intersections of curves on a surface. This is a fundamental and basic example in algebraic geometry, and since I’ve never studied intersection theory, it like seems a reasonable place to start. The references here are chapter 5 of Hartshorne’s Algebraic geometry and Mumford’s Lectures on curves on an algebraic surface.

1. Curves on surfaces

The subject of “curves on a surface” is the subject of Mumford’s book mentioned above; the purpose of this section is simply to set down the definitions.

Let ${k}$ be an algebraically closed field. A surface ${S}$ is a smooth projective surface over ${k}$. There is a classification of surfaces, but let’s just list a couple of basic examples: ${\mathbb{P}^2, \mathbb{P}^1 \times \mathbb{P}^1}$, (smooth) hypersurfaces in ${\mathbb{P}^3}$, and ruled surfaces.

Definition 1 curve on a surface ${S}$ is an (effective) divisor on ${S}$. Equivalently, it is a subscheme ${C \subset S}$ pure of codimension one, so locally cut out by one equation. (But ${C}$ is not necessarily smooth, or even reduced.)

The goal of this post and the next is to set up a basic intersection theory for curves on surfaces. Given two curves ${C, D \subset S}$, we’d like to define the intersection product ${C.D}$. There is one case where it is easy: suppose ${C}$ and ${D}$ meet only transversely. In other words, for each ${p \in C \cap D}$, we choose local equations ${f,g \in \mathfrak{m}_{S, p} \subset\mathcal{O}_{S, p}}$ for the subschemes ${C, D}$, and

$\displaystyle (f,g) = \mathfrak{m}_{S, p}.$

In particular, this implies that ${C, D}$ are nonsingular at all points of intersection. In this case, we would like to require

$\displaystyle C.D = \sum_{p \in C \cap D} 1 \quad (\text{if transverse intersection}). \ \ \ \ \ (1)$

Once we require the above condition and two more natural conditions, we will prove that the intersection product is uniquely determined:

• The equation (1) holds under transversality assumptions and if ${C, D}$ are smooth.
• The intersection product is additive. That is, given curves ${C_1, C_2, D}$, we have

$\displaystyle (C_1 + C_2). D = C_1.D + C_2.D,$

where ${C_1+C_2}$ is treated as an effective Cartier divisor.

• The intersection product is invariant under linear equivalence. If ${C, C'}$ are linearly equivalent curves, we want

$\displaystyle C. D = C'.D,$

so that the intersection product is invariant under deformation. In particular, this and the previous item show that the intersection product only depends on the line bundle associated to a divisor (and can make sense for any divisor, not necessarily effective).

Our goal is to prove:

Theorem 2 There is a unique pairing

$\displaystyle \mathrm{Pic}(S) \times \mathrm{Pic}(S) \rightarrow \mathbb{Z}$

satisfying the above three conditions. (more…)

There are a number of results in geometry which allow to conclude that a certain group vanishes or is bounded under hypotheses on the curvature. For instance, we have:

Theorem 1 If ${M}$ is a compact manifold of positive curvature, then ${H^1(M; \mathbb{R}) = 0}$.

Another such result is the Kodaira vanishing theorem, which enables one to show that certain cohomology groups of an ample line bundle on a smooth projective variety vanish in characteristic zero.

I’ve been trying to gain an understanding of such results, and it seems that there is a common technique in such arguments. The first strategy is to identify the desired cohomology group (e.g. ${H^p(M; \mathbb{R})}$) with the kernel of a Laplacian-type operator, by Hodge theory. The second step is to bound below the relevant Laplacian-type operator. In this post, I’d like to try to explain what’s going on, in a special case. (more…)

Yesterday I defined the Hilbert space of square-integrable 1-forms ${L^2(X)}$ on a Riemann surface ${X}$. Today I will discuss the decomposition of it. Here are the three components:

1) ${E}$ is the closure of 1-forms ${df}$ where ${f}$ is a smooth function with compact support.

2) ${E^*}$ is the closure of 1-forms ${{}^* df}$ where ${f}$ is a smooth function with compact support.

3) ${H}$ is the space of square-integrable harmonic forms.

Today’s goal is:

Theorem 1 As Hilbert spaces,

$\displaystyle L^2(X) = E \oplus E^* \oplus H.$

The proof will be divided into several steps. (more…)