March 19, 2013

The Picard scheme I

Posted by Akhil Mathew under algebraic geometry | Tags: , , , |
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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…)

 

March 18, 2013

Fundamental groups of unirational varieties

Posted by Akhil Mathew under algebraic geometry, topology | Tags: , , |
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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…)

 

January 27, 2013

Intersection theory on surfaces

Posted by Akhil Mathew under algebraic geometry | Tags: , , , |
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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 A 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…)

 

February 18, 2012

The Bochner technique

Posted by Akhil Mathew under differential geometry, topology | Tags: , , |
[4] Comments 

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…)

 

December 6, 2009

Decomposition of the space of square-integrable 1-forms

Posted by Akhil Mathew under analysis, complex analysis | Tags: , , , , |
1 Comment 

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…)