January 28, 2013

The Riemann-Roch and Hodge index theorems on surfaces

Posted by Akhil Mathew under algebraic geometry, differential geometry | Tags: , , |
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As in the previous two posts, let {S/k} be a smooth, projective surface over an algebraically closed field {k}. In the previous posts, we set up an intersection theory for divisors, which was a symmetric bilinear form

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

that gave the “natural” answer for the intersection of two transversely intersecting curves. Specifically, we had

\displaystyle \mathcal{L} . \mathcal{L}' = \chi(\mathcal{O}_S) - \chi(\mathcal{L}^{-1}) - \chi(\mathcal{L}'^{-1}) + \chi( \mathcal{L}^{-1} \otimes \mathcal{L}'^{-1});

the bilinearity of this map had to do with the fact that the Euler characteristic was a quadratic function on the Picard group. The purpose of this post is to prove a few more general and classical facts about this intersection pairing. As usual, Hartshorne’s Algebraic geometry and Mumford’s Lectures on curves on an algebraic surface are very helpful sources for this material; I also found Abhinav Kumar’s lecture notes useful.

1. The Riemann-Roch theorem

The Euler characteristic of a line bundle {\mathcal{L}} on {S} is a “topological” invariant: it is unchanged under deformations. Given an algebraic family of line bundles {\mathcal{L}_t} on {S} — in other words, a scheme {T} and a line bundle on {S \times_{k} T} which restricts on the fibers to {\mathcal{L}_t} — the Euler characteristics {\chi(\mathcal{L}_t)} are constant. This is one of the parts of the semicontinuity theorem on the cohomology of a flat family of sheaves. Over the complex numbers, one can see this by observing that the Euler characteristic of a line bundle is the index of an elliptic operator — more specifically, the index of the Dolbeault complex associated to {\mathcal{L}} — and can therefore be computed in purely topological terms via the Hirzebruch-Riemann-Roch formula.

In algebraic geometry, the fact that the Euler characteristic is a topological invariant is reflected in the following result, which computes it solely in terms of intersection numbers:

Theorem 1 Let {\mathcal{L}} be a line bundle on {S}. Then

\displaystyle \chi(\mathcal{L}) = \frac{1}{2} \mathcal{L}.( \mathcal{L} - K) + \chi(\mathcal{O}_S), \ \ \ \ \ (1)

where {K} is the canonical divisor on {S}. (more…)

 

May 18, 2010

The Riemann-Roch problem in number fields

Posted by Akhil Mathew under algebraic geometry, algebraic number theory, number theory | Tags: , , , |
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This post won’t be as cool as the title sounds. But I will prove something neat, and it will lead to neater things as time goes on (assuming I keep posting on this topic).

We will now return to algebraic number theory, following Lang’s textbook, and study the distribution of points in parallelotopes.

The setup is as follows. {K} will be a number field, and {v} an absolute value (which, by abuse of terminology, we will use interchangeable with “valuation” and “place”) extending one of the absolute values on {\mathbb{Q}} (which are always normalized in the standard way); we will write {|x|_v} for the output at {x \in K}.

Suppose {v_0} is a valuation of {\mathbb{Q}}; we write {v | v_0} if {v} extends {v_0}. Recall the following important formula from the theory of absolute values an extension fields:

\displaystyle |N^K_{\mathbb{Q}}(x)|_{v_0} = \prod_{v | v_0} |x|_v^{ [K_v: \mathbb{Q}_{v_0} ] }.

Write {N_v := [K_v: \mathbb{Q}_{v_0} ] }; these are the local degrees. From elementary algebraic number theory, we have {\sum_{v | v_0} N_v = N := [K:\mathbb{Q}].} This is essentially a version of the {\sum ef = N} formula.

The above formalism allows us to deduce an important global relation between the absolute values {|x|_v} for {x \in K}.

Theorem 1 (Product formula) If {x \neq 0},\displaystyle \prod_v |x|_v^{N_v} = 1.

 

The proof of this theorem starts with the case {K=\mathbb{Q}}, in which case it is an immediate consequence of unique factorization. For instance, one can argue as follows. (more…)