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Does anyone know of a global proof (involving no local argument) of Serre Duality at the level of varieties or manifolds (as opposed to schemes).

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Could you clarify what "no local argument" means? Is the Hodge theorem a "local argument" because being harmonic is a local condition? –  Ben Webster Nov 19 '09 at 15:47

5 Answers 5

up vote 8 down vote accepted

You might like the proof in section 5.3 of Voisin's book Hodge theory and complex algebraic geometry.

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Have you looked at the one in Griffiths and Harris? That's at least rather different from the general nonsense one in Hartshorne.

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Since a $\bar{\partial}$-Operator is an elliptic Operator, you can use elliptic theory in order to prove the Serre duality. In fact the Serre duality is a kind of corollary of the"fundamental theorem" (as I Know it). In fact, you do not need the Hodge theorem, since the Hodge theorem itself is a corollary of the theorem.

For a reference of this "fundamental theorem" (perhaps slightly reformulated) I would refer to one of the following:

  • Wells, Differential Analysis on Complex Manifolds;
  • Gilkey, Invariance Theory, the Heat Equation and the Atiyah Singer Index Theorem

(a complete proof for pseudodifferential operators)

  • Warner, Foundations of Differentiable Manifolds

(the theorem is included as an exercise on the last page

  • Kazdan, Lecture Notes on Applications of Partial Differential Equations to Some Problems in Differential Geometry (online available here)

(Corollary 2.5, for a sketch of the proof)

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I thought I'd offer a high-tech alternative for certain varieties. If $X$ is smooth and projective over a field $k$ then Bondal and van den Bergh give a proof here that $D^b(\mathrm{Coh}X)$ is saturated which is a strong representability condition on cohomological/homological functors to the category of $k$ vector spaces. It follows immediately that $D^b(\mathrm{Coh}X)$ has a Serre functor by using the fact that $Hom(A,-)^*$ is representable for every bounded complex of coherent sheaves $A$.

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I like the presentation from Analytic methods in algebraic geometry by Demailly. Here is the link: http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/eem2007.pdf.

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