Global proof of Serre duality 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).
 A: You might like the proof in section 5.3 of Voisin's book Hodge theory and complex algebraic geometry.
A: 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 in Generators and representability of functors in commutative and noncommutative geometry 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 $A$.
A: 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)
A: 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.
A: Have you looked at the one in Griffiths and Harris?  That's at least rather different from the general nonsense one in Hartshorne.
