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In the context of Kaehler geometry, we have the Kodaira embedding theorem. If you google "Kodaira embedding" you can get lots of good references. See also Huybrechts and Griffiths-Harris.

One reason is Givental's conjecture, which says that in the semisimple case, genus 0 GW invariants determine higher genus GW invariants. See this paper of Teleman, in which the conjecture is proved. …

Mukai's theorem states that if $X$ is an abelian variety, and $\check{X}$ is the dual abelian variety, then the Fourier-Mukai transform corresponding to the Poincare line bundle on $X \times \check{X} …

As I said in the comments, you should read AJ's answer to this question. If you haven't read Costello's paper "Topological conformal field theories and Calabi-Yau categories", then you should definit …

I googled a bit, and I found a paper "All toric local complete intersection singularities admit projective crepant resolutions" by Dimitrios I. Dais, Christian Haase, and Günter M. Ziegler. It might b …

There was some discussion about this (and other things) at the secret blogging seminar fairly recently: http://sbseminar.wordpress.com/2009/08/06/algebraic-geometry-without-prime-ideals/

My advice: spend a lot of time going to seminars (and conferences/workshops, if possible) and reading papers. Talk to people, read blogs, subscribe to the arxiv AG feed, etc. AG is a very large field, …

Frobenius manifolds arise in the study of quantum cohomology and mirror symmetry -- roughly they are manifolds (or varieties or whatever) such that the tangent spaces are Frobenius algebras (there are …

Let $C$ be a compact Riemann surface, let $C^2$ be the cartesian square of $C$, let $J(C)$ be the degree zero Jacobian of $C$, and let $\delta : C^2 \to J(C)$ be the map $(x,y) \mapsto [\mathcal{O}(x- …

I am interested in this claim: The $n$th symmetric power $C^{(n)}$ of a genus $g$ curve $C$ is isomorphic to the projectivization $\mathbb{P}(E_n)$ of the sheaf $E_n := \pi_\ast(P_n)$ over the Jacobi …

Yes! The geometric picture is very nice and very easy. It is explained on pages 408-409 of Griffiths-Harris. Here is roughly how it works: Let's work over $\mathbb{C}$ for simplicity. Think of $\mat …

In this question, Ariyan asks about the question of uniqueness of extensions of vector bundles when they exist. Sasha's answer suggests that extensions of vector bundles don't always exist. More pre …

Jacob Lurie's stuff seems to develop derived algebraic geometry via $E_\infty$ rings and/or maybe something like simplicial commutative rings. Ben Wieland's comment in this question indicates that Lur …

Not sure if I understand your question, but here are some generalities. There is a correspondence between line bundles and divisors (you can find this in any algebraic geometry book). There is a cano …

Stephen Griffeth's argument works over any field. The total space of a vector bundle is never proper (follows by, e.g., valuative criterion for properness). On the other hand, $P^n$ is always proper. …