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I am not sure whether this answers your question, but you should perahps have a look at Faber, Pandharipande: Relative Maps and Tautological Classes "The push-forwards of all Gromov-Witten cla …

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" …

Riemann-Roch in the version I know it: Let $(E,\bar{\partial})$ be a holomorphic bundle over a compact Riemann surface $M$. Then $$\operatorname{index}(\bar{\partial}) = \deg E -(g-1)\operatorname{ …

First a word of warning: I am not a trained algebraic geometer, so it is possible (likely) that these questions are inappropriate for MO, if so: my apologies. Said this: As far as I understand the ta …

The answer is yes!(at least if quaternionic holomorphic geometry counts) "Quaternionic holomorphic geometry" provides a very elegant description of surfaces in 3- and 4-dimensional space. The first pa …

First of all: this is from the "differential-geometric point of view" If you want to "classify" vector bundles chern classes are a very helpful tool. Well, it can happen that two bundles are "differe …

Let $V$ be a smooth manifold, $E \rightarrow V$ a vector bundle over $V$ and $\Gamma$ be a finite group acting nontrivially on $V$ and $E$. Let $s \in C^\infty(E)$ be a generic $\Gamma$-equivariant se …

WLOG the fan $\Sigma$ of our toric variety $X_{\Sigma}$ is simplicial. (So $X_{\Sigma}$ has at worst orbifold singularities and all cones $\sigma \in \Sigma$ are simplicial). The classical toric reso …

Let $M$ be a manifold (variety, scheme, your favorite object) and let $N_1,N_2$ be two submanifolds (subvarieties, closed subschemes, ideal sheafes, etc.) such that $N_1 \cap N_2 \neq \emptyset$. Deno …