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Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.

2 votes
0 answers
164 views

Chow variety of 1-cycles on abelian surface

It is an easy exercise to show that on a K3 surface, a smooth genus $g$ curve moves in a $g$-dimensional linear system. Nearly the same exercise shows that on an abelian surface, the corresponding li …
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7 votes
1 answer
427 views

Incorporating Divisors (D4-branes) into Donaldson-Thomas Theory?

Let $X$ be a Calabi-Yau threefold. Ordinary Donaldson-Thomas theory is formulated as a virtual count of ideal sheaves $\mathcal{I}$ with discrete invariants $\text{ch}(\mathcal{I}) = (1,0, -\beta, -n …
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7 votes
2 answers
460 views

Show Fiber Product of Rational Elliptic Surfaces is Calabi-Yau

In a handful of contexts people study Calabi-Yau threefolds formed by taking the fiber product of two rational elliptic surfaces. I can't find any detailed explanation of why such geometries are actu …
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18 votes
1 answer
1k views

Do all $\mathcal{N}=2$ Gauge Theories "Descend" from String Theory?

I asked this on PhysicsSE, but I think it also fits here as it's related to algebro-geometric connections to string and gauge theory. I'm thinking about the beautiful story of "geometrical engineer …
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