Questions tagged [complex-geometry]

Question 1

Let $S$ be a normal projective algebraic surface over a complex numbr field ${\Bbb C}$. Suppose that $S$ has a single isolated singular point $p \in S$. Let $U \colon= S\setminus p$ be a smooth open ...

Question 2

I'm trying to follow my lecture course in fiber bundles and I'm not sure whether my lecturer has made a mistake. Definition. Let $M$ be a complex manifold. Then define the picard group $\mathrm{Pic} (...

Question 3

I am currently reading through Malikov, Schechtman and Vaintrob's paper Chiral de Rham Complex. In the proof of Theorem 2.4, i.e. that the chiral de Rham complex extends the usual de Rham complex for ...

Question 4

I've been studying $\mathrm{Spin}^c$-structures on complex manifolds, in particular attempting to understand why every complex $n$-manifold with $n \geq 2$ has a canonical such structure. The usual ...

Question 5

I have a very hard time to understand something physicists call $A$ or $B$ twists in the context of topological string theory. A canonical reference seems to be this Witten's paper. Let $\Sigma$ be a ...

Question 6

Suppose X be a Riemann surface endowed with anti-holomorphic involution $\sigma_X$. $V$ is a holomorphic vector bundle on $X$ with holomorphic connection $D$. It is a fact that $\sigma_X^*\overline{V}$...

Question 7

Let us consider the complex projective space $\mathbb{CP}^n$ as a complex manifold. The holomorphic line bundle $\mathcal{O}(1)$ over $\mathbb{CP}^n$ may be defined as the dual bundle of the ...

Question 8

I'm working through Kerr's notes (#4 here) in algebraic geometry and am stuck on the following exercise: Show that the existence domain $M$ of $$ \mathfrak{F}(z) = \left( \prod_{i=1}^{2g+2} (z-\...

Question 9

It is well-known that modular curves parametrize elliptic curves with level structures. For the purpose of this question, I will work complex-analytically and describe analytically the moduli space $$\...

Question 10

I am stuck on a step in the proof of the proposition on pages 19-20 of Griffiths-Harris's "Principles of Algebraic Geometry" that may be obvious. Let $f: U → V$ be a holomorphic map between ...

Question 11

I have been learning about the Poincare residue in the context of the cohomology of projective hypersurfaces. For such a hypersurface, $X\subset\mathbb{P}^{n+1}$ with defining equation $F$, the ...

Question 12

All schemes in the following are assumed to be of finite type over $\mathbb C$. Let $\mathbb P^n$ be a projective space and $S$ a scheme. Denote by $$p : \mathbb P^n \times_{\mathbb C} S \to \mathbb P^...

Question 13

In Huybrechts' Complex Geometry, he defines a morphism of holomorphic vector bundles $\pi_E: E \to X$ and $\pi_F: F \to X$ to be a holomorphic map $f: E \to F$ such that $\pi_E = \pi_F \circ f$, the ...

Question 14

If you have a given symplectic form $\omega \in \Omega^{2}(M)$ on a smooth manifold $M$ it determines a space $\mathcal{J}(M,\omega)$ of compatible almost complex structures. Now, given a manifold $M$ ...

Question 15

Here"analytically isomorphic"means that the completion of the local rings of two points in some complex spaces are isomorphic. The smooth case is trivial so the only interesting case is that ...

Stack Exchange Network

Questions tagged [complex-geometry]

Canonical divisor of a singular surface.

First Chern class of a line bundle

Splitting of the Chiral de Rham differential for affine space

Why don't all complex manifolds admit spin structures?

How to understand the twists physicists use in topological string theory?

Induced connection on pull-back of conjugate bundle

Global holomorphic sections of $\mathcal{O}(1)$

Existence domain as a complex 1-manifold

Understanding modular curves as "moduli of Hodge structures"

Why does $f$ map components of the critical locus to a point? (Griffiths-Harris, pp. 19-20)

Explicit calculation of the Poincare residue for a pole of order 2

Why do we have $q_* \mathcal{O}_{\mathbb P^n \times S}(m) = H^0(\mathbb P^n, \mathcal{O}_{\mathbb P^n}(m)) \otimes \mathcal{O}_S$?

Morphism of holomorphic vector bundles

Families of Almost Complex Structures that Determine a Symplectic Form [closed]

what does analytically isomorphic mean for complex spaces？

Hot Network Questions