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Question 1

1vote

1answer

98views

First Chern class of a line bundle

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} (...

the_dude

774

askedNov 19 at 14:20

Question 2

2votes

0answers

42views

Splitting of the Chiral de Rham differential for affine space

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 ...

Siegmeyer of Catarina

301

askedNov 18 at 11:01

Question 3

6votes

1answer

149views

Why don't all complex manifolds admit spin structures?

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 ...

Baylee V

977

askedNov 13 at 1:05

Question 4

4votes

1answer

172views

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

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 ...

Gold

28.4k

askedNov 11 at 13:47

Question 5

0votes

0answers

46views

Induced connection on pull-back of conjugate bundle

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 $\sigma_X^*\overline{V}$...

Sandipan Das

1

askedNov 10 at 14:56

Question 6

1vote

1answer

110views

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

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 ...

Gawain

250

askedNov 6 at 20:51

Question 7

0votes

0answers

48views

Existence domain as a complex 1-manifold

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-\...

stone327

81

askedNov 4 at 21:28

Question 8

2votes

0answers

70views

Understanding modular curves as "moduli of Hodge structures"

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 $$\...

Horace4036

21

askedOct 25 at 19:16

Question 9

2votes

1answer

93views

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

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 ...

vimoe

21

askedOct 24 at 4:17

Question 10

8votes

0answers

244views

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

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 ...

CoffeeCrow

1,697

askedOct 23 at 2:32

Question 11

3votes

2answers

116views

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

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^...

Francis H.

968

askedOct 21 at 7:46

Question 12

1vote

0answers

66views

Morphism of holomorphic vector bundles

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 ...

The Surgeon of Death

652

askedOct 15 at 12:26

Question 13

1vote

0answers

61views

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

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 $M$ ...

Lucas Felizardo S. Gama

51

askedOct 15 at 1:17

Question 14

0votes

0answers

42views

what does analytically isomorphic mean for complex spaces？

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 ...

Xq Chern

1

askedOct 14 at 8:10

Question 15

3votes

0answers

113views

Hyperelliptic curve with binary octahedral symmetry

An old reprint by Wolfart mentions the hyperelliptic curve (Riemann surface) $y^2=x^8+14x^4+1$ of genus $3$ with binary octahedral symmetry, and various follow-up papers include it in tables. I am ...

Mikhail Katz

49.2k

askedOct 12 at 7:59

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Stack Exchange Network

Questions tagged [complex-geometry]

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？

Hyperelliptic curve with binary octahedral symmetry

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