Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

learn more… | top users | synonyms

2
votes
0answers
13 views

Which complex manifolds admit a collection of compatible 'charts' $\varphi_{\alpha} : U_{\alpha} \to \mathbb{C}^n$ which cover $X$?

A complex manifold of dimension $n$ is a $2n$-dimensional topological manifold $X$ together with a complex atlas which is a collection of compatible charts $\varphi_{\alpha} : U_{\alpha} \to B(0, 1) ...
1
vote
0answers
185 views

Top current research topics in Complex Algebraic and Differential Geometry

I am currently a first year PhD candidate and I need some help figuring out the top current research topics in Complex Algebraic and Differential Geometry (can be in both, or just one of the two). I ...
1
vote
1answer
138 views

Tangent bundle of P^n and Euler exact sequence

I'm thinking about the Euler exact sequence for complex projective space, and I'm a bit confused. In the topological category, one has $$ T\mathbb{P}^n \oplus \mathbb{C} \cong L^{n+1} $$ where $L$ ...
0
votes
1answer
20 views

First Chern Class of divisors on compact Riemann surfaces

let $X$ be a compact Riemann surface and $D$ a divisor on $X$. I'm looking for a argument for the statement $c_1(\mathcal{O}_X(D)) = \deg(D)$, where $\mathcal{O}_X(D)$ is the associated line bundle to ...
6
votes
0answers
168 views

Exercise $3.1.7$ from Huybrechts' Complex Geometry: An Introduction

I am trying to do the following question: Show that $L$, $d$, and $d^*$ acting on $\mathcal{A}^*(X)$ of a Kähler manifold $X$ determine the complex structure of $X$. Here $L$ is the Lefschetz ...
2
votes
0answers
45 views

On a method to compute dimension of moduli space of Riemann surfaces

Consider genus $g$ Riemann surfaces, and thier moduli space $\mathcal M_g$. To determine dimension of $T\mathcal M_g$, start with a complex structure, which in some coordinates can be written ...
1
vote
1answer
41 views

Meromorphic function with bounded order of zeros and poles

The following problem has been bothering me for a long time; Let $X$ be a compact Riemann surface of genus $g$. Is there a non-zero meromorphic function on $X$ with all of its poles and zeros have ...
1
vote
0answers
42 views

Cartier divisors of schemes

In his notes, Ravi Vakil only defines the notion of an effective Cartier divisor. Furthermore, the Wikipedia page only defines the notion of an effective Cartier divisor for a general scheme. ...
1
vote
0answers
21 views

What does the equation $\tau \tau^* = \sigma^* \sigma$ represent in the ADHM construction of vector bundles?

I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction. At the heart of this is the complex $$ V ...
2
votes
1answer
52 views

Complex structures on punctured disks.

Let $X$ be a smooth surface diffeomorphic to the punctured unit disk $\{(x,y)\in \mathbb{R}^2 \ | \ 0<x^2+y^2<1\}$ in the plane. It admits a lot of non equivalent complex structures, for example ...
4
votes
1answer
72 views

Almost complex structure which fails to be compatible

Let $V$ be a real vector space equipped with a scalar product $\langle, \rangle$ (i.e. a positive definite symmetric bilinear form). We say that an endomorphism $J: V \to V$ is an almost complex ...
2
votes
2answers
130 views

Are there many almost complex structures on a (complex) manifold?

I guess one can have many almost complex structures on a manifold, can someone give me an example? How about when the manifold is complex? is the almost complex structure induced by the complex ...
2
votes
1answer
47 views

Giving Holomorphic structure to Complex line bundles on projective space and torus

I am doing questions from [Huybrechts, Complex Geometry, An Introduction] page 143 questions 3.3.7 and 3.3.8. Basically the questions ask: For question 7, for any complex line bundles on the ...
0
votes
0answers
30 views

Two generating meromorphic functions seperate points on a compact Riemann surface?

Problem Suppose $z,f$ are two meromorphic functions on a compact Riemann surface $M$, whose meromorphic function field is $\mathbb C(M)=\mathbb C(z,f)$, where $\mathbb C(M)$ is a finite extension of ...
3
votes
1answer
98 views

Can one give me some concrete examples explaining Picard's Great Theorem

Picard's Great Theorem Every non-constant entire function attains every complex value with at most one exception. Furthermore, every analytic function assumes every complex value, with possibly one ...
1
vote
0answers
35 views

Degrees of spaces of polynomials

Let $I$ be an ideal in $K[x_1,\dots,x_n]$ where $K$ is a char $0$ field. Let $Z(I)$ be a set of discrete points whose cardinality is exponential in $n$ and spanning $n$ dimensions. Let $P$ be the ...
12
votes
1answer
201 views

Proof of holomorphic Lefschetz fixed point formula using currents in Griffiths and Harris

I am trying to understand the proof of the Holomorphic Lefschetz fixed point formula on page 426 in Griffiths and Harris. However, I find their use of currents extremely confusing. They seem to go ...
5
votes
1answer
416 views

Proper mapping theorem

My professor mentioned a proper mapping theorem after the name of Remmert which says: Let $X$ and $Y$ be complex manifolds, $f:X \to Y$ be a proper holomorphic map, and $V \subset X$ be a complex ...
1
vote
1answer
286 views

The image of a map $H^2 \setminus \Delta \to \mathbf{R}^4$ where $H$ is the upper half-plane and $\Delta$ is the diagonal

Let $H = \{z \in \mathbf{C}: \operatorname{Im} z > 0\}$ be the upper half-plane, and let $D = H^2 \setminus \Delta$, where $\Delta = \{(u,u) \in H^2\}$ is the diagonal. Define $\varphi: D \to ...
2
votes
1answer
286 views

Differential forms and a chain rule

Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$. Let $Q\in U$ ...
1
vote
0answers
28 views

Homogeneous polynomials and line bundles

In Huybrechts' Complex Geometry text, he makes the following claim: (Claim) "Any homogeneous polynomial $s \in \mathbb{C}[z_o,\dots, z_n]_k$ of degree $k$ defines a linear map ...
3
votes
1answer
35 views

Transcendence Degree of the Function field of $\mathbb{C}$

What is the transcendence degree of the field of meromorphic functions over $\mathbb{C}$? By a cardinality argument (meromorphic functions are determined by their image under a countable dense subset ...
2
votes
1answer
42 views

Basis for a line bundle

I have a basic confusion. The following is drawn from Huybrechts' Complex Geometry text: "Let $L$ be a holomorphic line bundle on a complex manifold $X$ and suppose that $s_0,\dots, s_n \in H^0(X, ...
1
vote
0answers
41 views

Undergraduate Complex Analysis: Use of Rouche's Theorem

We are asked to prove $ f = z^{3}e^{1-z} = 1 $ has exactly 2 roots inside $|z| = 1$ We've tried creating functions $p$ and $q$ where $p + q = f$, $p$ with 2 roots inside our boundary, and using ...
3
votes
0answers
35 views

explicit (holomorphic) map revealing blow-up as a connected sum with $\overline{\mathbb{CP}}^n$

I am trying to understand the statement that a blow-up of a complex manifold $M$ at a point $p$ is equivalent to the connected sum of $M$ with $\overline{\mathbb{CP^n}}$ and, being a physicist, I need ...
13
votes
4answers
540 views

Where can I learn about complex differential forms?

So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd ...
3
votes
0answers
33 views

Why is this function on the Riemann surface holomorphic?

Forster defines analytic continuation of a germ of a holomorphic function at a point on a Riemann surface as follows. Suppose $ X$ is a Riemann surface, $a\in X$ and $\phi\in\mathcal{O}_a$ is a ...
3
votes
2answers
52 views

Hermitian manifold counterexample

I'm trying to come to come to grips with the notion of a hermitian manifold. Although I know some examples of hermitian manifolds, I am more interested in counterexamples: naturally occurring ...
0
votes
1answer
20 views

Any real analytic Frobenius theorem used in the proof of integrable almost complex manifolds to arise from complex manifolds?

The reference book is S.S.Chern's Complex Manifolds Without Potential Theory. It's used in integrability condition for an almost complex structure to arise from a complex structure. It's claimed that ...
0
votes
1answer
22 views

In the Riemann sphere 1 is not summe of holomorphics map vanishing on 0 and $\infty$

I want to prove (if it's right) that in the Riemann sphere one can not write the constant function 1 as a summe of two holomorphics map, one vanishing in 0 and one vanishing in $\infty$.
3
votes
0answers
31 views

Holomorphic vector fields on complex projective spaces

Question: is there a simple description of holomorphic vector fields on the complex projective space $\mathbb{C P}^n$ ? More precisely : for $n=1$, the holomorphic vector fields are of the form $P(z) ...
2
votes
0answers
41 views

Why is $H^1(X, \mathcal{O}) \neq 0$ for $X = (B(0, 1)\times B(0, 2)) \cup (B(0, 2)\times B(0, 1))$?

In this MathOverflow answer, David Speyer says that \begin{align*} X &= \{ (z,w) \in \mathbb{C}^2 : (|z|, |w|) \in [0,1) \times [0,2) \cup [0,2) \times [0,1) \}\\ &= (B(0, 1)\times B(0, 2)) ...
0
votes
0answers
13 views

antiholomorphic involution action on resolved orbifold of tori

Consider the space $\tilde X= T^2 \times T^2 \times T^2 / G$ where $T^2$ is a torus and $G=Z_2 \times Z_2$ acts as $\theta_1:(z_1,z_2) \mapsto (-z_1,-z_2)$, $\theta_2:(z_2,z_3) \mapsto (-z_2,-z_3)$ ...
1
vote
1answer
20 views

Unitary transformation of Fubini-Study metric

I am trying to solve a problem in Introduction to Complex Geometry by D. Huybrechts, question 3.1.6 which is the following: let $A\in GL(n+1, \mathbb{C})$ be a $\mathbb{C}$-linear transformation ...
2
votes
1answer
40 views

Simpleminded example: flasque sheaves

Consider the sheaf $\mathcal{F}$ of polynomial functions on $\mathbb{R}^2$ endowed with the usual topology. A sheaf is said to be "flasque" (or "flabby") if, given $V \subset U$ both open sets, the ...
3
votes
2answers
36 views

Working in complex number field

I have to draw the graphic of "group of points given by the equation $$(|z^2|-3|z|+2)(z^4+4)=0$$ I solved the first part by factoring and obtaining $|z|=1$ and $|z|=2$ so in the graphic I have the ...
1
vote
0answers
36 views

deformation space inside cohomology

For which smooth projective varieties $X$ is $H^1(X,T_X)$ (canonically ) contained in $H^\cdot(X,\mathbb C)$? If $K_X$ is trivial this is true. But are there other type of varieties?
4
votes
2answers
286 views

Equivalent definitions of Kähler metric

Two different ways to define a Kähler metric on a complex manifold are: 1) The fundamental form $\omega = g(J\cdot,\cdot)$ is closed, ie, $d\omega=0$; 2)The complex structure $J$ is parallel with ...
0
votes
1answer
37 views

Dual Lefschetz Operator and Contraction with the Fundamental Form

Let $M$ be a Kahler manifold, with metric $g$, Kahler form $\omega$, Lefschetz operator $L$, and dual Lefschetz operator $\Lambda$. $\Lambda$ and contraction with $\omega$ both map $k$-forms to ...
2
votes
0answers
57 views

Why is this map of sheaves surjective?

Let $M$ be a complex manifold. Let $\mathcal{L}$ be a holomorphic line bundle over $M$, with $\pi:M\longrightarrow\mathcal{L}$ being the projection map. Any section $s$ of $\mathcal{L}$ over $M$ is a ...
2
votes
1answer
230 views

Is there a geometric projection for every complex function?

I was wondering about the best way to visualize complex functions. As they're $$ {\mathbb R}^2 \rightarrow {\mathbb R}^2\ ,$$ I think best way are complex plane image/grid transforms like they used in ...
1
vote
1answer
2k views

Cross product in complex vector spaces

When inner product is defined in complex vector space, conjugation is performed on one of the vectors. What about is the cross product of two complex 3D vectors? I suppose that one possible ...
1
vote
0answers
65 views

first chern class

If $M$ is a Fano manifold, and $K_M$ is the canonical line bundle of $M$. If $L$ is an ample line bundle over $M$, and $c_1(L)=\lambda c_1(M)$, for some positive number $\lambda$. What is the relation ...
1
vote
0answers
39 views

On the Chern connection

It is well kown that If $\,\bar\partial_E$ defines a holomorphic structure on the complex vector bundle $E \to X$ and $K$ is a hermitian metric on (the fibres of) $E$, there is a unique connection ...
1
vote
0answers
48 views

on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, semi-simple); call $T$ its maximal torus, its Lie algebra $\operatorname{Lie}(G)=\mathbf g$, its Cartan subalgebra ...
2
votes
1answer
47 views

Rational functions on varieties

To give a rational function $f$ in the function field $K(X)$ of a smooth projective curve $X$ is equivalent to giving a finite morphism $f:X\to \mathbb P^1$. What is the precise analogue of this ...
0
votes
2answers
39 views

Real part of a complex number divided: $\Re\frac{z+1}{z-1}=0$

$\Re\frac{z+1}{z-1}=0$ I've tried so many methods, they all end up with two variables $a, b$. I tried setting $z= a+ib$. This give me the equality $2\cdot\Re\frac{z+1}{z-1} = ...
7
votes
2answers
199 views

Why should automorphism groups of compact hyperbolic curves be finite

Let $X$ be a compact connected Riemann surface of genus at least two, or let $X$ be a smooth projective connected curve over an algebraically closed field of characateristic zero. Then Hurwitz proved ...
0
votes
1answer
12 views

Disconnecting a complex vector space

Can a (complex) dimension $n$ subspace disconnect a (complex) dimension $n+1$ vector space ? If the answer is no, what if we replace "vector space" by "manifold" ?
2
votes
0answers
29 views

Is log-general type an intrinsic property of a variety

Let $X$ be a smooth quasi-projective variety over $\mathbb C$. Let us say that $X$ is of log-general type if for some choice of smooth compactification $\bar X$ with normal crossings boundary divisor ...