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

8
votes
0answers
409 views

Arithmetic and geometric genus

There are two notion of genus in algebraic geometry, namely arithmetic genus $p_a=(-1)^{\dim X}(\chi(\mathcal{O}_X)-1)$ and geometric genus $p_g=\dim H^0(X,\Omega^{\dim X})$. I keep forgetting ...
8
votes
0answers
165 views

Interesting applications of Kähler Identities

The Kähler identities give commutation relations between the Lefschetz operator ($\alpha \mapsto \omega \wedge \alpha$), the differential operators $\partial, \bar{\partial}$ and its adjoints ...
7
votes
0answers
110 views

Why do algebraic varieties contain curves passing through two given points

Let $X$ be an algebraic variety over the complex numbers. My definition of an algebraic variety is a finite type separated $\mathbf C$-scheme. Someone told me that such varieties have the following ...
6
votes
0answers
80 views

Hilbert's syzygy theorem in the analytic setting

If $X$ is a projective variety then Hilbert's syzygy theorem says that any coherent sheaf of $\mathcal O_X$ modules has a finite resolution by locally free modules. By GAGA, I believe this should ...
6
votes
0answers
39 views

Chern classes of free quotient manoflds

Let $X$ be a compact complex manifold. Assume that a finite group acts on $X$ freely. Then the quotient $X/G$ is again a compact complex manifold. I wonder if there is a good way to compute Chern ...
5
votes
0answers
63 views
+250

Formula for decomposing a form into $(p,q)$ forms

Let $L: \mathbb{C}^n \to \mathbb{C}$ be a real linear map. In other words, $L(a\vec{v}_1+b\vec{v_2}) = aL(\vec{v}_1)+bL(\vec{v}_2)$ for all $a,b \in \mathbb{R}$. Then $L$ decomposes uniquely into a ...
5
votes
0answers
56 views

lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
5
votes
0answers
124 views

Finding two curves with intersection included in a specified neighborhood

Let $V$ be a complex projective surface and $U$ an analytic neighborhood of $x\in V$. How can I prove that there are two smooth curves $C_1$ and $C_2$ s.t. $C_1\cap C_2\subset U$ ?
5
votes
0answers
57 views

Automorphism on a kahlerian variety which acts as -id on $H^2(X,\mathbb{Z})$

I've read this proposition: "there is no automorphism (biholomorphic map) of a Kahlerian complex manifold $X$ which acts on $H^2(X,\mathbb{Z})$ as $-$identity" so I tried to prove this statement and ...
5
votes
0answers
54 views

Branch locus along a smooth curve

Let $f : S \to X$ be a dominant morphism of smooth complex surfaces. Let $C \subset S$ be a smooth curve such that $df$ is of rank $1$ along $C$ and that in a neighborhood of $C$, $df_x$ is an ...
5
votes
0answers
122 views

Which is the correct universal line bundle: the tautological bundle or its dual?

With topological line bundles over $\mathbb{C}$, one learns that every line bundle is a pullback of the universal line bundle, which is the tautological line bundle over $\mathbb{C}P^\infty.$ In ...
5
votes
0answers
260 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 ...
5
votes
0answers
243 views

understanding this differential operator on a tensor product

I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
4
votes
0answers
31 views

2-forms represented by a first Chern class?

Let $M$ be a complex manifold and $\omega$ be a 2-form on $M$. Is there a good way to see whether $\omega$ is represented by the first Chern class of a line bundle on $M$? In other words, when is it ...
4
votes
0answers
71 views

Why $H^{1,1}(X,\mathbb{C}) = Pic(X) \otimes \mathbb{C}$ for Calabi-Yau 3-folds?

Let $X$ be a Calabi-Yau 3-fold, that is $\omega_X = 0$ and $h^{1,0}=h^{2,0}=0$. Let $\operatorname{Pic}(X)$ be the group of line bundles on $X$. Then why the following isomorphism is true ...
4
votes
0answers
36 views

Continuous complex functions.

We are given with a map $g:\bar D\to \Bbb C $, which is continuous on $\bar D$ and analytic on $D$. Where $D$ is a bounded domain and $\bar D=D\cup\partial D$. Then $\partial(g(D))\subseteq g(\partial ...
4
votes
0answers
100 views

Fundamental group of a space under a group action

The short version: Why does a Borcea-Voisin threefold has trivial fundamental group? Well, what is a Borcea-Voisin threefold? These are named after Borcea and Voisin, who introduced a method of ...
4
votes
0answers
59 views

Negative self intersection and section of the conormal sheaf for a singular complex curve

Let $M$ be a complex $2$-dimensional manifold and $C$ be a compact complex curve in $M$ (possibly singular). Let us suppose that there exists a holomorphic function $f\in\mathcal{O}(M)$ such that ...
4
votes
0answers
41 views

Properties of divisors when moving from char 0 to char p.

Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
4
votes
0answers
105 views

Is there a relation between Super Riemannian manifolds and Kähler manifolds?

(This question has a physics motivation). Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kähler manifolds, or at ...
4
votes
0answers
93 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 ...
4
votes
0answers
123 views

Is $\bar{\partial}_E + \bar{\partial}_E^*$ a Dirac operator?

I have previously asked about Weitzenböck identities and received some great answers on MathOverflow. One question which has arisen from the post is the following: Let $E$ be a hermitian ...
4
votes
0answers
252 views

Why is an analytic variety irreducible provided that the set of its smooth points is connected?

Here is a proposition from Griffith and Harris, Principles of Algebraic Geometry. Let $V$ be an analytic variety on a complex manifold $M$, let $V^*$ be its points of smooth points. If $V^*$ is ...
4
votes
0answers
196 views

Hirzebruch surfaces

How can I express the 2nd Hirzebruch surface, $F_{2}$ in terms of $SO(3)$. Is it true that $F_{2}$ is the total space of a bundle with fibre SO(3) over $\mathbb{R}_{+}$?
3
votes
0answers
84 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 ...
3
votes
0answers
67 views

Cotangent bundle of complex manifold is Calabi-Yau manifold

We say that a complex manifold $M$ is Calabi-Yau if the canonical bunlde is trivial $K_M=0$. How can we prove that the total space of the cotangent bundle of a compact complex manifold $N$ is ...
3
votes
0answers
65 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 ...
3
votes
0answers
77 views

Reference about Moduli Spaces of Riemann surfaces

I'm looking for some (introductory, and in any case not too technical) reference (book, lecture notes, papers) regarding moduli spaces $\mathcal{M}_g$ and $\mathcal{M}_{g,n}$ of (punctured) Riemann ...
3
votes
0answers
68 views

Explicitly realizing Riemann surfaces as a quotient of the upper-half plane

Let $\Sigma_g$ be a Riemann surface of genus $g \ge 2$. Then it is known that $\Sigma_g$ is (holomorphically) a quotient of the upper-half-plane (or unit disk) by a group $\Gamma$ of hyperbolic ...
3
votes
0answers
34 views

Linear Complex Structure and Kähler Angles

I am trying to read Donaldson's paper on symplectic submanifolds http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214459407 and am getting a bit ...
3
votes
0answers
98 views

Every holomorphic map between Kähler manifolds is harmonic

I was reading the Wikipedia article on harmonic maps and saw the following statement in the 'examples' section: Every holomorphic map between Kähler manifolds is harmonic. I am not that familiar ...
3
votes
0answers
71 views

How should I imagine the cup product on a topological space/manifold/variety

Let $X$ be a "space" with a vector bundle $E$. Let $n$ be the dimension of $X$. Then there is a map $$H^1(X,E)\otimes \ldots\otimes H^1(X,E)\to H^n(X,\Lambda^n E).$$ This map is given by taken the ...
3
votes
0answers
46 views

Maps between total spaces of holomorphic vector bundles

I am wondering what is possible and what is not possible regarding maps between the total spaces of holomorphic vector bundles. Let me outline a situation that is a bit more concrete, to help focus ...
3
votes
0answers
40 views

$SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$

I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and ...
3
votes
0answers
96 views

Continuing a Divisor

I need some help, to find the strategy for solving the following problem: Given an analytic surface $S^0$, its compacitfication $S$ and a horizontal Divisor $D^0$ on $S^0$, I have to continue $D^0$ to ...
3
votes
0answers
44 views

Holomorphic analogue of geodesics

Let $X$ be a complex manifold with a Hermitian metric. Is there a "complex" analogue of geodesics on $X$ which is of any interest? For example, is anything known about holomorphic maps $f : \mathbb C ...
3
votes
0answers
175 views

Anti-holomorphic involution of $\mathbb{P}^1$

I wonder if anti-holomorphic involution of $\mathbb{P}^1$ is, up to change of coordinate, given by either $$ z\mapsto \overline{z}, \ \ \,z\mapsto -\overline{z}, \ \ \ or \ \ \ z\mapsto ...
3
votes
0answers
100 views

Length of $\frac{\partial }{\partial z}$ in Kaehler geometry.

I am taking a Kaehler geometry course this semester. The book we use is Tian's Canonical Metrics in Kaehler Geometry. I got a little confused about the calculation there in. For example, ...
3
votes
0answers
175 views

Why the moduli space of complex structure in a compact complex manifold is of finite dimension

I admit the statement in the title might be much too unclear. I just heard from my teacher that we can form a finite dimension moduli space of all the complex structure in a compact complex manifold. ...
2
votes
0answers
19 views

Automorphisms of Complex Projective Space

Each automorphism $\mathbb{C}P^n\rightarrow \mathbb{C}P^n$ (where $\mathbb{C}P^n$ is regarded as a complex manifold) is induced by a linear map $\mathbb{C}^{n+1}\rightarrow \mathbb{C}^{n+1}$. I know ...
2
votes
0answers
26 views

some ring theory applied to holomorphic functions

I'd like to know if my understanding of this business is correct. Let $U \subset \mathbb{C}^n$ be open and connected. The set $\mathcal{K}(U)$ of meromor­phic functions on $U$ is a field. Is it true ...
2
votes
0answers
19 views

relation between structure group of holomorphic ,antiholomorphic, complex tangent bundle

I Know that a complex tangent bundle $T_{\mathbb{C}}M$ can be written as direct sum of holomorphic and anti- holomorphic tangent bundles $i. e.$ $T_{\mathbb{C}}M = T_{\mathbb{C}}'M \oplus ...
2
votes
0answers
54 views

All almost complex structures on a manifold

I read the statement of the Newlander-Nirenberg theorem, which says that "any integrable almost complex structure is induced by a complex structure". To make sense of the statement, I was wondering ...
2
votes
0answers
42 views

SU(2) Lefschetz decomposition of Jacobian of Riemann surfaces

Start with a (closed and oriented) Riemann surface with $g$ handles $\Sigma_g.$ I'm interested in (the cohomology of) its Jacobian $Jac(\Sigma_g)=T^{2g},$ in particular how the $SU(2)$ or ...
2
votes
0answers
23 views

Comparing the norm of a trace of a curvature tensor with the full norm

Let $V$ and $E$ be complex vector spaces of dimensions $n$ and $r$, equipped with hermitian inner products $\omega$ and $h$ respectively. Let $R$ be a curvature-type tensor, that is an element of ...
2
votes
0answers
78 views

Miranda book Proposition 3.10 pag. 40

Let $F$ and $G$ be two holomorphic maps between Riemann surfaces $X$ and $Y$. If $F=G$ on a subset $S$ of $X$ with a limit point in $X$, then $F=G$. Choose two charts $(U_\alpha,\phi_\alpha)$ and ...
2
votes
0answers
43 views

geometry of Automorphism groups

It is proved that Aut group of a compact complex manifold is a lie transformation group. How do we show that a automorphism group of a tubular geometry is a lie group?
2
votes
0answers
62 views

If the intersection of the preimage of a generic point of a flat morphism with an open set is dense, does it imply the open set is dense?

Let $f: E \rightarrow M $ be a flat morphism of varieties (over $\mathbb{C}$) and $E^{\prime}$ an open subset of $E$. Assume that both $E$ and $E^{\prime}$ have pure dimension $2n$, where $n$ ...
2
votes
0answers
70 views

Can every complex space be covered by a finite number of Stein spaces?

Can every complex space be covered by a finite number of Stein spaces?
2
votes
0answers
77 views

Product of two Kähler manifolds

Let $M$ and $N$ be Kähler manifolds. I know that the cartesian product $M\times N$ is a kahler manifold. I was wondering which form has the Kähler form on $M\times N$. Here's what I thought: Let ...