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

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787 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
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217 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 ...
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145 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 ...
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248 views

Another Algebraic de Rham Cohomology question…

NOTE: scroll down to read my latest edit first if you're reading this for the first time :) My aim is to calculate the de Rham cohomology of the variety $U = \text{Spec} \ A$, where: $$A = ...
6
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202 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 ...
6
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105 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
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66 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 ...
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44 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 ...
6
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327 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 ...
5
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319 views

Higher dimensional analogue for Riemann Hurwitz formula

There are few questions like here and here already asked about this. But I don't have the background to understand the answers there. I am just beginning to learn classical algebraic geometry, and ...
5
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52 views

Is there a natural ring structure on $\operatorname{Pic}(\mathbb{CP}^1)$?

The set of isomorphism classes of holomorphic line bundles on a complex manifold $X$ is a group under tensor product. This group is called the Picard group and is denoted $\operatorname{Pic}(X)$. We ...
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107 views

Not taking holomorphic bundles for granted

When we say something to the effect of, "Consider a holomorphic bundle $V$ on a complex manifold $X$...", we are saying that the transition functions of $V$ are holomorphic functions with respect to ...
5
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76 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
129 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
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0answers
70 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 ...
4
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0answers
101 views

Analytic variety is a countable union of complex manifolds

In an article on real analytic manifolds I came across the following remark: Let $W$ be a purely $k$-dimensional analytic subvariety of a domain in $\mathbb{C}^n$ and let $S$ be its singular locus. ...
4
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45 views

About the Chern class of infinite complex Grassmannian

I learned that any characteristic class of rank-$k$ complex vector bundles on paracompact spaces is determined bijectively by a cohomology class in $H^*(Gr_k^\infty(\mathbb C))$, the cohomology ring ...
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101 views

Complex structures on Riemann surfaces

Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq H^0(M;K^2)$. Considering $\alpha$ as a map $T^{0,1} M \to T^{1,0} M$, the bundle $$ \{v + \alpha(v) \mid v \in T^{0,1} M\} ...
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64 views

Is there an algebraic description of the ring of analytic functions on the real projective line?

Apologies for the long question. Let $X=\mathbf P^1(\mathbf R) \subseteq \mathbf P^1(\mathbf C)$ be the real projective line. Let $\mathcal O_X$ be the sheaf of real-analytic complex-valued functions ...
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0answers
54 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
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87 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 ...
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129 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 ...
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0answers
132 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 ...
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76 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 ...
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44 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
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0answers
163 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
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387 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 ...
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260 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. ...
4
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0answers
210 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 $\mathbb{R}_{+}$?
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68 views

Correspondence between line bundles and $U(1)$-bundles: a mistake from the physicists?

I am reading a paper written by physicists and they say the following: Let $(L,h,\nabla)$ be an holomorphic line bundle equipped with a Hermitian metric $h$ and Chern connection $\nabla$. If ...
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31 views

Complete intersections with respect to different sections

Let us consider the Whitney sum $E$ of holomorphic line bundles $L_1,\dots, L_k$ on a smooth (projective) variety $X$. For a generic global section $s$ of $E$, the zero locus $Z_s := s^{-1}(0_E)$ ...
3
votes
0answers
72 views

defining the canonical divisor

I am just starting to learn some basic algebraic geometry, but I am very confused with some definitions. So I hope I am not asking something that is completely trivial. Suppose $X$ is a normal ...
3
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0answers
50 views

“Pulling back a function from a neighborhood of 0 in $\mathbb{C}^2$ to $\mathbb{U}$ is analogous to computing the derivative of that function.”

I'm reading on Riemann Surfaces and after defining $$\mathbb{U} \equiv \{(z,\ell)\in\mathbb{C}^2\times\mathbb{P}_1;z\in\ell\},$$ and $B\ell_0:\mathbb{U}\to\mathbb{C}^2$, the author mentions that, ...
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0answers
47 views

Cohomology vanishing on projective manifold, want to show that a line bundle $L$ is ample

I have some questions regarding the proof of the following theorem. Let $X$ be a projective manifold and $L$ a line bundle on $X$. Then $L$ is ample if and only if for all coherent sheaves ...
3
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0answers
40 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) ...
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0answers
68 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 ...
3
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0answers
42 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
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43 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) ...
3
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34 views

Holomorphic $K$-theory

$K$-theory is traditionally defined for arbitrary compact Hausdorff spaces. If instead we require the base to be a complex manifold and work with only holomorphic vector bundles, in what ways would ...
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0answers
38 views

Gaussian curvature of a complex projective curve

Let $X \subset \mathbb CP^2$ be a complex curve inheriting metric from $\mathbb CP^2$. Suppose that locally $X$ is given by a holomorphic map $z \to [h_1(z) \colon h_2(z) \colon h_3(z)]$. What is the ...
3
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0answers
84 views

Example for divisors, line bundles and meromorphic functions on $\mathbb{CP}^2$

I have been studying divisors using Griffiths/Harris (chapter 1.1) as well as Huybrechts (chapter 2.3). However, I cannot seem to find any very easy worked examples - i.e. $\mathbb{CP}^1$ or ...
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43 views

Natural action on Hilbert Schemes

I'm reading chapter 4 of Nakajima's Lectures on Hilbert Schemes of Points on Surfaces and I'm a little confused. He says the natural action of $\Gamma$ on $\mathbb{C}^2$ gives a natural action on the ...
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56 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 ...
3
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0answers
80 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 ...
3
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0answers
107 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
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0answers
46 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
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0answers
153 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
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0answers
76 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
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0answers
53 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
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0answers
44 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 ...