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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An almost complex structure on real 2-dimensional manifold

Why an almost complex structure on real 2-dimensional manifold is integrable?
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1answer
186 views

Why $S^1\times S^{2m-1}$ carries a complex structure.

Let $S^n$ denotes $n$-sphere, then why $S^1\times S^{2m-1}$ carries a complex structure.
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53 views

Submanifold of a Kobayashi hyperbolic manifold

Let $M$ be complex manifold which is Kobayashi hyperbolic. Let $N$ be a submanifold of $M$ obtained as the zeroes of an analytic submersion $f : M \rightarrow R$, $R$ complex manifold. Question : ...
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46 views

moduli of lattices

Consider the set $M$ of all (rank $g$) lattices in $g$-dimensional complex affine space $C^g$. Does M identify in some way with Siegel upper half space $H_g$? Let's say a lattice has CM if it has ...
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41 views

Kahler condition related to Ricci curvature formula of a Hermitian, holomorphic vector bundle over a complex manifold

I read a local formula like this: Under some sort of Kahler condition, $$Ric(h)=-i\partial\bar\partial \log \det(h_{\alpha\bar\beta})$$ where $h_{\alpha\bar\beta}$ is the matrix of the Hermitian ...
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1answer
136 views

Application of Riemann Roch

I have read that thanks to Riemann Roch theorem, if get $\Sigma$ a compact Riemann Surface of genus $g$ there exists a conformal branch covering $\phi: \Sigma \rightarrow S^2$ of degree less than $g+1$...
8
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1answer
107 views

Zariski vs analytic cohomology of $\mathcal O_X^\times$

Let $X/\mathbf C$ be a smooth proper variety. Is it true that $H^1(X, \mathcal O_X^\times) = H^1(X^{an}, \mathcal O_{X^{an}}^\times)?$ GAGA doesn't apply, because $\mathcal O_X^\times$ is not coherent ...
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51 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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1answer
101 views

Exact sequence of sheaves of holomorphic functions

This is from Exercise 2.4.P. June 2013 version of Ravi Vakil's Math 216 notes. The idea is to show $\mathscr{O}_X \xrightarrow{\text{exp}} \mathscr{O}^*_X$ is an epimorphism. It seems ...
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100 views

Difference between isomorphisms, deformations, diffeomorphisms and homeomorphis,s

I am learning about complex algebraic varieties and have encountered several equivalence relations: isomorphism, birational, deformation equivalent, diffeomorphic and homeomorphic. I consider all of ...
3
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1answer
228 views

Learning roadmap for classical algebraic geometry (italian school)

Can someone suggest a learning roadmap for classical algebraic geometry as develop by the great italian school? Severi has a few books but in italian. I would like to know what are the best English ...
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1answer
118 views

The pullback of a nontrivial line bundle is nontrivial?

Let $X$ be a complex manifold and $E$ a holomorphic vector bundle over $X$ of rank $2$. Let $\mathbb{P}(E)$ denote the projectivization of $E$, with the natural map $p: \mathbb{P}(E) \rightarrow X$. ...
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1answer
241 views

pullback is injective on picard groups?

Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map ...
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1answer
43 views

Do we need a metric to define plurisubharmonic functions?

There are various notions of 'harmonicity' on various manifold. Sometimes, I am counfuesed by the definitions. For real manifold, the harmonic manifold is defined by $\Delta f=0$, where $\Delta$ is ...
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264 views

Univalent functions whitch are not starlike or convex

$S$ denotes the univalent function class, $S=\{f \in \mathcal{A}:f \in H_{u}(\mathrm{U})\}$. $S^{\ast}=\left\{f\in\mathcal{A}:\operatorname{Re}\frac{zf^{\prime}(z)}{f(z)}>0, \;z\in\mathrm{U}\right\}...
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60 views

classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with Fubini-Study metric, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\overline w, \overline z] \qquad \eta :[z:w]\...
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105 views

Ample tangent bundle

I am looking for definition of ample tangent bundle or positive tangent bundle and why on a complex manifold , positive bisectional curvature means ample tangent bundle?
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269 views

Explicit Kähler forms and Kähler cone of one-point blowup of $\mathbb{CP}^2$

I am interested in understanding the Kähler cone of the one-point blowup of $\mathbb{CP}^2$, also known as the first Hirzebruch surface. Let's call this manifold $\Sigma_1$, and call its Kähler cone $\...
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61 views

What will be the integration region?

Where $\Omega_s$ is new integration region, due to change in geometry integration region will also change. Also note that $\Omega_l$ is $\Omega$ with $|x|<l$ is the integration region for $\...
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89 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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1answer
124 views

Why is the projection map proper?

It might be a silly question. I got stuck there. For the context, see S.K.Donaldson's Riemann Surfaces, chapter 4, section 4.2.3. Suppose $P(z,w)=a_0(z)+a_1(z)w+\dotsb+a_n(z)w^n\in\mathbb C[z,w]$ is ...
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226 views

How many differential forms on the complex plane?

I am puzzled by the fact that the two differential forms $$\begin{array}{cc} dz=dx+i dy , & d\overline{z}=dx-i dy \end{array} $$ are $\mathbb{C}$-linearly independent, even if the underlying ...
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17 views

Begin study of convex algebraic sets in complex projective space

Where should I begin the study of convexity of (semi-)algebraic sets? In other words, projective varieties defined by polynomials of complex variables. The long-term goal is to study optimization in ...
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50 views

Parametric ideal decomposition

Let $x = \{x_{1},\dots, x_{n}\}$ be a set of variables and let $a = \{ a_{1}, \dots, a_{m}\}$ be a set of parameters. Let $\{f_{1}(a,x), \dots, f_{s}(a,x)\} \subset \mathbb{C}[a,x]$ be a set of ...
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1answer
85 views

Equivalent Definition of a Hermitian Metric on an Almost Complex Manifold?

For an almost-complex manifold $M$ with almost-complex structure $J$, we say that a metric $h:T_p(M;{\mathbb R}) \times T_p(M;{\mathbb R}) \to {\mathbb R}$ is Hermitian if it holds that $$ h(J(v),J(w)...
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1answer
206 views

Chern classes via connections

Let $M$ be a smooth real manifold and $B$ an Hermitian vector bundle over it. Then one can define Chern classes as $$c(B)=\sum c_i(B)t^i=\det \left( I+\frac{it\Omega}{2\pi} \right) \in H_{DR}^*(M),...
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1answer
114 views

Is every analytic hypersurface in $\mathbb{C}^n$ cut out by one holomorphic function?

Is every analytic hypersurface in $\mathbb{C}^n$ cut out by one holomorphic function?
3
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1answer
97 views

Complex structure on $\mathbb{C}\mathbb{P}^2\# \dots \# \mathbb{C}\mathbb{P}^2$

I know the chern classes-related theorem that states that $\mathbb{C}\mathbb{P}^2\# \dots \# \mathbb{C}\mathbb{P}^2$ ($k$ times) has no almost complex structure (hence no complex structure) if and ...
5
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1answer
174 views

Closed subschemes of projective space

We work over the complex numbers. If $X$ is an integral closed subscheme of $\mathbb P^n$, then there exists a homogeneous ideal $I$ of $A=\mathbb C[x_0,\ldots,x_n]$ such that $X = V_+(I) =$ Proj$(A/...
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2answers
464 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 $L$ ...
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128 views

Cauchy's Theorem by Differential Geometry

Is there a prove of Cauchy's theorem footing on the topology of the complex plane (homotopy, differential forms, etc.)? More specific consider a differentiable Banach space valued complex function. ...
2
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1answer
53 views

Numerical equivalence of divisors on fibered surface

Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $g > 2$ and $\pi_i\colon C\times C \to C$ the projection onto the the $i$-th factor. Let $f_i \in \mathrm{Num}(C\times C)$ be the ...
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3answers
113 views

A question on differential topology

Let $\mathbb{C}P(1)$ denote the complex projective line. I am attempting to show that there does not exist a nonzero holomorphic differential $1$-form on $\mathbb{C}P(1)$. My intuition is as follows:...
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Kahler identities on symplectic manifold

In Hutchings and Taubes's lecture note on Seiberg-Witten equation, a Weitzenbock formula is given, and the authors states that the Weitzenbock formula, proven in Donaldson's book using Kahler ...
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1answer
76 views

A complex manifold isn't a sympletic manifold

I want to think about an example of a complex manifold which isn't a sympletic manifold. I consider it in this way: $X=\mathbb{C}^2-\{0\}$, a group $\mathbb{Z}$ acts on X by $(n,z)=2^nz$, then I think ...
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1answer
110 views

Relation between Kähler potential and Hermitian metric

Let $(M,\omega)$ be a Kähler manifold and $h$ be its Hermitian form, then in local sense we can write $$\omega=\partial\bar\partial\log h,$$ and also if $f$ be the Kähler potential then we can write $$...
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1answer
113 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 ($k-2$)...
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1answer
218 views

Question on Intersection Theory of Effective Divisors

I am reading Section 1.1C of Lazarsfeld "Positivity in Algebraic Geometry I" and I need help understanding one line. On Page 17, Remark 1.1.13(iii), he says the following: If $D_1,..., D_n$ are ...
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1answer
606 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 ...
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1answer
260 views

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 ...
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0answers
96 views

Application of Kodaira Embedding Theorem

I am going to give a talk on Kahler manifold. In particular, I will outline a proof of the Kadaira Embedding theorem. I also wish to give some applications of the theorem. One of the application ...
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49 views

Space of almost complex structures on a compact manifold

According to the book by Huybrechts, Complex Geometry: An Introduction, this is a nice space and may be regarded, after some form of completion, as an infinite-dimensional manifold. How is this done, ...
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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 ...
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1answer
92 views

Existence of finite morphism to projective line

As we all know, Belyi's theorem says: A complex curve $X$ is defined over a number field, if and only if there exists a finite morphism $t:X\to \mathbb{P}^1_\mathbb{C}$ of varieties over $\mathbb{C}$ ...
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2answers
225 views

Complex manifold with no divisors

I read in Griffith Harris P132 that a complex manifold of dimension greater than one can have no divisors on it at all. I want to find examples. Is there an example? Does the Hopf manifolds $S^1\times ...
3
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1answer
515 views

Picard group and jacobian of a singular curve

I found somewhere written that the jacobian of a reducible curve is the product of the jacobians, which seems quite reasonable to me. Where can I find some proof of it and a description of the Picard ...
4
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1answer
172 views

Problems understanding the construction of Hitchin moduli space in his paper “The self-duality equations on a riemann surface”

First, if this post must be broken up in separate questions, please tell me so. I thought it would be better if I simply posed my questions in one thread, as they are directly related to each other. ...
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1answer
72 views

Branch locus of a projection of algebraic set

Let $X$ a algebraic cone in $\mathbb{C}^n$ with $\dim_0 X=p$. def.: if $f:A\to B$ is smooth map between smooth manifolds, then $br(f)$ is the points $x$ that $df_x$ is not surjective. def.: if $\pi: ...
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131 views

Deformation polarized abelian variety

In "$C$ is not algebraically equivalent to $C^{-}$ in its Jacobian" by Ceresa and then in "On the periods of certain rational integrals, II" by Griffiths (which is quoted as a reference in the first ...
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66 views

Requirements on holomorphic embedding in terms of the associated line bundle

Consider a holomorphic embedding $f$ of a genus $g$ Riemann surface $\Sigma_g$ into a generic quintic $Q \subset \mathbb{CP}^4$ This is the same as giving the (very ample) line bundle over the curve $...