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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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 $\mathbb{...
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194 views

A reference on Kodaira-Spencer deformation theory

I am looking for an introduction to Kodaira-Spencer deformation theory. I have a background in Teichmüller theory, but I know almost nothing of (and am not so interested in) algebraic geometry. Do ...
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1answer
70 views

Is it possible to realize a general compact Riemann surface in $\mathbb CP^2$?

Let $X$ be a compact Riemann surface with smooth boundary $\partial X$. Is it always possible to realize $X$ as a complex submanifold of $\mathbb CP^2$? In other words, is it true that there exists a ...
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121 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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1answer
76 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 ...
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1answer
86 views

Isometries of the plane and fixed lines

I am given that for all reflections $g$ there are infinitely many lines $L$ satisfying $g(L) = L$ which makes perfect sense (just take lines perpendicular to the axis of reflection). I am asked to ...
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75 views

degree of determinant line bundle of pullback in terms of Chern class

I have a holomorphic map $f: C \to X$ where $\dim_\mathbb{C}C=1$ and $\dim_\mathbb{C}X=2$. Why is $\text{deg}\det f^*TX=-c_1(X)\cdot[C]$? I'm really curious to how that negative sign shows up.
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1answer
114 views

Blowup of a (very) simple singularity

Take the action of $\mathbb{Z}_2$ on $\mathbb{C}^2$ given by $(-1) \cdot (z,w) = (-z,-w)$ and of course, $(1)\cdot (z,w) = (z,w)$. If you look at the resulting quotient space $\mathbb{C}^2 / \mathbb{Z}...
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1answer
40 views

Global n-form on Calabi-Yau

I am now reading these lectures by Stefan Vandoren on complex geometry. Everything is fine in general, hiwever I am confused with how he defines 1-form on a Calabi-Yau 1-fold (or 2-form on CY$_2$). ...
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157 views

Colimit and push forward of quasi-coherent sheaves in the analytic setting

I'm currently working on my master thesis and have some unresolved questions about quasi-coherent sheaves. Since I'm new to algebraic geometry, they might be rather trivial. I'm working in the ...
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1answer
287 views

holomorphic map between compact Riemann surfaces

Why nonconstant holomorphic map between compact riemann surfaces is surjection? I don't understand: if open-closed connected subset X of connected space Y, then X is all Y??
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1answer
80 views

3 points collide in $\mathbb{C}^2$

In Nakajima's book, "Lectures on Hilbert Schemes of Points on Surfaces", he gives an explicit description of the corresponding ideal for two points colliding in $\mathbb{C}^2$. This basically ...
5
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1answer
191 views

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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47 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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42 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 $g+1$...
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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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0answers
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 ...
4
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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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0answers
101 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 ...
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1answer
230 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
121 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
253 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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1answer
283 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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61 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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1answer
274 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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2answers
230 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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18 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
86 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
208 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?
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1answer
99 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
180 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
481 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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2answers
129 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. ...
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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
115 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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80 views

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
77 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
111 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
114 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
228 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 ...