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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Real Quadratic Forms, Complex Quadratic forms, and the Inertia Theorem.

I am very confused about the classification of quadratic forms. Scroll to the last paragraph for my question. Here is what I know: A $\bf \text{real}$ quadratic form is obtained from any bilinear ...
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Why was this terminology of holomorphism used here?

page 6 in this notesays: A note on index convention: $i$,$j$, and $\bar{i}$, $\bar{j}$ index over the holomorphic and antiholomorphic components. I never knew there were holomorphic ...
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Equivalence of a vector bundle being trivial on $\mathbb{P}^1$

I am looking for various statements about a vector bundle $E$ of arbitrary rank being trivial on the complex projective line, $\mathbb{P}^1$. In particular, some arguments about cohomology would be ...
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Moduli Space of elliptic fibration

Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration. I know how it is done for the generic Weierstrass ...
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35 views

What is $\partial \bar{\partial}$ and when is it non-zero?

I often see $\partial \bar{\partial}$ arising in complex geometry, what is it (explain in relatively simpler language)? And sometimes $\partial \bar{\partial}T$ is non-zero (for example, the ...
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Questions about the connected component of a relative Picard Scheme.

Let $X$ be a smooth, projective surface (i.e. $2$-dimensional connected variety) over $k=\mathbb{C}$. Denote by $\mathrm{Pic}_{X/k}$ the associated relative Picard scheme. We write ...
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120 views

Is the homology class of a compact complex submanifold non-trivial?

Let $X$ be a connected complex manifold (not necessarily compact). Let $C \subset X$ be a compact complex $k$-dimensional submanifold (for some $k>0$). Is it true, in this generality, that the ...
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48 views

When is a quasiprojective variety Kobayashi hyperbolic?

I am looking for some (simple and well-known) sufficient conditions on a quasiprojective variety to be Kobayashi hyperbolic. I realize that in this generality it may be a complicated (maybe even ...
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1answer
20 views

Proving the principle symbol is globally defined

I want to prove the principle symbol is globally defined as an element \begin{align} \Gamma(T^* M / \{0\}, \textrm{Hom}(\pi^*E, \pi^* F) ) \end{align} To more specify, let me explain the definition ...
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1answer
46 views

Riemann Surface of $w^{2}=\sqrt{1-z^{2}}$

I'm working in the problem of finding branch points and build the Riemann Surface of the following complex function: $$ w(z)=\sqrt{1-z^{2}} \,\, . $$ I'm reading lots of texts about how to do this, ...
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35 views

Doubt with an illustration of algebraic curves and Riemann surfaces

The complex equation $w - z = 0$, $z$, $w \in \mathbb{C}$, represents a complex curve (also called $1$-dimensional complex manifold). This complex curve corresponds to the complex plane $\mathbb{C}$ ...
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k-symbol differential opeartor L, and its independent of choices.

This material is in O. Well's Differential analysis on complex manifold, page 115. Let, $(x,v) \in T'(X)$ ($T^*(X)$ with deleted zero section) and $e \in E_x$, Find $g\in \epsilon(X)$ and $f\in ...
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23 views

Is there a description of variation of (mixed) Hodge Structures in terms of a Deligne operator?

A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition \begin{equation} ...
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19 views

Hermitian vector space and relation of associated operators.

Here what i want to do is prove proposition 1.1 in chapter 5, on Wells, Differential analysis on complex manifold, The propositions are follows For Hermitian vector space of complex dimension $n$. ...
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96 views

(Anti-) Holomorphic significance?

What are holomorphic and anti-holomorphic components? Why don't we call them complex components and their conjugates? What is holomorphic coordinate transformation?
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1answer
67 views

Simplifying the Kahler form

In the link here, p.4, it says that, given a fundamental 2-from $\mathcal{K}$ $$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge d\bar{z}^{\bar{j}},$$ a manifold is said to be Kahler if this ...
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77 views

Is there an elementary way to see that there is only one complex manifold structure on $R^2$?

Is there an elementary way to see that there is only one complex manifold structure on $\mathbb{R}^2$? (Up to biholomorphism, naturally.) Elementary in the sense of not appealing to the ...
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1answer
37 views

What does $dz^2$ mean?

I'm reading a paper ("La Formule de Verlinde" by Christoph Sorger) and at a certain point, the author switches from algebro geometric language to complex geometric language. He uses the symbol $dz^2$, ...
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65 views

Primitive cohomology, example request

$X$ is a compact Kähler manifold or smooth projective variety. is there an example that a primitive class $0\neq [\omega]$ of $H^{p+q}(X, \mathbb{C})$ is wedge product of other two primitive classes: ...
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55 views

Complement of the zero section in degree $k$ line bundle over $\mathbb{P}^n$

Consider the tautological line bundle $\mathcal{O}(-1)$ over $\mathbb{P}^n$. Let $L^{-k}$ denote the total space of $\mathcal{O}(-k)$ with $k \in \mathbb{N}$. Then it is claimed that the complement of ...
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41 views

Holomorphic section is determined by arbitrarily small neighborhood?

Let $X$ be a connected complex manifold and $E\to X$ a holomorphic vector bundle. Suppose that $s:X\to E$ is a holomorphic section such that $s(x)=0$ for all $x$ in a non-empty open set $U\subset X$. ...
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Constructing transition function of given manifold

This is extension of my previous question Moduli space about $CP^{N-1}$ and $T^* CP^{N-1}$., meaning of $\mathcal O(-1)$ in algebraic geometry? . What i have been considered are followings First ...
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32 views

Tangent cone from differential geometric point of view?

For a plane curve define by the equation $f = f_h + f_{h+1} + \ldots + f_n$, where the $f_i$ are the homogeneous parts of degree $i$ (in the variables $x - a$ and $y - b$), and $f_h$ is the first ...
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37 views

Riemann - Hurwitz Formula for topology.

I am quite confused about the notion of branch points at infinity? and even in general the idea of branch points? I know branch points to be where points diverges to infinity. Could someone please ...
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114 views

What exactly does it mean to take something modulo an equivalence relation?

For instance, the complex projective space is defined as $\mathbb{C}{\mathbb{P}^n} = \left( {{\mathbb{C}^{n + 1}}\backslash \left\{ 0 \right\}} \right)/ \sim $ Where the equivalence relation is ...
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34 views

differing notions of the degree of a smooth projective plane curve

[Everything here will be over $\mathbb{C}$] Hello, one definition of a smooth projective plane curve $X$ is, for $X \subset \mathbb{P}^{2}$.. $deg(X) =$ maximum number of intersection points (without ...
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103 views

If a complex Lie group has the structure of an algebraic group, is this structure unique?

If $G$ and $H$ are algebraic groups over $\mathbb{C}$, and $f : G \rightarrow H$ is an isomorphism of complex Lie groups (i.e. a biholomorphic group isomorphism), then must $f$ be algebraic? If not, ...
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66 views

Global sections of projective subspaces

I have a general question with a specific application. The cohomologies of line bundles on $\mathbb{P}^n$ are known and in particular, $H^0(\mathbb{P}^n, \mathcal{O}(d))$ is canonically isomorphic to ...
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1answer
33 views

Möbius transformations are differentiable

Let's consider the Möbius transformation $$ f:\mathbb{C}\cup \{\infty\}\rightarrow\mathbb{C}\cup\{\infty\},z\mapsto \frac{az+b}{ -\overline{b}z+\overline{a}}, $$ with $a,b\in\mathbb{C}$ such that ...
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40 views

Applications of the Hurwitz Theorem on Number of Automorphisms?

So I am giving a talk in which we'll prove the semi-famous Hurwitz Theorem: Let $S$ be a compact Riemann surface of genus $g \geq 2$. Then $|Aut(S)| \leq 84(g-1)$, the group of holomorphic ...
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Finitely Many genus-g Quotients of Compact Riemann Surface

I hear there is a semi-famous theorem from my advisor, but he didn't know the name and I was unable to find it online. Does anybody know of the following? Let $S$ be a compact Riemann surface. Then ...
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75 views

Space of Extensions of Holomorphic Line Bundles

On a compact complex manifold $X$, fix two holomorphic line bundles $L$ and $L'$. Consider a holomorphic vector bundle $V$ of rank 2 which fits in an exact sequence $$0\to L\to V\to L'\to0$$ I would ...
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1answer
34 views

Conformal metric

I'm trying to solve the following : Let $D$ be a simply connected region stricly included in $\mathbb{C}$. Let $\mathbb{D}$ be the open unit disk. Let $f \in \text{Hol}(D)$ be a bounded function ...
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1answer
18 views

Annihilator ideal of image sheaf

Let $(f,\tilde{f}): (X, \mathscr{O}_X) \rightarrow (Y, \mathscr{O}_Y)$ be a holomorphic map between complex spaces, such that $f_*(\mathscr{O}_X)$ is $\mathscr{O}_Y$-coherent. Define $\mathscr{I} = ...
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How to show that in a 6 dimensional manifold $\ast_6 A = - J \wedge A$ for $A^{1,1}$ primitive $1,1$ complex form and $J$ k\"ahler form

Given a 6 dimensional manifold, of complex dimension 3, take the Hodge star operator $\ast_6$ and a primitive (1,1)-form $A_2$ (i.e. such that $J \wedge \ast_6 A = J_{mn}A^{mn}=0$ and also $J\wedge J ...
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About singularity

Let $X$ be a normal affine variety over $\mathbb{C}$. Q1. Let $x\in X$ be a singularity with a Cartier divisor $x\in D$. Then $\mathcal{O}_{X,x}$ is Cohen-Macaulay if and only if $\mathcal{O}_{D,x}$ ...
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confusion about the notion of $\bar{\theta^j}$ and $\bar{\theta_i^j}$

Let $(M,J,g)$ be an almost Hermitian manifold, and $\{e_i\}$ be $(1,0)$-vector field basis, $\{\theta^i\}$ be its dual basis. We have $$g=g_{i\bar{j}}\theta^i\otimes\bar{\theta^j}$$ If connection ...
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Condition for a complex vector bundle to be holomorphic?

Suppose that $(E,\pi,M)$ is a complex vector bundle. I've seen it suggested in a few places that if $E$ and $M$ are complex manifolds, and $\pi$ is a holomorphic map, then $(E,\pi,M)$ is in fact a ...
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1answer
25 views

Are there closed Riemann surfaces without non-constant holomorphic functions?

I came across the Handbook of Teichmuller Theory, and they talk about "closed Riemann surfaces with non-constant holomorphic functions". Are there Riemann surfaces without those functions?
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Are inhomogeneous coordinates … COORDINATES?

It might seem a silly question but I'm asking the following: Take the complex projective line, are the inhomogeneous coordinates sufficient to have an atlas where the transition functions are ...
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61 views

cohomology of a tangent bundle

Suppose that $C$ is a complex riemann surface of positive genus lying in a complex algebraic surface of general type. Let $T_C$ the tangent bundle to the curve $C$. Is there a way to compute the ...
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Why care about (local) rational functions in algebraic geometry?

I'm trying to get a feel for basic algebraic geometry. One of the first things I read about is the localization of a polynomial ring at a point, yielding a ring of rational functions defined locally ...
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Extending a d-closed (p,q) form of a fibre of complex analytic family

Let $\phi: X \to B$ be a family of complex manifolds. Fix a point $0 \in B$ and $X_0 := \phi^{-1}(0)$. For any $\alpha \in A^{p,q}(X_0) := \{C^\infty (p,q)\text{ forms on }X_0\}$ such that $d \alpha = ...
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Is there a complex surface into which every Riemann surface embeds?

Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma \to \mathbb{CP}^3$. It follows from the ...
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Complex differential forms on $CP^n$

Why Complex projective spaces don't admit some differential forms? To be more specific, I know that the space of complex forms is decomposed as direct sum of holomorphic and anti-holomorphic part; ...
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56 views

Base change of topogical spaces VS Base change of schemes

In algebraic geometry, we have the following famous base change theorem [Hartshorne III Theorem 12.11]: Let $f:X\to Y$ be a projective morphism of noetherian schemes, and let $\mathcal F$ be a ...
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Definition of a Subtorus

Let $V$ be a finite vector space over $\mathbb{C}$ and consider a lattice $L$ of $V$ i.e a discrete subgroup of $V$ of maximal rank. Consider the torus $T=V/L$. Definition: Let $S\subset T$ be a ...
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1answer
70 views

When do vector bundles decompose into line bundles?

I know that over $\mathbb{C}\mathbb{P}^1$, every vector bundle decomposes as a direct sum of line bundles. When else does this happen? My question is, what assumptions do I need to put on a complex ...
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Equality $H^i(K,\mathcal{F}_{|K})=\varinjlim_{U\supset K}H^i(U,\mathcal{F}_{|U})$ for a constructible sheaf

The setting is the following. I have a complex algebraic variety $X$, and $\mathcal{F}$ is a constructible sheaf on it (i.e. there is a stratification of Zariski-locally closed subsets $X=\sqcup_{i ...
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Line bundles with no meromorphic/holomorphic sections

I am studying about the natural map (homomorphism) from the divisors of a complex manifold $X$ to the holomorphic line bundles on $X$, $$Div(X)\longrightarrow\;Pic(X)$$ where each divisor $D$ is sent ...