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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6
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1answer
138 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 ...
2
votes
1answer
42 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 $dz^2$, ...
2
votes
0answers
78 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: $...
3
votes
1answer
65 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 ...
3
votes
1answer
42 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 $U\subset X$. ...
1
vote
0answers
18 views

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 ...
2
votes
0answers
36 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 ...
0
votes
0answers
42 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 ...
1
vote
3answers
121 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 ...
0
votes
1answer
37 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 ...
10
votes
1answer
111 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, ...
2
votes
0answers
73 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 ...
1
vote
1answer
42 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 $|a|...
4
votes
2answers
52 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 ...
4
votes
1answer
49 views

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 ...
3
votes
2answers
96 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 ...
2
votes
1answer
39 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 ...
0
votes
1answer
23 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} = \...
1
vote
0answers
36 views

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 \...
0
votes
0answers
39 views

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 $\mathcal{O}_{D,x}$ ...
1
vote
0answers
18 views

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 $D$...
7
votes
0answers
98 views

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 ...
0
votes
1answer
27 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?
0
votes
1answer
28 views

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 ...
2
votes
0answers
73 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 ...
1
vote
0answers
38 views

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 ...
1
vote
0answers
38 views

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 = ...
21
votes
1answer
334 views

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 degree-...
2
votes
1answer
88 views

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; ...
1
vote
0answers
61 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 ...
2
votes
0answers
25 views

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 ...
2
votes
1answer
89 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 ...
3
votes
0answers
34 views

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 \...
3
votes
0answers
86 views

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 ...
3
votes
0answers
91 views

Sections of the canonical bundle

This is maybe a stupid question. Let $M$ be a simply-connected complex (kahler?) manifold, is it true that the canonical bundle $K_M$ has always (global) sections? For example, we know that an ...
4
votes
0answers
120 views

Clarification about the definition of Calabi-Yau manifold

There are a lot of different definitions of a Calabi-Yau manifold. Roughly, we can divide them in two sets, see Wikipedia https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold . I will refer to ...
2
votes
0answers
74 views

Understanding Complex Differentials (forms)

In the study of Riemann surfaces, many books bring in their discussions, the complex differentials or differential forms, and there my understanding gets stopped. I personally interacted with many ...
1
vote
1answer
48 views

Étale morphism has all its Deck transformation homotopic to identity

Is there an example that étale morphism (of degree $d,d<\infty$) $\pi: X\rightarrow Y$, s.t. all its Deck transformations homotopic to $Id_X$,except the trivial one, where $Y$ is general Enriques ...
4
votes
0answers
77 views

$h^{0, 1}$ of K3 surface (a priori non-Kahler)

I am trying to understand paper by Siu "Every K3 surface is Kahler". Let $M$ be a K3 surface. Siu wrote $H^1 ( M , \mathscr{O}_M ) =0$ without any references. It is written on fifth page. Maybe I ...
2
votes
1answer
25 views

Vocabulary question: singularity for an analytic map

I have a question that is purely on vocabulary. My native language is not english, so I would like to know the usual convention for the following. When people say "let $f: X \to Y$ be an analytic map"...
4
votes
2answers
187 views

complex submanifolds in complex euclidean space

Assume all manifolds are without boundaries. In Euclidean space $\mathbb{R}^n$, there are many submanifolds (Whitney Embedding Theorem). In complex Euclidean space $\mathbb{C}^n$, are there any ...
3
votes
0answers
80 views

vanishing theorem in algebraic geometry

This is a general question: As we know there are lot of vanishing theorem like Fujita vanishing, kodaira Nakano vanishing, vanishing for big nef line bundle, Kollár vanishing, etc. Those just for ...
5
votes
1answer
106 views

Why can the zero set of a collection of holomorphic functions be written as the zero set of finitely many?

I've been reading the first few sections of Griffiths and Harris, and they state without proof (about halfway down page 14 in my 1994 Wiley Classics Library edition) that the zero locus of an ...
4
votes
0answers
49 views

entension of a holomorphic map on projective algebraic manifolds

I'm reading a paper.It says if $M$ is a projective algebraic variety and $f:N-S\rightarrow M$ is a holomorphic map,where $N$ is a connected complex manifold and $S\subset N$ is a proper analytic ...
2
votes
1answer
57 views

Corresponding toric variety for n-simplex

Let $P $ be a Delzant polytope and $X_P $ be a corresponding Toric variety. I want to see if $P=\sum $ be a n-simplex then $X_P=\mathbb P^n$
3
votes
0answers
43 views

When a current is actually a holomorphic form?

If a current $f$ of bidegree $(p,0)$ (acting on forms of bidegree $(n-p,n)$) satisfies $\bar{d}f=0$, is it true that $f$ is a holomorphic differential form? In general, do we have any standard ...
2
votes
0answers
31 views

Dual Lattice is a Lattice for the Dual Tori.

Let $T=V/L$ be a complex tori with lattice $L$. Consider the set $$ \overline{\Omega} = \{ h:V \to \mathbb{C} \text{: h} \text{ antilinear } \}$$ I am reading Birkenhake, Christina; ...
4
votes
1answer
73 views

what are fundamental groups of (almost) complex manifolds?

Are there any restrictions on the fundamental group of an even-dimensional manifold admitting an almost complex structure? integrable almost complex structure? or can I construct any of these with a ...
2
votes
1answer
46 views

Orthogonal complex structures on $\mathbb{R}^4$

How does one see that (the space of isomorphism classes of) orthogonal complex structures on $\mathbb{R}^4$ are a 2-sphere? It seems that one has to take a quotient of $O(4,\mathbb{R})$ by $U(2,\...
1
vote
1answer
87 views

Calculating Betti and Hodge number for product of a curve

How can we compute the Betti numbers and Hodge numbers for $S=C\times C/\sigma$? (Where $\sigma$ is swapping the two factors.)