# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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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 $\mathscr{I} = \... 0answers 36 views ### How to show that in a 6 dimensional manifold$\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 \...
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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 $D$...
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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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $h^{0, 1}$ 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 ...
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### 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"...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $X_P=\mathbb P^n$
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### 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 ...
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### 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; ...