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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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
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109 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 ...
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62 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 ...
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1answer
47 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 ...
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66 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 ...
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1answer
24 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 ...
3
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1answer
166 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
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77 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 ...
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1answer
101 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 ...
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48 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
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1answer
55 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 $X_P=\mathbb P^n$
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38 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 ...
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27 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
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1answer
66 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 ...
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1answer
43 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 ...
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1answer
77 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.)
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46 views

Find the biholmorphicity on two complex tori

This is related with the textbook, James Morrow and Kunihiko Kodaira, Complex Manifolds page 13, complex tori. For $M\in \mathbf{C}$, Let \begin{align} G = \{ g| g:z \rightarrow g(z) = z+ ...
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79 views

Exercise 2.2.4 from Complex geometry by Daniel Huybrechts

This question is the poini ix) of the exercise $2.2.4$ coming from page 68 of the book Compex Geometry by Daniel Huybrechts. Here i write the question: we have two holomorphic vector bundle $E,F$ ...
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2answers
126 views

$H^2(X,\mathbb{Z})$,$H_2(X,\mathbb{Z})$of smooth complex projective variety$X$

is there some example that: $X$ is a smooth complex projective variety, second singular cohomology $H^2(X,\mathbb{Z})$ has nontrivial torsion subgroup?
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1answer
49 views

Image of the Riemann-sphere

Let $S$ be the Riemann-sphere (the unit sphere in $\mathbb{R}^3$) and $\psi: S \rightarrow \mathbb{C}$ be defined by $$\psi(x_1, x_2, x_3)=\frac{x_1 + ix_2}{1-x_3}.$$ Let $\pi$ be a plane in ...
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1answer
123 views

Integral $(1, 1)$ forms and holomorphic line bundles

Let $X$ be a complex manifold. We say that a cohomology class in $H^2(X,\mathbb{C})$ is integral if it lies in the image of the natural morphism $j : H^2(X,\mathbb{Z}) \longrightarrow ...
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1answer
82 views

Local $\partial \bar{\partial}$-lemma..

I am trying to prove the local $\partial \bar{\partial}$ lemma. This says that for a polydisc in $\mathbb{C}^{n}$, a form in $A^{p,q}(U)$ being $d$-closed implies that it is $\partial ...
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30 views

étale morphism of germs of algebraic schemes over $\mathbb{C}$

$\cdot$ Let $f:(x\in X)\rightarrow(y\in Y)$ be a morphism of germs of algebraic schemes over $\mathbb{C}$. We say that $f$ is étale at $x$ if we can express $(x\in X)$ as ...
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2answers
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How do I express this vector field, given “in terms of $\mathbb{C}$”, in the standard notation for derivations?

In a set of lecture notes (not available online) that I'm currently working through, one is given the following set-up: Consider rotational vector fields on the plane ...
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1answer
59 views

Is any 2$m$-dimensional manifold almost complex?

In Nakahara's book "Geometry, Topology and Physics" (Ch. 8, about the almost complex structure) they write: Note that any 2$m$-dimensional manifold locally admits a tensor field $J$ [type (1,1)] ...
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1answer
124 views

Complex structure of a torus

Given the definition of complex structure for a complex manifold: the real $(1,1)$ type tensor $J_p : T_p M \rightarrow T_p M $ defined by $$ J_p \left(\frac{\partial}{\partial x^\mu} \right) = ...
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40 views

Why generalized vectors can be written locally as sum of vectors and 1-forms?

I would like to understand better this point. In generalized complex geometry the generalized bundle $E$ is defined as a non-trivial fibration of the cotangent bundle $T^*M$ over the tangent bundle ...
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46 views

What does “free morphism” mean in the context of moduli space of stable maps?

Let $\overline{M}_{0,0}(X, \beta)$ denote the moduli space of genus zero stable maps into $X$ that represent the homology class $\beta \in H_2(M, \mathbb{Z}) $. What does it mean to say that ...
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33 views

A curve and its translate are linearly equivalent in a complex torus?

Let $X=\frac{V}{\Lambda}$ be a complex torus of dimension $2$. Suppose $C$ is a curve in $X$. Consider $C+a=C_1$, the curve which is a translate of $C$ by $a\in X$. Are $C$ and $C_1$ linearly ...
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How one can show the equivalence relation from the two systems are equivalent? [In Complex Manifold]

This is related with the textbook, "Complex Manifold" by James Morrow and Kunihiko Kodaira. From their defintion 2.3, Two systems $\{ z_i\}_{i\in I}$, $\{ w_j \}_{ j \in J}$ are ...
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259 views

Is there a proof of Bézout's theorem via residue theory?

Let's define intersection numbers as follows. Consider a collection $f_1,\dots, f_n$ of holomorphic functions on some neighborhood of zero in $\mathbb C^N$ cutting out divisors $D_1$, all of which ...
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1answer
21 views

Question about notation for Hermitian metrics on complex manifods

The standard notation for a Hermitian metric looks like this: $$\sum ds^2 = \sum dz_i \otimes d\overline{z_j}.$$ The conjugate confused me for a while until I explained it to myself as follows. In ...
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Any reference where the following “partial integration” or “partial residue” concept is defined?

I am studying the on-shell diagrams techniques to compute scattering amplitudes, let's take as a common reference this paper: http://arxiv.org/abs/1212.5605. Trying to put the content of the article ...
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1answer
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almost complex embedding of $S^2$ and $S^6$ into $\mathbb{C}^N$

In Which Spheres are Complex Manifolds? , I find that $S^2=\mathbb{C}P^1$ is a complex manifold and $S^6$ is an almost complex manifold. Are there references about: What is the smallest integer $N$ ...
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1answer
83 views

Smooth section of Hodge bundle ($F^pH^k$) can be viewed as a smooth form of type$F^pH^k(X,C)$ over$ X$,$ X\rightarrow B$ is an analytic family.

I think it is due to Kodaira. could someone explain the idea that Kodaira come up with this. maybe I shouldn't say"can be viewed as". I really mean the smooth form restrict on each fibre is just the ...
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1answer
55 views

How to understand the Mobius transform as a group action?

The group $SL(2,R)$ acts on the upper half-plane by the formula $$ \left(\begin{array}{cc} a & b \\ c & d \end{array} \right) z = \frac{az + b}{cz + d} .$$ It is indeed straightforward to ...
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Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there. $\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has ...
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71 views

Line bundles on $\mathbb{P}^2 \times \mathbb{P}^2$

I'm curious as to what are all the line bundles on $\mathbb{P}^2 \times \mathbb{P}^2$? I might be mistaken but I believe they are classified as $$ \mathcal{O}_{\mathbb{P}^2 \times \mathbb{P}^2}(p,q) ...
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1answer
49 views

Is the canonical bundle topologically trivial?

Suppose $X$ is a $n$-complex dimensional complex manifold, we can form its canonical bundle $K_{X,\mathrm{hol}}=\bigwedge^n\Omega_{X,\mathbf{C}}$. Usually this bundle is not holomorphically trivial. ...
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1answer
60 views

$H^{p,0}$ is isomorphic to the space of holomorphic forms?

In Voisin's "Hodge Theory and Complex Algebraic geometry I" Corollary 7.6: $\textbf{Corollary 7.6}$ For every $p\leq n$, $H^{p,0}(X)$ is isomorphic to the space of holomorphic forms of degree $p$ on ...
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1answer
65 views

The complex structure of a complex torus

A complex torus $X$ of complex dimension $1$ is just a quotient of $\mathbb{C}$ by a lattice $\Lambda=\mathbb{Z}\oplus \tau\mathbb{Z}$ where $\tau=a+ib$ is a complex number with $b>0$. The complex ...
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2answers
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Sum of Betti numbers of a degree $d$ hypersurface of $\mathbb{P}^n_{\mathbb{C}}$

Let $X \subseteq \mathbb{P}^n_{\mathbb{C}}$ be the zero set of a homogeneous polynomial of degree $d$. Assume that $X$ is smooth. Is there a way to find the sum of the Betti numbers of $X$ (i.e. the ...
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1answer
74 views

Complex-valued differential forms.

Let $X$ be smooth (real) manifold and let $T^{*}(X)_{\mathbb{C}}$ denote the complexification of the cotangent bundle. We define the complex valued differential r-forms on $X$ to be the smooth ...
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Complexification of proper scheme

Let $X$ be a proper scheme over $\mathbb{C}$. We define $X_{\mathbb{R}}$ to be a scheme over $\mathbb{R}$, which is the same topological space as $X$ with structure sheaf generated by real and ...
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1answer
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Todd genus as homomorphism from complex cobordism

It's well-known that the Todd genus/arithmetic genus $\chi(\mathcal{O}_X)$ (or probably preferably $\int_X \text{td}(T_X)$ so as to define it in purely terms of the complex structure) is a genus in ...
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1answer
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Definition of complex submanifold

For smooth manifolds, we can define an embedded submanifold to be either (1) a subset locally cut out by "slice" charts, or (2) a subset that is a manifold in the subspace topology and admits a smooth ...
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51 views

Vector bundles on Hirzebruch surface $\mathbb{F}_2$

I would like to know a classification for all holomorphic vector bundles on the second Hirzebruch surface $\mathbb{F}_2$. Is this known? What is known? In particular, I'm looking for holomorphic ...
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1answer
43 views

Transform to flatten a parametric curve (polynomial)

Given a polynomial parametrized by $p(t)=(x(t),y(t))$ such that $y(t)=p(t)$, $x(t)=t$, and $p(t)= \sum_{i=0}^na_it^i$, for real coefficients $a_i$, is there some transformation I can take such that ...
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65 views

Equivalence of (almost) complex structures

Preamble: An almost complex structure on a manifold $M$ is an endomorphism $J : TM \to TM$ such that $J^2 = -1$. An almost complex structure $J$ is said to be integrable if the Nijenhuis tensor, ...
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Question on the matrix of a Kaehler Metric in Normal Coordinates

I am currently studying normal coordinates on a Kaehler manifolds: Let $h$ be a Kaehler metric on a complex manifold $M$ and let $p \in M$. Let $(z_1,..,z_n)$ be a coordinate chart such that $h$ is a ...