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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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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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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$H^2(X,\mathbb{Z})$,$H_2(X,\mathbb{Z})$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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51 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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127 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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91 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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31 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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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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66 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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169 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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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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33 views

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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271 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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24 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
99 views

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
90 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
73 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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373 views

Does $\Bbb{CP}^{2n} \mathbin{\#} \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there. $\Bbb{CP}^{2n+1} \mathbin{\#} \Bbb{CP}^{2n+1}$ supports a complex structure: ...
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87 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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50 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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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
74 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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110 views

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
88 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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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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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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55 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
64 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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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 ...
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61 views

Question about Normal Coordinates on a Kaehler Manifold

I am currently reading FY Zheng's textbook, "Complex Differential Geometry". In section 7.4 Proposition 7.14, he is trying to prove thata metric $h$ Kaehler is equivalent to the statement, "For any $p ...
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Existence of real structure on CY m-fold

Suppose $M$ is Calabi-Yau $m$-fold with complex structure $J$, Kahler form $\omega$, metric $g$ and holomorphic $m$-form $\Omega$. What are the conditions on $M$ for the existence of a map $\sigma: M ...
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1answer
63 views

Two different descriptions of a complex torus

I came across the following description of a (1-dimensional) complex torus while learning about Calabi-Eckmann manifolds: For a fixed $\alpha \in \mathbb{C} \setminus \mathbb{R}$, the subgroup $Z = ...
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33 views

Complex number z satisfies both the inequality $|z-ai|=a+4$ and the inequality $|z-2|<1$

The number of integral values of $a$ for which at least one complex number z satisfies both the inequality $|z-ai|=a+4$ and the inequality $|z-2|<1$. I supposed $z=x+iy$ and put in both equations, ...
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2answers
61 views

Definition of a complex fiber

We define a real hypersurface as a subset $M\subset\Bbb C^n$ which is locally defined as the zero-locus of some $r\in\mathcal C^2(\Omega,\Bbb R)$ ($\Omega\subseteq\Bbb C^n$ open). Then let $z_0\in M$. ...
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Algebraic methods to compute the cohomology ring of the complex topology of a variety?

Suppose $V$ is an affine (resp. projective) subvariety of the affine (resp. projective) space $\mathbb A_\mathbb C^n$ (resp. $\mathbb P_\mathbb C^n$) with vanishing ideal $I\subseteq\mathbb ...
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Compactifying $\mathcal{O}_{\mathbb{P}^1} (-2)$

I have the total space of $\mathcal{O}_{\mathbb{P}^1} (-2)$ and I see that a "standard" way to compactify is to add the trivial line bundle, $\mathcal{O}_{\mathbb{P}^1}$, and then projectivize. That ...
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Spinors and forms

In this link http://benasque.org/2009gph/talks_contr/074Herdeiro.pdf page 15, it was said that: "Use spinorial geometry techniques: One takes the space of Dirac spinors to be the space of ...
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Why don't we use the constant sheaf as the dualizing sheaf in Verdier Duality?

The Verdier dual of a complex of sheaves $F$ on a complex manifold $X$ is defined in the derived category $D_b(X)$ as $\mathcal{RHom}(F,\mathbb{D}_X)$, where $\mathbb{D}_X$ is the dualizing "sheaf" ...
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A triangle and its median in complex plane.

Let $z_1$, $z_2$, $z_3$ be vertices of the triangle $\triangle ABC$. And given that $|z_1|=|z_2|=|z_3|$. Then the median through $A$ cuts the circumcircle at which point? We need to get the answer in ...
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51 views

How to tell complex structures apart

Complex structures are rigid, yet weirdly flexible. For example, the Riemannian mapping theorem says that every non-empty simply connected open subset of $\mathbb{C}$ that is not $\mathbb{C}$ is ...
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169 views

Global Residue Theorem in $\mathbb{CP}^2$.

Consider the following meromorphic form defined on $\mathbb{CP}^2$: Be $\phi_{1,2}$ the chart given by $\phi_{1,2}(Z_1,Z_2) = (1,Z_1,Z_2)$, in these coordinates define $\omega= \frac{1}{Z_1 Z_2} dZ_1 ...
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39 views

Explicit descriptions of the modular curves $Y(5)$ and $Y_1(5)$

The modular curves $Y(5)$ and $Y_1(5)$ associated to the congruence subgroups $\Gamma(5)$ and $\Gamma_1(5)$ are of genus 0. As such, they are nothing but the Riemann sphere $\mathbb P^1$ minus a ...
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90 views

Triviality of a vector bundle is an open condition

Is the following statement true (and if not, are there additional assumptions that make it true?) A vector bundle on a variety which is trivial if restricted to a closed subvariety is trivial on ...
2
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1answer
32 views

real coordinates of a complex manifold

I have a naive question about real coordinates of a complex manifold. Let's consider 1-dimensional case for simplicity. Let $X$ be a Riemann surface and $z$ be a local complex coordinate. Then one ...
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55 views

A question about current and Dirac measure

$0$ can be seen as a divisor of $\mathbb{C}$, and the current $[0]$ is defined as $[0](\varphi)=\varphi(0)$. Why is this reasonable?