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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How to get a polygon from a translation surface $(X,\omega)$

Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence $\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces ...
3
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1answer
269 views

Affine space structure on the space of Hermitian connections

I'm reading Gauduchon's paper Hermitian connections and Dirac operators. For a fixed almost-Hermitian manifold $(M, g, J)$ let $\mathcal A(g, J)$ be the space of connections $\nabla$ s.t. $\nabla g = ...
42
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0answers
2k views

Complex structure on $S^6$ gets published in Journ. Math. Phys

A paper by Gabor Etesi was published that purports to solve a major outstanding problem: Complex structure on the six dimensional sphere from a spontaneous symmetry breaking Journ. Math. Phys. 56, ...
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0answers
97 views

Ricci flow in complex analysis [closed]

Occasionally, I find a paper http://arxiv.org/abs/math/0505163 written by Chen, Lu and Tian. In this paper, the uniformalization theorem was proved by Ricci flow. I think it is a very interesting ...
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0answers
79 views

Looking for examples of holomorphic maps to $\mathbb{P}^1$ with certain property

I would like to know any example of nonconstant holomorphic map $f:X\to\mathbb{P}^1$ such that $K_X\cong f^*\mathcal{O}(2n)$ for some positive integer $n$, where $K_X$ is the canonical bundle of $X$. ...
2
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88 views

Do smooth manifolds create colimits for complex manifolds?

Suppose we have a diagram $D$ in the category $\textrm{Diff}_\mathbb{C}$ of complex manifolds, and suppose this diagram has a colimit $L$ after inclusion in the category $\textrm{Diff}$ of smooth ...
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1answer
161 views

Almost complex structure and nontrivial idempotents

Is there a compact Reiemannian manifold $M$ for which the following complex $C^{*}$ algebra does not have a nontrivial idempotent: $A=Hom(E,E)$ where $E$ is the complexification of $TM$. Of ...
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1answer
349 views

Artin approximation of a diagram

Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods ...
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247 views

Classification of complex Kronecker foliations

Let $\theta \in \mathbb{C}$ be a fixed complex number. The submersion $f:\mathbb{C}^{2}\to \mathbb{C}\; \text{with}\; f(x,y)=y-\theta x$ defines a complex foliation on $\mathbb{C}^{2}$. Consider the ...
3
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1answer
205 views

Tate twist and comparison between Betti and de Rham cohomology

In Deligne's paper "Hodge cycles on abelian varieties" (see page 11 of http://jmilne.org/math/Documents/Deligne82.pdf) he says that the following diagram fails to commute by a factor of $(2 \pi i)^m$, ...
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0answers
110 views

Lie derivative and taking trace

Let $(M,\omega)$ be a complex Kahler manifold, and $g$ is a smooth function such that $\int_Mg\omega^n=0$. It is obvious that there exists a smooth function $f$ such that $\triangle_\omega f=g$. ...
23
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2answers
4k views

The error in Petrovski and Landis' proof of the 16th Hilbert problem

Edit:For a recent progress on the Hilbert 16th problem see the following note which consider an infinite dimensional nature for this (apparently 2 dimensional) amazing problem . Best wishes for ...
1
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0answers
89 views

Moduli space of line-bundle holomorphic structures and sections over a Kahler-Hodge manifold

Let $(M,\omega)$ be a Kahler manifold with Kahler integral two-form $\omega$ and let $(L,h)$ be a rank-one complex vector bundle over $M$ equipped with a fixed hermitian metric $h$. I am interested in ...
3
votes
1answer
146 views

Blow-up as polar coordinates?

While doing some explicit calculations involving a blow-up of the plane in a point, I realised what I was doing was basically writing things in polar coordinates. Somewhat astonished that I hadn't ...
4
votes
1answer
433 views

Topology of algebraic varieties

Let $X$ be a projective variety (lets say normal and irreducible) with the topology coming from being a subspace of $\mathbb{P}^N$ (and not the Zariski topology). Surely one can then define the ...
2
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0answers
69 views

How to study Kähler metrics singular along a submanifold of codim 2?

Let $M$ be a compact complex manifold, $S\subset M$ a submanifold of codimension $2$, let $\omega$ be a k\"ahler metric on $M\setminus S$. Then we know by Reese Harvey's paper "Removable singularities ...
5
votes
1answer
236 views

Algebraic spaces which are automatically schemes

Let $S$ be a scheme, and let $f:X\to S$ be a morphism of algebraic spaces. If $f$ is smooth proper curve of genus at least two, then $X$ is a scheme. (Here I mean that $f$ is a smooth proper morphism ...
4
votes
2answers
301 views

Families of Fano varieties over non-hyperbolic curves

Let $C$ be a non-hyperbolic (smooth quasi-projective connected complex algebraic) curve. That is, $C$ is isomorphic to $\mathbb P^1, \mathbb A^1, \mathbb G_m$, or an elliptic curve. Let $f:X\to C$ be ...
7
votes
1answer
406 views

Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{n-1}$

Let $f,g:\mathbb{C}^n\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^n\to \mathbb{C}^n$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and ...
2
votes
1answer
137 views

Constant spinors from constant forms

Let $(X,g)$ be a $m$-dimensional complex, hermitian, spin manifold and let us denote by $S_{\mathbb{C}}$ its complex spinor bundle. Then: $S_{\mathbb{C}}\simeq \Lambda_{\mathbb{C}}(X)$ Let $\nabla$ ...
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1answer
717 views

Hamiltonian potentials of holomorphic vector fields on modifications of Kahler manifolds

let $(M,\omega)$ be a compact Kähler manifold. Let $\mathfrak{g}=H^{0}(M,T_{M})$ be the Lie algebra of holomorphic vector fields on $M$.We can decompose $\mathfrak{g}$ as ...
6
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1answer
166 views

Pontryagin Forms and Special Holonomy

Let $(M,g)$ be a Riemannian manifold. Recall that the $k^{th}$-Pontryagin class is a topological invariant which, by classical Chern-Weil theory, can be represented using the so-called Pontryagin ...
0
votes
1answer
149 views

The injection of direct image sheaf

Let $f:X \longrightarrow Y$ be a smooth holomorphic fibration between K\"ahler manifolds and $L$ be a holomorphic line bundle on $X$. Let $m$ be a positive integer. We denote by ...
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0answers
99 views

Geometric transitions of Calabi-Yau threefolds

Let me start with the definition of geometric transition: Let $Y$ be a Calabi–Yau 3–fold and $\phi: Y \rightarrow \overline Y$ be a birational contraction onto a normal variety. If there exists a ...
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2answers
145 views

Special connection of vector bundle over real manifold

Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: ...
4
votes
1answer
188 views

Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$). Let $\beta \in H_2(X_k, \mathbb{Z})$ be the homology class given by $$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$ ...
3
votes
1answer
197 views

What are the indecomposable classes on a del-Pezzo surface?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$). Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by $$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$ ...
3
votes
0answers
254 views

What's the name of this branched covering?

I've come across a double cover of $\mathbb P_1(\mathbb C)$, ramified at $[1:1]$ and $[-1:1]$ in homogeneous coordinates, given as the quotient by the natural $\mathbb Z/2\mathbb Z$-action generated ...
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0answers
17 views

Intersection of an irreducible curve with an exceptional divisor

Suppose $A$ is an irreducible curve on the blowup of $\mathbb{P}^2$. Then if $A$ is not equal to $E$ (where $E$ is the exceptional divisor) it has to intersect it positively. However, $A.E = c_1 ...
3
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0answers
124 views

Varieties acted upon faithfully by an abelian variety

Let $X$ be a smooth projective variety over the complex numbers. Suppose that some positive-dimensional abelian variety $A$ acts faithfully on $X$. Examples of such varieties $X$ are provided by ...
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vote
1answer
146 views

Cohomology of Homogeneous Complex Manifolds

Let $M$ be compact $G$-homogeneous manifold, equipped with the equivariant complex structure, when $G$ is a semi-simple algebraic group. The obvious example is every flag manifold. In that case, all ...
1
vote
1answer
135 views

Can a rigid CY threefold have infinitely many automorphisms

Let $X$ be a rigid Calabi-Yau threefold. Does $X$ have only finitely many automorphisms? N.B. A smooth projective threefold $X$ over $\mathbb C$ is a rigid Calabi-Yau variety if $h^i(X,\mathcal O_X) ...
2
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0answers
66 views

Is there any topological information encoded by the zero locus of a complex Hessian?

On a Kahler manifold one can always find a function $K$ so that (up to some constant) the Kahler form $\omega$ can be written as $\omega = \partial \bar{\partial}K$ (sometimes, under conditions that I ...
7
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0answers
76 views

Equation of the curve corresponding to a polarization of an abelian surface

Let $\mathbb{C}^2/\Lambda$ be a polarized abelian surface. I think it is well-known how to write down the equation of the divisor corresponding to the polarization, in terms of theta functions etc. ...
5
votes
1answer
319 views

Factors of automorphy from Chern connection

This question is inspired by a recent question about holomorphic bundles and factors of automorphy. Suppose $X$ is a compact, complex manifold whose universal cover $\widetilde{X}$ is Stein (the ...
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0answers
61 views

What are “minimal lines” in complex geometry?

Schwarz reflection across a real analytic arc $C$ in $\mathbb{R}^2$ is usually defined analytically. Thinking of the arc as the image of an interval in $\mathbb{R}$ under an invertible holomorphic map ...
0
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0answers
143 views

Are del-Pezzo surfaces complete intersections?

Let $X_k$ be $\mathbb{CP}^2$ blown up at $k$-points (where $k$ is from $0$ to $8$). I think it is known that $X_k$ can be embedded in $\mathbb{CP}^n$ for some $n$. $\textbf{Question:}$ Can $X_k$ ...
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0answers
60 views

Does the monodromy of such VHS have to be trivial

Consider a variation of polarized Hodge structure on a punctured disk. Suppose that connection preserves Hodge filtration (which is much stronger, than Griffiths transversality). Moreover assume that ...
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0answers
80 views

Almost complex structure gluing

Consider smooth complex manifold $X$ of complex dimension $n$ and its smooth submanifold $Z$ of codimension $k$. Denote the normal bundle of the inclusion $Z\subset X$ by $\nu.$ One can blow up $X$ ...
3
votes
1answer
97 views

Is every positive $(n-1,n-1)$ form almost decomposable?

Suppose $(X,\omega)$ is a compact hermitian manifold of complex dimension $n$, and $\alpha$ is a smooth $(n-1,n-1)$ positive form, i.e., $\alpha = a \omega ^{n-1} + \displaystyle \sum _{i=1} ^m ...
1
vote
1answer
116 views

Tensoring by Line Bundles to Produce Holomorphic Sections

Inspired by the line bundle case, I have the following question: Given an equivariant holomorphic vector bundle over complex projective space, is it true that tensoring it by line bundles often ...
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0answers
237 views

When a Spherical variety is $K$-stable

Let $X=G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then we call $X$ spherical. My question is When a Spherical variety is $K$-stable? Is ...
3
votes
2answers
290 views

Smooth paths on affine varieties

I have the following question which is in some way related to an application of Randell Isotopy Theorem to complex hyperplane arrangements. Let $h,k\geq1$ be integer numbers and let ...
2
votes
0answers
157 views

Characterization of certain analytic vector fields on $S^{2}$

Let $X$ be a real analytic vector field on $S^{2}$ which satisfies: $X$ has a finite number of singularities on $S^{2}$ The equator is invariant under flow of $X$ 3.$g_{*}X=\pm X$ where $g$ ...
5
votes
0answers
171 views

de Rham cohomology group and Dolbeault cohomology group on compact complex analytic spaces

My question is: On a compact complex analytic space,since Hodge Theorem becomes invalid,is it true that the de Rham cohomology group $H^p_{DR}(M)$ and the Dolbeault cohomology group ...
11
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3answers
607 views

Conformal Welding Reference

I'm looking for a reference for the following fact: given two Riemann surfaces and an identification of their boundaries, once I topologically glue the surfaces together there exists a unique ...
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1answer
169 views

Singular homology of the zero loci of polynomials

I am very sorry but apparently I am really weak in cohomology flavored questions. I try to reformulate my problem in a very simple and hopefully clear way. This question is related with a problem in ...
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0answers
51 views

A holomorphic vector bundle structure for $\Omega^{(0,1)}(M)$

For a complex manifold $M$, the complexified tangent space $\Omega^1(M)$ splits into a direct sum $\Omega^1(M) = \Omega^{(1,0)}(M) \oplus \Omega^{(0,1)}(M)$. As is well-known $\Omega^{(1,0)}(M)$ is ...
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1answer
251 views

A geometric construction of the complex projective plane?

The paper Kötter's synthetic geometry of algebraic curves, (N. Fraser, Proceedings of the Edinburgh Mathematical Society 7, 46–61, 1888) opens with a sketch of what appears to be a synthetic ...
2
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0answers
110 views

Holomorphic extension of a section of a line bundle

let $(X,g,\omega)$ be a non compact complete K\"ahler manifold of dimension $m\geq3$. Let $\nabla$ the covariant derivative wrt $g$ and $Riem$ the curvature tensor of $g$. Suppose that ...