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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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 ...
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1answer
299 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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32 views

Is a toric blow-up in codimension 2 a real toric blow-up?

Let $X, Y$ be toric projective algebraic varieties over $\mathbb{C}$. Suppose that $X$ and $Y$ are $\mathbb{Q}$-factorial and smooth in codimension two (e.g. they have terminal singularities). Let ...
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126 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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55 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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62 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 $X$ ...
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1answer
87 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 ...
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30 views

Non-Trivial Kernel of the Dolbeault--Dirac on a Compact Kahler Manifold

Let $V$ be a holomorphic vector bundle over a compact Kahler manifold. For a choice of Hermitian structure on $V$, let $\Delta$ be the Laplacian on $V \otimes \Omega^{(0,\bullet)}$. ...
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1answer
96 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
227 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 ...
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2answers
285 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
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154 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
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159 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 ...
7
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1answer
344 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 ...
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3answers
601 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 ...
1
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1answer
166 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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1answer
687 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 ...
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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 ...
7
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1answer
245 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 ...
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1answer
116 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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104 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 ...
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59 views

Convenient Basis Presentation of Lefschetz Decomposition

Let $V$ be an almost-complex vector space, equipped with a symplectic element $\omega \in V^{(1,1)}$. In terms of a basis $b^+_i \in V^{(1,0}$, $b^-_i \in V^{(0,1}$, does there exist a "simple" ...
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0answers
109 views

Characterizations of regular holonomic D-modules

I'm looking for references for the various characterizations of regular holonomic D-modules, in particular proofs of their equivalence. For instance, some characterizations I've seen (in the analytic ...
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1answer
131 views

Closed parallel (1,1)-forms on compact Kähler manifolds

Let $(X,\omega)$ be a compact Kähler manifold. We know one example of a closed parallel (1,1)-form, namely, $\omega$ itself. Are there obstructions for the existence of non-vanishing closed parallel ...
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30 views

Applicability of Scheja's theorem

First, let me begin by recalling Scheja's (cohomology extension) theorem. Let $X$ be a complex manifold of dimension $n$ and $Z\subset X$ a complex submanifold of dimension $d.$ Let $\mathcal{F}$ be ...
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1answer
77 views

A question of direct image of relative canonical bundle

Proposition: Let $f:X\rightarrow Y$ be a smooth holomorphic fibration between K\"ahler manifolds, and $L$ be a holomorphic line bundle. Then there exists a Zariski open set $Y_0\subset Y$ such that ...
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7answers
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Is there a complex structure on the 6-sphere?

I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a ...
3
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2answers
197 views

What are the easiest examples of irreducible, but not big, monodromy representations

Let $\rho: \pi_1(S,s_0) \to GL(V)$ be the monodromy representation associated to a local system of $\mathbb Q$-modules $\mathbb V$ with $\mathbb V_{s_0} = V$. Let $H$ be the Zariski closure of the ...
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2answers
283 views

References for the moduli space of complex structures

I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex ...
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0answers
63 views

Curvature of vector bundles associated to holomorphic fibrations

Let $D=U\times \Omega$ in $\mathbb C^m\times\mathbb C^n$ be a pseudoconvex domain and $\phi$ is a strictly plurisubharmonic function on $D$. We suppose that $\phi$ is smooth up to the boundary. Now, ...
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1answer
226 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 = ...
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3answers
1k views

flatness in complex analytic geometry

It is always a pain to move back and forth between definitions in algebraic geometry and complex analytic geometry. Dictionary is much easier when are working with (family of) smooth varieties but the ...
4
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1answer
100 views

Singularities of the moduli stack of polarized hyperkahler varieties

Inspired by the recent question on singularities of the moduli stack of Calabi-Yau threefolds (Singularities of the moduli stack of Calabi-Yau threefolds) I'd like to ask the following question. Is ...
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1answer
161 views

Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory

I'm trying to read this paper of Bogomolov and Tschinkel http://arxiv.org/pdf/math/9902092.pdf about potential density of rational points on elliptic K3 Surfaces. I got quite stuck in Corollary 3.27 ...
4
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1answer
283 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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1answer
962 views

Primes that are the sum of three squares

This is in some sense an extension of the earlier MO question, "Gaussian prime spirals." Gaussian primes in the complex plane, $a+b i$, require $a^2 +b^2$ prime off the axes. The generalization to ...
6
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1answer
251 views

Singularities of the moduli stack of Calabi-Yau threefolds

Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack? In many cases I ...
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1answer
486 views

Must an algebraic variety with trivial tangent bundle be an abelian variety?

Suppose $X$ is a proper algebraic variety with trivial tangent bundle $T_X$ (not only canonical bundle $K_X$), is it true that $X$ is an abelian variety? (For the complex manifold case this is not ...
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0answers
84 views

Canonical metric of toric Kahler manifolds

Let $X$ be non-compact toric Kahler manifold associated to a Delzant polygon $P$ and $g$ be the canonical Kahler metric constructed by Guillemin. Is it true that the real part of $g$, as a ...
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2answers
419 views

Classification of complex structures on $\mathbb{R}^{2n}$

Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a ...
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0answers
52 views

Good covering of a (singular) curve in a complex surface

Let $W$ be a $2$-dimensional complex manifold and $C\subset W$ a compact complex curve (possibly singular). I would like to know a reference for the following fact: there exists a collection ...
2
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0answers
109 views

Simply connected Kahler manifold without any effective divisor

Does anyone know an example of a simply-connected compact Kahler manifold without an effective divisor? Does anyone know a reference on this topic? Thanks!
5
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2answers
214 views

Examples of surface automorphisms with no periodic points

Consider a smooth projective complex surface $S$ with an automorphism $g:S\to S$. A point $p$ is periodic if it has finite orbit under iterates of $g$. What are some examples of surface ...
3
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1answer
131 views

intersection of holomorphic curve with hyperplane

Let $f : \mathbb{C} \rightarrow \mathbb{C}^n$, $n>1$ be an entire function. Assume for simplicity that $f(0)=0$. Let $B$ be the closed ball of centre $O$ and radius $R$. Is there an upper bound ...
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1answer
101 views

How local is the exponent in the definition of a function with analytic/algebraic singularities?

In Demailly's Analytic Methods in Algebraic Geometry (available on his web page), the definition of a (plurisubharmonic) "function with analytic singularities" is a (plurisubharmonic) function $ u: ...
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3answers
1k views

When is a holomorphic submersion with isomorphic fibers locally trivial?

A justly celebrated theorem by Ehresmann states that a proper smooth submersion $\pi: X\to S$ between smooth manifolds is locally trivial in the sense that every point $s\in S$ downstairs has a ...
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2answers
927 views

A question about Marsden-Weinstein reduction theory

Let $G$ be a compact Lie group and $\frak g$ be its Lie algebra. Then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J \colon M\to \frak g^*$ be its moment map then the ...
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0answers
223 views

Algebraic K-theory of complex varieties

Maybe this question is trivial, but I was not able to find an answer. The question is this: Consider the algebraic K-theory of smooth complex projective varieties (such that the K-theory and the ...
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0answers
71 views

Compact locally conformal Kahler manifolds with non-zero Euler characteristic

I would like to know if there exist eight-dimensional compact manifolds such that: It has SU(4)-structure (and hence it is spin). It is locally conformal Kahler (and not Kahler). Its Euler ...
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74 views

Equivalence of holomorphic line bundles from Kahler potentials

Let $(M.\omega)$ be a Kahler manifold with fundamental form $\omega$. Then $\omega$ is closed and by the $\partial\bar{\partial}$-lemma on every contractible open set $U\subset M$ we can write ...