**3**

votes

**0**answers

126 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 ...

**3**

votes

**0**answers

264 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 ...

**1**

vote

**1**answer

148 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) ...

**0**

votes

**1**answer

160 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 ...

**2**

votes

**0**answers

70 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**

votes

**0**answers

81 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. ...

**2**

votes

**0**answers

64 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**

votes

**0**answers

150 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 $X_k$ ...

**1**

vote

**0**answers

63 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 ...

**0**

votes

**0**answers

86 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

**1**answer

101 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

**1**answer

135 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 ...

**3**

votes

**2**answers

296 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 ...

**5**

votes

**0**answers

203 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 ...

**1**

vote

**1**answer

173 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 ...

**1**

vote

**0**answers

56 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 ...

**8**

votes

**1**answer

274 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**

votes

**0**answers

114 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
...

**1**

vote

**0**answers

123 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 ...

**0**

votes

**0**answers

33 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 ...

**1**

vote

**1**answer

143 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 ...

**0**

votes

**1**answer

241 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 ...

**0**

votes

**1**answer

100 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 ...

**0**

votes

**0**answers

66 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" ...

**3**

votes

**2**answers

214 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 ...

**5**

votes

**2**answers

327 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 ...

**0**

votes

**0**answers

66 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, ...

**4**

votes

**1**answer

113 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 ...

**1**

vote

**1**answer

174 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 ...

**6**

votes

**1**answer

265 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 ...

**1**

vote

**0**answers

94 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 ...

**15**

votes

**1**answer

531 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 ...

**16**

votes

**2**answers

452 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 ...

**0**

votes

**0**answers

57 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**

votes

**0**answers

114 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**

votes

**2**answers

229 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**

votes

**1**answer

141 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 ...

**0**

votes

**0**answers

77 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 ...

**1**

vote

**1**answer

161 views

### generalization of fundamental theorem of algebra for several complex algebra [closed]

I am looking for a generalization to fundamental theorem of algebra for several complex variables functions or systems. If such theorem exists, it should concisely relates the number of zeros of ...

**0**

votes

**0**answers

83 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
...

**1**

vote

**0**answers

231 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 ...

**2**

votes

**2**answers

179 views

### Connected complement manifold

I'm working on some problem in algebraic geometry. I need a reference to the following result:
Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$
be a non ...

**0**

votes

**0**answers

60 views

### How exactly do we construct the $T^2\times \mathbb{R}$ toric Calabi-Yau three-fold?

I am trying to understand why and how the functions $r_{a}(z) = |z_1|^2 - |z_3|^2$, $r_{b}(z)=|z_2|^2 - |z_3|^2$ and $r_{c}(z)=\Im(z_1z_2z_3)$ "generate" the toric CY threefold $T^2 \times \mathbb{R}$ ...

**3**

votes

**0**answers

115 views

### Writing down gerbes explicitly over the projective line

Let $X = [\mathbb P^1/(\mathbb Z/2\mathbb Z)]$, where we take the trivial action of $\mathbb Z/2\mathbb Z$ on $\mathbb P^1$. Is this DM stack over $\mathbb C$ a gerbe over $\mathbb P^1$? Is it the ...

**1**

vote

**1**answer

104 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: ...

**3**

votes

**0**answers

177 views

### Does the higher cohomology of a quasi-coherent sheaf on a Stein manifold vanish?

It is a well-known result in algebraic geometry that if $X$ is an affine scheme and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then the higher cohomologies of $\mathcal{F}$ vanish, i.e.
$$
...

**0**

votes

**0**answers

48 views

### Optimal bound in $L^2$ product on compact Kahler manifold

Let $X$ be a compact Kahler manifold of dimension $n$, equipped with a Kahler metric of volume $1$. There exists a constant $C \geq 1$ such that for any smooth functions $f,g$ on $X$ we have
$$
\int_X ...

**2**

votes

**1**answer

194 views

### Smoothing transverse intersections

Let $S$ be a complex surface with ample canonical class. Let $C_1$ and $C_2$ be smooth complex curves in $S$ that intersect transversally at $n $ points. Furthermore, assume that the self-intersection ...

**1**

vote

**0**answers

87 views

### holomorphic curves invariant by lattices

Suppose I have an entire function $f : \mathbb{C} \longrightarrow \mathbb{C}^n$ for
$n \geq 1$.
Let $C$ be the curve $f(\mathbb{C})$ in $\mathbb{C}^n$.
Let $\Lambda$ be a lattice in $\mathbb{C}^n$ ...

**0**

votes

**1**answer

190 views

### Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$

I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19):
Wirtinger's Inequality.
Let $L$ be a complex linear space and let $M$ be a real
even-dimensional ...