# Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties, and, by extension, of almost complex structures. It is a part of both differential geometry and algebraic geometry.

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### Kelly's theorem for quadratic polynomials

Let $f_1, \ldots, f_m$ be homogeneous irreducible quadratic polynomials in $\mathbb{C}[x_1, \ldots, x_n]$.
Assume that these polynomials are pairwise coprime.
Denote $P:= f_1 \cdot f_2 \ldots \...

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### Continuous function on a complex space that is holomorphic on the complement of a closed subspace

Let $X$ be a complex analytic space and $Y\subseteq X$ a closed complex subspace. Suppose that $f:X\to\mathbb{C}$ is a continuous function that is holomorphic on $X\setminus Y$. Is $f$ holomorphic on $...

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### Holomorphic line bundles with torsion Chern class [closed]

Suppose you have a holomorphic line bundle $L$ such that $L^{n}$ is a trivial holomorphic line bundle and the base complex manifold $M$ has no torsion cohomology classes in second degree (i.e. $H^{2}...

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### Rationality of the moduli space of genus g curves

I'm not an expert in this topic, so please excuse my negligence. I'd also appreciate references to the literature. Throughout, I will work over the complex numbers, although the analogous questions ...

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### Status of global spherical shell conjecture for minimal complex surfaces?

A class VII surface is a compact complex surface $M$ such that $b_1(M)=1$ and $kd(M)=-\infty$. Class VII surfaces with vanishing second Betti number have been classified by Bogomolov (and are either ...

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### Kähler form on complex projective algebraic variety [closed]

I am not very familiar with the notion of projective algebraic varieties, I work mostly from an algebraic topology/differential geometry point of view, but I am trying to find a prove for the ...

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### Does there exist a leaf of this holomorphic foliation with non trivial holonomy?

Let's $\mathcal{F}$ be the holomorphic foliation of $\mathbb{C}^2$ tangent to the kernel of $\alpha=(sin x) dx -(cos x)dy$.
Are all leaves of $\mathcal{F}$ simply connected? If the answer is no, ...

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### Embedding of a complex submanifold in projective space

Suppose you have a projective manifold $M$, a very ample bundle $\scr L$ and a transverse holomorphic section $s \in H^0(\scr L)$. Then the zero set of $s$ is a complex submanifold $S_M$.
Can we ...

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### Set of sections whose zeroes avoid a given divisor is (Zariski) dense?

Let $X$ be a smooth complex projective variety of dimension $n$, and let $\mathcal{F}$ be a globally generated rank $n$ vector bundle on $X$. Let $D$ be a smooth divisor on $X$.
Is it true that ...

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### What separates a cyclic polytope from a projective polytope?

I am having trouble understanding the difference between a cyclic polytope and a convex projective polytope as positive geometries.
The link https://arxiv.org/pdf/1703.04541.pdf is the source of ...

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### Are there non-projective, but algebraic, hyperkahler varieties?

Let $k$ be an algebraically closed field of characteristic zero. I am not sure what the right definition of a hyperkahler variety over $k$ is, but I think the following might be close enough.
...

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### The Ricci Form and the First Chern Class

Let $(M, \omega)$ denote a compact Kähler manifold. Since $d\omega =0$, $\omega$ represents a cohomology class in $H^2(M, \mathbb{R})$. Let $\rho$ denote the Ricci form of $M$, in local coordinates, ...

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### Structure of the Kähler cone

In Calabi's Extremal kahler metrics paper, MR0645743, on page 262, the author mentioned that "It is conjectured that the structure of Kähler cone is determined by a finite number of real analytic ...

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### Degeneration of a metric

I want to understand how the metric degenerate on a family of projective varieties (mainly for abelian varieties.).
Let $X$ be a smooth projective variety over $\mathbf{C}$.
Let $B$ be a smooth ...

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### When is the space of holomorphic sections of the tensor product of two line bundles given by the span of the tensor product of the basis?

Let $S$ be a compact complex manifold and $L_1, L_2 \longrightarrow S$
be two holomorphic line bundles. Under what conditions (hopefully something that is easy to check) on $L_1$ and $L_2$ is the ...

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### (Real) holomorphic vector fields on compact Kähler manifolds

I am trying to prove Proposition 2.1.1 of Gauduchon's note on Kähler extremal metrics (page 67). In order to show that, for compact Kähler manifolds, the complex Lie algebra of real holomorphic vector ...

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### Criteria for a coherent sheaf pushing forward from the universal cover

I would like to prove (or find a counterexample to) the following statement:
Let $X$ be a complex analytic scheme and let $\pi: Y \to X$ be its universal cover. Let $F$ be a coherent sheaf on $X$ and ...

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### Can an algebraic variety over a field $k$ be the union of proper closed subsets $(S_i)_{i\in I}$ with $I < k$

Let $k$ be an algebraically closed field (of characteristic zero, if it helps).
Let $X$ be an algebraic variety over $k$. Let $I$ be an index set such that the cardinality of $I$ is smaller than the ...

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### Examples of surfaces with negative Kahler curvature operator

Compact ball quotients are examples of compact Kahler surfaces with negative curvature operator.
Are there any other examples ? What about nonpositive (other than the product of two Riemann surfaces ...

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### Extending polarization (or Kähler metric) from the central fiber to the nearby fibers (or the total space)

Let $\pi: \mathcal{X}\to B$ be a complex analytic family of compact complex manifolds, i.e. $\pi$ is a surjective, proper submersion between complex manifolds. For simplicity, we assume $B$ is the ...

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### Siu's arguments on Calabi-Yau theorem?

In Siu's lecture note Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, he shows the $C^0$ and $C^2$ estimates of the complex Monge-Ampère equation on a Riemannian ...

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### About the isotriviality of pencils of plane curves

Let $F$ and $G$ be coprime complex homogeneous polynomials in three variables of the same degree $d\geq 4$. Suppose that a general member of the pencil $\{F+tG=0\}\subset \mathbb{P}^2$ is smooth.
...

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### Obstructions to simultaneous resolution

The following question on simultaneous resolutions is a follow-up to earlier questions posed here (e.g. Resolution of singularities for flat families.).
What I'm interested in is an "obstruction ...

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### Can one construct a ramified cover over prescribed divisors?

Suppose $X/\mathbb C$ is a smooth projective variety of dimension $d>1$, and $D \subset X$ is a simple normal crossing divisor. For any $k>0$, can I construct a ramified $K$-th cover $f: \tilde ...

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### Can an abelian variety dominate a variety of general type?

Let $X$ be a projective (not necessarily smooth) normal variety of general type over $\mathbb{C}$. Let $A$ be an abelian variety and let $A\to X$ be a surjective morphism.
Is $X$ zero-...

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### Cycle class of zeroes of a global section

Let $\mathcal{F}$ be a locally free sheaf of rank $n$ on an $n$ dimensional complex manifold $X$. If the zero locus of a generic global section of $\mathcal{F}$ is $0$ dimensional, then its cycle ...

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### The logarithm of Kähler metric is not globally defined

In reducing the existence of Kähler-Einstein metrics to the complex Monge Ampere equation, the logarithm $$-\log \det (\omega + \partial \overline{\partial} \phi)$$ appears, where $\omega$ is a Kähler ...

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### Dolbeault cohomology of complex manifolds

Let $X$ be a compact complex manifold of dimension $n$ and let $D$ be a smooth hypersurface. Then the restriction map gives rise to a homomorphism in Dolbeault cohomology:
$$
\rho:H^{p,q}_{\bar{\...

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### Differential topology on arbitrary fields

What do the differential topology theories on arbitrary fields have in common?
Different differential topology theories
There is "ordinary" differential topology on real manifolds, with its rich ...

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### How to see the Hodge filtrations vary holomorphically directly

I have asked this question on Stack Exchange, "Why the Hodge filtrations vary holomorphically", but I have not got any reply, therefore I have revised this question a little bit, and post it here on ...

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### Do we have a $4$-term decomposition of $\bar{\partial}_M$ for a holomorphic fiber bundle $M\to B$?

First let us consider a Riemannian fiber bundle, i.e a fiber bundle $\pi: M\to B$ of oriented Riemannian manifolds. We denote by $T(M/B)$ the bundle of vertical tangent vectors and assume that the ...

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### A question on Steenbrink's paper, limit of Hodge structures

Steenbrink in his paper "Limit of Hodge Structures", (supplemented by the book "Mixed Hodge Structures" by Peters and Steenbrink) discuss the limit mixed Hodge structures for a fibration over the unit ...

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### $S^{2}$-bundles over complex projective varieties

Is there an example of a smooth complex projective variety and an $S^{2}$-bundle over it which is not diffeomorphic to a complex projective variety?

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### On the Choice Content of Carathéodory's Conformal Mapping Theorem

The Schoenflies theorem, as a variant of the well-known Jordan curve theorem, states that the interior and the exterior planar regions determined by a simple closed curve (aka Jordan curve) in $\...

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### Do non-constant maps specialize to non-constant maps?

Let $R$ be a dvr with fraction field $K$ and residue field $k$. Let $\mathcal{X}\to \mathcal{Y}$ be a morphism of $R$-schemes such that $\mathcal{X}_K\to \mathcal{Y}_K$ is non-constant.
Is the ...

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### Criterium for algebraicity of an analytic map

Let $X$ and $Y$ be algebraic varieties over $\mathbb{C}$. Let $f:X^{an}\to Y^{an}$ be a holomorphic map.
Is the following statement correct?
If there is an algebraic variety $V$ over $\mathbb{...

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### Do the adjoints of the Lefschetz operators always commute?

Let $M$ be a projective nonsingular complex variety. Let $I$ and $J$ be two complex structures on M. We then have the corresponding Kälher classes $\omega_I$ and $\omega_J$ in $H^2(M, \mathbb{R})$, ...

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### Biholomorphic maps between cotangent bundles with non-standard complex structures

Let $X$ be a compact Kähler manifold. Let $\omega_i$ (i=1,2) be Kähler forms on $X$. Assume that $\psi:X\rightarrow X$ is a diffeomorphism such that $\psi^*\omega_2=\omega_1$.
Recall that each $\...

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### Equivalence of complex structures on flag manifold

Let $G$ be a compact Lie group and $T$ a maximal torus of $G$. One way to construct a complex structure on $G/T$ is to choose a nilpotent subalgebra $\mathfrak{n}^+$ corresponding to some choice of ...

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### A general definition of an equisingular family of singular varieties?

This question is about the existence of a definition. I'm far from being an expert in the field in question I apologize in advance for any inaccuracies or stupid and wrong assumptions.
Let $X$ be a ...

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### Lagrangian foliation for a holomorphic symplectic manifold

I am interested in gathering as many examples as possible for Lagrangian foliations of holomorphically symplectic manifolds $(X, \omega)$, where $X$ is a $2n$-dimensional complex manifold equipped ...

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### Proof of Chow's theorem in special case

I'm trying to prove Chow's theorem (closed analytic subvarieties of projective space are algebraic) in a special case by the following differential geometric method (imitating the argument for the ...

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### What is the “dual” of the space of currents?

On a smooth maniflod $M$ of dimension $n$, a current of degree $n-p$ is a functional on the space of compactly supported differential $p$-forms which is continuos. We denote the space of currents of ...

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### The algebraic structures on $H^{1,1}(X,\mathbb C)$ induced by Kahler classes

Let $X$ be a compact Kähler manifold of dimension $n$. Each Kähler class $\omega$ on $X$ defines an adjoint Lefschetz operator $\Lambda$, and using this we can make $H^{1,1}(X,\mathbb C)$ into an ...

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### Classifying $PGL(n,\mathbb{C})$-bundles over a compact Riemann surface

Let $X$ be a connected compact Riemann surface. How does one go on proving that the set of PGL($n,\mathbb{C}$)-bundles over $X$ is topologically classified by $\pi_1(PGL(n,\mathbb{C}))$? Is it true ...

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### Reference to the conjecture about injectivity of Abel-Jacobi map

Suppose $k$ is a number field, and $\sigma:k \rightarrow \mathbb{C}$ is an embedding. Then there is the (generalised) Abel-Jacobi map
\begin{equation}
\text{CH}^j(X)_0 \rightarrow \frac{H^{2j-1}((X \...

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### Distribution of random hyperplanes in projective spaces

Let $X\subset \mathbb{CP}^{N-1}$ be a smooth subvariety of dimension $n$. Assume that $X$ is not contained in a hyperplane of $\mathbb{CP}^{N-1}$.
Let $\mu$ be a smooth probability measure on $X$. ...

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### Explanation of proposition 6.7 (a) of Fulton's Intersetion Theory

Suppose $X$ is a smooth variety over a field $k$ of characteristic zero, and $Z$ is a smooth subvariety of codimension d. Now let $\tilde{X}$ be the blow-up of $X$ at $Z$, and let the exceptional ...

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### Translates of a line bundle on a complex $n$-torus

Suppose $\mathbb T:=V/\Gamma$ is a complex $n$-torus (i.e., $V$ is an $n$-dimensional $\mathbb C$-vector space and $\Gamma$ is a rank $2n$ lattice in $V$). Fix a holomorphic line bundle $L\in\text{Pic}...

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### Complex algebraic submersions

Let $X$, $Y$ and $Z$ be smooth complex algebraic varieties and let $f:X\to Y$ and $g:X\to Z$ be two morphisms. Suppose that $f$ is surjective, that $df_x:T_xX\to T_{f(x)}Y$ is surjective for all $x\in ...