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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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87 views

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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1answer
174 views

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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1answer
247 views

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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0answers
61 views

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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1answer
211 views

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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0answers
61 views

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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2answers
398 views

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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1answer
52 views

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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0answers
26 views

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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1answer
315 views

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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421 views

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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122 views

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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0answers
107 views

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 ...
2
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2answers
191 views

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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144 views

(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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2answers
429 views

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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1answer
131 views

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 ...
4
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1answer
106 views

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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0answers
94 views

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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218 views

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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1answer
236 views

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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1answer
199 views

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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0answers
150 views

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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1answer
257 views

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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2answers
172 views

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 ...
5
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1answer
219 views

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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0answers
180 views

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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0answers
171 views

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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0answers
147 views

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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47 views

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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1answer
251 views

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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1answer
137 views

$S^{2}$-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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195 views

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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123 views

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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0answers
113 views

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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112 views

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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123 views

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 $\...
4
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1answer
132 views

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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166 views

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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0answers
110 views

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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0answers
240 views

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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2answers
266 views

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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125 views

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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135 views

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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1answer
169 views

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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0answers
117 views

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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0answers
101 views

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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0answers
71 views

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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0answers
109 views

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 ...