# Questions tagged [complex-geometry]

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

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### complex K3 surfaces with automorphisms of given orders

Concerning complex K3 surfaces, there are various methods to show the non-existence of an automorphism of certain orders. The usually way is to investigate the action of the automorphism on the space $...

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### Cartan–Remmert reduction of an algebraic variety

Let $V$ be a normal connected algebraic (say, quasi-projective) variety over complex numbers. Assume that underlying complex analytic space $V^\text{an}$ is holomorphically convex, and thus admits the ...

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### Diffeomorphism problem for complex surfaces?

I'm sure the following is well known by the right people, I'm just hoping for some pointers. I know about Markov's theorem that the diffeomorphism problem for general 4-manifolds is undecidable.
Let $...

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### Examples of compact non-Kähler complex manifolds with Kodaira dimension zero

Let $X$ be a minimal compact non-Kähler complex manifold. Suppose that Kodaira dimension $\kappa(X)=0$.
Is there a known example where the canonical bundle is not holomorphically torsion?
For ...

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### Faithfully flat descent in complex analytic geometry

A common technique for constructing objects (sheaves) and morphisms in algebraic geometry is faithfully flat descent. Roughly speaking this consists on constructing an object or a morphism "...

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### Do we have an equivariant Newlander-Nirenberg theorem for finite group action?

Let $X$ be a $2n$-dimensional $C^{\infty}$-manifold, let $G$ be a finite group acting smoothly on $X$. Let $J: TX\to TX$ be a map such that $J^2=-id$ and
$g_*Jg^{-1}_*=J$ for any $g\in G$.
We can ...

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### Number of holomorphic line bundles with holomorphic sections for a fixed cohomology class

Let $X$ be a smooth compact Kähler (or more strongly projective) manifold and $\alpha$ an element of its Néron-Severi group. Let $\mathrm{Pic}^{\alpha}(X)$ denote the subset of the Picard group $\...

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### Is Stenzel's Ricci-flat metric on $T^*\mathbb{CP}^n$ hyperkahler?

In a well-known paper, Stenzel constructed complete Ricci-flat Kahler metrics on the total spaces of cotangent bundles of $S^n$, $\mathbb{RP}^n,$ $\mathbb{CP}^n$, $\mathbb{HP}^n$, and $\mathbb{OP}^2$.
...

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### What does mean módulo 2pi? [migrated]

I was reading a paper and it have a equation inside absolute value with a small 2pi on the right corner , the paper explains
|.|2pi denotes modulo 2pi

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### Prescribed intersection of varieties

Every variety here is complex analytic, or complex algebraic if it solves anything.
Given a germ of a (possibly singular, nor necessarily irreducible) hypersurface $(H,0)\subset(\mathbb{C}^{n+1},0)$ ...

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### Betti numbers of threefolds with trivial canonical class

I am interested in a simply-connected compact complex manifold $M$ of dimension three with trivial canonical class.
Note that if it is K"ahler, then it is a Calabi-Yau threefold.
Its independent ...

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### Does nefness in analytic setting depend on Hermitian metric?

I'm reading Demailly's book 'Analytic Methods in Algebraic Geometry'.
Let $X$ be a compact complex manifold with a Hermitian metric. A line bundle $L$ is said to be nef if for every $\epsilon>0$, ...

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### Minimal Betti numbers of simply-connected threefolds with trivial canonical class

By a threefold, I mean a compact complex manifold of dimension three.
For a simply-connected threefold with trivial canonical class, its Betti numbers satisfy:
$$b_2 \ge 0, b_3 \ge 2.$$
I am wondering ...

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### Cohomology rings of complex varieties and combinatorics

It is a classical fact that the cohomology ring (with complex coefficients) of a complex smooth projective manifold is a bigraded algebra satisfying (1) Poincare duality; (2) hard Lefschetz theorem; (...

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### Do we have a simple proof for this criterion for basepoint freeness?

The following is a criterion by Fujita (On the structure of polarized varieties with $\Delta$-genera zero).
Consider a complex smooth projective variety $X$ of dimension $n$ and an ample divisor $H$, ...

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### Why $H^2(X,\mathrm{End}(L))=H^2(X,\mathcal O_X)$?

$\DeclareMathOperator\End{End}$Let $X$ be a compact complex manifold, and $E$ be a holomorphic vector bundle over $X$.
In Chan & Suen's paper A differential-geometric approach to deformation of ...

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### Are two notions of generalized solution of Monge-Ampere equation equivalent?

Let $\Omega\subset \mathbb{C}^n$ be an open subset. Let $u\colon \Omega\to \mathbb{R}$ be a continuous plurisubharmonic (psh) function. The theorem of Chern-Levine-Nirenberg defines a non-negative ...

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### Determine the coefficient of the exceptional divisor

Consider the following setting: suppose that $X$ is a smooth variety and let $f:X\rightarrow \Delta$ be a smooth morphism outside the origin $0$. Let the central fiber $X_0$ be a reduced (Cartier) ...

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### Obstruction map for holomorphic line bundle $\operatorname{Ob}_L:H^1(X,\mathcal O)\to H^2(X,\mathcal O)$

$\DeclareMathOperator\Ob{Ob}$In Chan & Suen's paper A differential-geometric approach to deformation of pairs $(X, E)$ p.20, the authors define a Kuranishi map (or obstruction map) for the ...

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### Threefolds with the same Betti numbers and the same Chern numbers

By a threefold, I mean a compact complex manifold of dimension three.
My question is a simple one:
Are there known INFINITELY many non-homeomorphic threefolds that have the same Betti numbers and the ...

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### Étale cover of diffeomorphic projective manifolds

Let $f\colon X' \to X$ be an étale morphism of degree $>1$ between two complex projective manifolds. Suppose $X'$ and $X$ are diffeomorphic to each other and $f$ induces an isomorphism of $\mathbb{...

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### Relative automorphism groups of holomorphic Lagrangian fibrations

Let $p\colon X\to B$ be a Lagrangian fibration on an irreducible holomorphic symplectic manifold over $\mathbb C$. Assume that the base $B$ is smooth, hence $\mathbb P^n$. Define the relative tangent ...

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### Current progress on rationality problem for complex hypersurfaces

How is the current progress on rationality problem for complex hypersurfaces $X\subset\mathbb{P}^{n+1}$ with $n\geq 3$?
There are many hypersurfaces are shown to be unrational, such as smooth cubic ...

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### Complex manifold with conjugate complex structure

Let $(M,J)$ be a complex manifold with complex structure $J$. It is clear that $(M,-J)$ is also a complex manifold.
Under what condition is $(M,J)$ biholomorphic to $(M,-J)$?

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### Kähler metric with two compatible complex structures

Let $(M^4,g)$ be a complete $4$-dimensional Riemannian manifold such that two almost complex structures $I$ and $J$ are compatible with $g$ and $\nabla_g I=\nabla_g J=0$.
Can we prove that $(M,g)$ is ...

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### Do non-projective K3 surfaces have rational curves?

Define a compact Kähler surface $X$ to be a K3 surface if $X$ is simply connected, $K_X \simeq \mathcal{O}_X$, and $h^{0,1}=0$. If $X$ is projective, then a theorem typically attributed to Bogomolov ...

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### Sufficient condition for moduli space of slope-stable bundles to be non-empty

I'm trying to find to a statement concerning the non-emptiness of the moduli space of slope stable vector bundles over a Kähler surface in the literature.
Let $X$ be a Kähler surface. Let $\mathscr{M}(...

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### Uniformization of Riemann surfaces with cone singularities

Let $\Sigma$ be a Riemann surface (not necessarily compact), and $x_1, \cdots, x_k$ a set of points on $\Sigma$. Let $n_1, \cdots, n_k$ be a sequence of integers, each of which is $\geq 2$, and such ...

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### Cohomology of the base of an elliptic fibre space

Work over $\mathbb{C}$. Let $\Phi : X \to S$ be an elliptic fiber space, where $X$ is a smooth projective threefold with $H^1(\mathcal{O}_X)=H^2(\mathcal{O}_X)=0$, and $S$ is a smooth projective ...

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### Fundamental group of a smoothing of a complex surface

Let $X_0$ be a compact complex algebraic surface with an isolated singularity and let $X_t$ be a smoothing of $X_0$ over the disc. How can we compute the fundamental group of $X_t$ say in terms of the ...

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### Is the sign of the intersection between an effective divisor and a curve preserved via birational maps?

Let $f \colon X \to X'$ be a birational map of smooth complex projective varieties with nef
canonical bundles. Then the complement of the maximal open subset $U$ (resp. $U'$) where $f$ (resp. $f^{-1}$)...

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### Does $F_{A}^{0,2}=0$ for a connection $A$ on $TM$ almost complex give a complex structure?

First, some motivation. Let $X$ be a complex manifold, and $A$ a Hermitian connection on some complex vector bundle $E$ over $X$. It is known that the existence of $A$ such that the $(0,2)$-part of ...

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### Holomorphic function on $\mathbb C^n$ [closed]

I take $F$ from $\Omega\subset \mathbb C^n$ to $\mathbb C^n$ to be a holomorphic function such that
$$| \det(J_F)|\leq 1,$$
where $J_F$ is the Jacobian matrix of $F$.
My question: Is there any ...

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### When the whole space $H^2(X,\mathbb Q)$ can be represented by $c(L_t)$ of $X_t$?

Let $X$ be a compact complex manifold, $\pi:\mathcal X\to B$ be a holomorphic family of $X$ with $X_t=\pi^{-1}(t),t\in B$, and $X_0=X$.
Let $L_t$ be a holomorphic line bundle over $X_t$, then its ...

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### Resolving complexes of coherent analytic sheaves

Background
Throughout, let $X$ be a smooth complex manifold.
It is a classical fact that a coherent analytic sheaf admits a local resolution by locally free sheaves (also known as a local syzygy). ...

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### How do we define the type of a singularity on a cubic surface?

Nine different types of singularities are possible on a cubic surface, according to Wikipedia. How exactly is the "type" of singularity defined? I know that the number corresponding to the ...

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### Is there a non-singular cubic surface that has a point where four lines intersect?

Every non-singular complex projective cubic surface has $27$ lines. Many such surfaces contain points where three lines intersect (called Eckardt points). There are even surfaces with many Eckardt ...

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### Proper journal for a preprint in complex geometry

I ran into the very cryptic paper Proper analytic embedding of $\mathbb{C P}^1$ minus a Cantor set into $\mathbb C^2$ by Orekvov on proper holomorphic embedding of the complement of a Cantor set $C$ ...

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### Steenbrink spectral sequence and modifications of the central fibre

If $f: X \to S$ is a proper map from a complex manifold to a disc, $Y=f^{-1}(0)$ is a divisor with strictly normal crossings and the action of monodromy on $X_t=f^{-1}(t)$ for some (hence any) $t \neq ...

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### Complex manifolds whose tangent and cotangent bundles are isomorphic as complex vector bundles

Question. What are examples of compact complex manifolds whose tangent and cotangent bundles are isomorphic as complex vector bundles?
A family of examples are, of course, holomorphically symplectic ...

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### Representing homotopy classes of Kähler manifolds by harmonic maps

Let $(M,g_M)$ be a compact Kähler manifold with negative bisectional curvature. Let $\alpha : (S,g_S) \to M$ be a continuous map from a compact Riemannian manifold $(S,g_S)$.
Is $\alpha$ homotopic to ...

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### topological Euler characteristic of canonical divisor

Let $X$ be a complex smooth projective variety with trivial topological Euler characteristic $\chi_{\text{top}}(X)=0$. We assume that $D$ is a smooth irreducible divisor in the linear system $|K_X|$ ...

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### One-dimensional family of complex algebraic K3 surfaces

Let $X$ be an algebraic complex K3 surface, we know that $X$ is deformation equivalent to a smooth quartic surface or more generally a K3 surface with Picard number $1$ (a very general K3 surface in ...

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### Existence of covering isomorphism

Let $C,D$ be two non-compact complex algebraic smooth curves. Suppose that two unramified regular finite maps $p_1, p_2: C \rightarrow D$ are given and have the same degree. Is there always an ...

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### Non-Kähler pseudo-Kähler manifolds

A pseudo-Kähler manifold is a complex manifold $(X, I)$ endowed with a non-degenerate closed $(1, 1)$-form $\omega$. In that case, the symmetric tensor $g(\cdot, \cdot) = \omega(\cdot, I \cdot)$ is a ...

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### Vector subbundles of a given one in $\mathbb{CP}^1$

I apologize if this question is not suited for MathOverflow. This has been crossposted in MathStackExchange here and it is related to some open questions on that site that remain unsolved.
I would ...

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### An inf-sup estimate for holomorphic functions

Is the following true?
Conjecture? Let $U \subset \mathbb{C}^n$ be open and $\eta : U \to \mathbb{C}$ be holomorphic. Denote by $B(z,r)$ the usual ball of radius $r$. There is a constant $\kappa<\...

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### Request for Acta Math Sinica 1984 paper

The mathscinet reference for the paper I am after is here:
MR807424 53C55 (32H99)
Chen, Zhi Hua; Yang, Hong Cang Estimation of the upper bound on the Levi form of the distance function on Hermitian ...

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### Riemann uniformization theorem (limit case)

Let $\mathbb D_r=\{z\in\mathbb C:|z|\le r\}$ be the closed unit disk of radius $r$,
let $\mathring {\mathbb D}_r=\{z\in\mathbb C:|z|< r\}$ be its interior,
and let $\mathbb A_r=\mathbb D_r\setminus ...

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### Irreducible components of a general singular fiber correspond to irreducible components of the hypersurface consisting of singular fibers

I already asked this on math.SE, but didn't receive any response.
The following question arose when studying Hwang and Oguiso's Characteristic foliation on the discriminant hypersurface of a ...