# 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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### 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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1 vote
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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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### 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 ... 106 views

### 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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### 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 $|K_X|$ ...
1 vote
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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 ... 126 views

### 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 ...
1 vote
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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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