# 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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### What are we to deduce from a structure theorem of this type concerning totally geodesic maps?

I apologise in advance for the vague nature of the question, but some insight would be greatly appreciated.
I'm reading a paper of Lei Ni concerning structure theorems for Kähler manifolds. Here is an ...

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### Are there known examples of almost complex manifolds admitting neither a symplectic nor a complex structure?

I have seen the the example of $S^6$ being touted around here and there but it does not seem to be generally confirmed that there is no complex structure on it.

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### do cohomologically Kähler classes extend to Kähler classes?

Let $f: X \to S$ be a proper morphism from a complex manifold to a small disc which is smooth away from $Y = f^{-1}(0)$, an snc divisor. A class $\omega \in H^2(Y)$ is called cohomologically Kähler if ...

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### Non Abelian Hodge theory: underlying structure holomorphic vector bundles

Let $X$ be a compact Riemann surface. We fix a complex vector bundle $E$ of rank $n$ and degree $d$ (unique up to diffeomorphism). From results coming originally (I think at least) by Simpson,...

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### Union of all the entire curves in a complex manifold

Let $X$ be a connected closed complex manifold. Let $S$ be the set of non-constant holomorphic functions $f:\mathbb{C}\to X$.
If $\bigcup\limits_{f\in S}f(\mathbb{C})$ is a proper subset of $X$ can it ...

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### Is there a “minimal” Whitney stratification of a complex hypersurface?

Let $X\subset \mathbb C^n$ be a complex hypersurface (given by $F=0$ where $F$ is a polynomial). It is known then that $X$ admits a Whitney stratification. This is a decomposition of $X$ into smooth ...

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### Smooth toric compactification of $\mathbb C^n$

By a compactification $(X,Y)$ of $\mathbb C^n$, we mean an irreducible compact complex space $X$ and a closed analytic subspace $Y\subset X$ such that $X\setminus Y$ is biholomorphic to $\mathbb C^n$. ...

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### Is the Ueno fibration smooth?

Let $A$ be an abelian variety over $\mathbb{C}$ and let $X\subset A$ be a closed subvariety. Let $X\to Y$ be the Ueno fibration. (That is, $Y$ is of general type and a closed subvariety of $A/B$ where ...

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### Whitney stratification of a $\mathbb C^*$-invariant hypersuface in $\mathbb C^n$

Let $H\subset \mathbb C^n$ be an irreducible hypersurface invariant under a diagonal $\mathbb C^*$-action with positive weights ($H$ is given by a quasi-homogeneous polynomial). Consider the Whitney ...

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### Existence of coframe for Hermitian metric on complex manifold?

I am reading page 28 of the 1994 version of Principles of Algebraic Geometry by Griffith. Let $M$ be a complex manifold of dimension n, Griffith defined a Hermitian metric to be a positive definite ...

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### Computing the invariants of ball quotient surfaces

The two-dimensional complex unit ball $B$ has group of biholomorphic automorphisms $PU(2,1)$.
If $Γ$ is an arithmetic subgroup of $PU(2,1)$, the quotient $Γ\text{\\}B$ is an orbifold.
Taking its ...

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198 views

### Rigidity lemma up to cover

Let $X,Y,Z$ be [Edit: normal] proper varieties over $\mathbb{C}$. Let $W\to X\times Y$ be a finite flat surjective morphism, and let $W\to Z$ be a morphism.
Fix $x \in X$ and suppose that $\{x\}\times ...

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218 views

### Possible number of components of anticanonical sections of projective manifolds

Let $M$ be a connected projective complex manifold with a smooth anticanonical divisor $D$ ($D \sim -K_M$).
Let $k$ be the number of components of $D$.
Some cheap thoughts give:
If $M$ is a Fano ...

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106 views

### Classifying spaces for holomorphic bundles

I am not very experienced in the topic so the question may be naive.
Is there a way to "homotopically" classify holomorphic G-bundles for a complex Lie group G?
So the first subquestion: is ...

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184 views

### Historical proof of Leschetz Hyperplane Theorem

I browse in Phillip Griffiths' Slides
on historical development of
Hodge-theory and these include a sketch of the original approach
with Lefschetz used to study complex surfaces in his famous
...

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115 views

### Oka-Grauert principle, up to the boundary

Let $Z\subset \mathbb{C}^n$ a domain of holomorphy with smooth boundary $\partial Z$ and closure $\bar Z$. There is a natural notion of holomorphic vector bundle over $\bar Z$, given in terms of ...

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### Holomorphic sections to anti-holomorphic sections

Let $X$ be a compact Kähler manifold and $L$ be a holomorphic line bundle on $X$ with a Hermitian metric $h$. I am trying to give a norm preserving isomorphism between the space of holomorphic ...

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### Volume of singular Kahler metric

Let $X$ be a compact complex manifold of complex dimension $n$ and let $\omega$ be a smooth Kahler form on it. Let $Y \subset X$ be a complex (possibly singular) hypersurface and let $u: X \setminus Y ...

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### Geometry of elements with prescribed multiplicity eigenvalues

Let us take $G=\operatorname{Gl}(n,\mathbb{C})$ (considered as a linear algebraic group). Let us take $x \in G$: we know that its orbit $\mathcal{O}_x$ under the conjugation action is isomorphic (as ...

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### Quaternion-Sasakian manifolds and special holonomy Sasakian manifolds

Two well-known slogans are
A Sasakian manifold is the odd dimensional analogue of a Kähler manifold
and
A $3$-Sasakian manifold is the odd dimensional analogue of a hyper-Kähler manifold
Does this ...

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440 views

### Non-Abelian Hodge theory

Let $X$ be a compact Riemann surface. I would like to find a somehow complete reference for the proof of the so called non-Abelian Hodge correspondence relating Dolbeaut, Betti and Higgs bundle moduli ...

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121 views

### Localizing the intersections of cubics

For Hermitian matrices $A,B \in \mathbb{C}^{n \times n}$, can one readily compute a set of cones that separate the maxima of $$\frac{x'Ax}{x'Bx}$$ among $x$ with unit-norm components?
i.e. where do ...

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### Estimate for the diameter of the image of a holomorphic disk by the area of the holomorphic disk

Let $(M, J, g)$ be a compact almost complex manifold with a Riemannian metric $g$ that preserves the almost complex structure $J$. I want to prove that a holomorphic disk $u: D^2\to M$ of a small area ...

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356 views

### Comparison of two monodromies

Let us consider a smooth projective curve $\Sigma_b$ of genus $b$, and assume that there is a surjective group homomorphism $$\varphi \colon \pi_1 (\Sigma_b \times \Sigma_b - \Delta) \to G,$$ where $\...

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### Koszul-Malgrange theorem on non-compact manifolds

Does the Koszul-Malgrange Theorem hold on non-compact manifolds?
That is, given a connection on a complex vector bundle over a non-compact complex manifold with a connection whose curvature has ...

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### How do I remember which power of the Lefschetz operator $L$ corresponds to the $k$th Primitive cohomology group?

Let $X$ be a compact Kähler manifold with $L$ denoting the Lefschetz operator $L(\bullet) = \bullet \wedge \omega$. The primitive cohomology groups are defined, for $k \in \mathbb{N}$, by $$P^k(X, \...

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### Quaternonic Kähler Chern connection

For a Riemannian manifold, the natural connection is of course the Levi-Civita connection. For a complex manifold, the natural connection is the Chern connection, which coincides with the Levi-Civita ...

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### Lifting holomorphic automorphisms along the principal bundle

I have a holomorphic principal bundle,
$$E\xrightarrow{H} B$$ defined by an action of a contractible (non-compact) Lie group $H$ (in my case $H\cong\mathbb{C}^l$). Here E and B are complex manifolds, ...

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### Mixed Hodge structure cohomology of fibration

Let $X$ be a smooth complex algebraic variety. From Deligne's work, we know that the have a Mixed Hodge structure over its (rational) compactly supported cohomology $H^{*}_c(X,\mathbb{Q})$. With this, ...

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### Existence of uniformly bounded Darboux chart

In Donaldson's paper Symplectic submanifolds and almost-complex geometry, he mentioned that for each point $p$ in a compact almost-Kähler manifold $(V,\omega ,J)$, there exists a Darboux chart $\...

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### Invariant ideal generated by invariant elements

Let $G$ be a complex reductive group acting linearly on $\mathbb{C}^n$ and let $X$ be a $G$-invariant closed subvariety of $\mathbb{C}^n$. Is $X$ the zero-set of finitely many $G$-invariant functions?
...

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### Hartogs' theorem for real-analytic subvarieties

One version of Hartogs' extension theorem is the following (see, e.g. [1], Theorem 5B, p. 50).
Theorem. Let $U \subset \mathbb{C}^n$ be open and let $X \subset U$ be a complex-analytic subvariety of ...

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### The period map and the Kodaira--Spencer map

Let $f : X \to B$ be a non-isotrivial holomorphic submersion with connected fibres between compact Kähler manifolds of positive relative dimension. Suppose the fibres of $f$ are Calabi--Yau $(c_{1,\...

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### Behaviour of canonical divisor under a glueing of locally birational maps

Let $Y$ be a smooth projective variety with a finite affine cover $\{U_1,...,U_r \}$. Suppose that we have a family of birational maps $\pi_i:V_i \to U_i$ with $V_i$ smooth quasi-projective for each $...

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### Holomorphic sectional curvature and Kobayashi hyperbolicity

Let $(M,g)$ be a compact Hermitian manifold. Let $\text{HSC}(g)$ denote the holomorphic sectional curvature of $g$. The implication $$\text{HSC}(g) < 0 \implies M \ \text{is Kobayashi hyperbolic}$$ ...

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### Boundary Maslov index of holomorphic disks in Calabi-Yau manifolds

Let $u \colon \Sigma^2 \to M^{2n}$ be a holomorphic disk (so $\Sigma = \{z \in \mathbb{C} \colon |z| \leq 1\}$) in a compact Calabi-Yau manifold $M$ of real dimension $2n$ with boundary on a ...

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### Saturation of sheaves

Let $(X, \mathcal{O}_X)$ be a complex manifold, which we can take to be projective. A coherent subsheaf $\mathscr{F}$ of some sheaf $\mathscr{G}$ is said to be saturated in $\mathscr{G}$ if the ...

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### blow up of a log resolution

Suppose that $(X, \omega)$ is a Kähler manifold with a semi-ample canonical line bundle(meaning there exists $m$ such that $mK_{X}$ is base point free). Then we have a canonical map $\Phi: X \to \...

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### Branched covers of the sphere branched over few points

Let $X$ be a compact Riemann surface of genus $g\geq 2$. By the Riemann-Roch theorem, $X$ is a branched cover of the sphere, branched over finitely many branched values. What is the smallest number of ...

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### Change in Connection on a complex Line bundle

Let's say $M$ is a compact Kähler manifold and $L$ is a complex line bundle on $M$. Now let's say $A$ be a connection or equivalently a hermitian metric on $L$. Hence one can have the operators
$\bar\...

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### Looking for a criterion for a subset of a complex variety to be of measure zero

Suppose $f:X\longrightarrow Y$ is a surjective morphism of smooth complex varieties. Let $S$ be a subset of $X(\mathbb{C})$. I'm wondering if there are results that roughly say that if the ...

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### Examples of complex manifolds for which the logarithmic cotangent bundle is big, but the cotangent bundle is not big

Let $(X,D)$ be a log pair, with $X$ a projective manifold (or quasi-projective) and $D$ a divisor with simple normal crossings.
I'd like to construct an example, or be pointed to a reference, for an
...

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### An integration identity on $\mathbb{P}^{n-1}$

Let $\omega_{\text{FS}}$ denote the Fubini–Study metric on $\mathbb{P}^{n-1}$ with unit volume, and let $[w_1 : \cdots : w_n]$ be standard unitary homogeneous coordinates. On page 5 of Yang–Zheng's ...

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### Expansion around a singular point of a multivalued meromorphic function (due to Riemann/Cauchy)

In Riemann's publication about Abelian functions
'Theorie der Abelschen Functionen' (Here the original paper in german)
at the end of Chapter 4, part 2 is clamed that for every Riemann
surface $T$ and ...

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### Are smooth Schubert varieties Kähler? [closed]

Schubert variety $V$ is a special type of (possibly singular) subvarieties of a Grassmannian. Since the Grassmannians are Kähler manifolds (in fact projective varieties) are we able to conclude that ...

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### Analytic or holomorphic extension of the ellipse perimeter function

Let ${\mathbb{R}^2}^+=\{(x,y)\in \mathbb{R}^2\mid x>0, y>0\}$.
Let $P:{\mathbb{R}^2}^+\to \mathbb{R}$ be the function with $P(a,b)=$ $\text{The perimeter of ellipse}\;\; \frac{x^2}{a^2}+\frac{y^...

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### Hyperbolic 3-manifolds inside algebraic varieties

I have a hyperbolic 3-manifold $M$ that I'd like to see sit inside a complex algebraic variety $V$ as beautifully and snugly as possible. I don't want to specify attributes of "beauty" (e.g.,...

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### Approximation of a complex manifold by an algebraic variety

What are some natural notions of distance $d$ between two complex manifolds of dimension $n$? For any of these notions what are the current best results on approximation of a complex manifold $M$ by a ...

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### Why does the bisectional curvature blow up?

Suppose we have $X = X_1 \times X_2$, where $X_1$ is a Calabi-Yau manifold (i.e. $c_1(X_1) = 0$) and $X_2$ is a compact Kähler manifold of $c_1(X_2) < 0$. We can consider the metric $\omega(t, x_1, ...

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### Flatness of holomorphic maps

Let $X,Y$ be two normal complex spaces of same dimension and $\phi:(X,\mathcal{H}_X) \longrightarrow (Y,\mathcal{H}_Y)$ be a holomorphic map such that the morphism $\mathcal{H}_Y \longrightarrow \phi_*...