# 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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### Complex Monge-Ampere equation with degenerate right hand side

Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation:
$(\omega_0 +i \partial \bar \partial \varphi)^...

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### Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide

There are two definitions of intersection multiplicity of two complex algebraic curves. One is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t^k) )$ be the local ...

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### Examples of Stokes data

I'm trying to learn Stokes data but can't find an example to get my teeth into it.
Background. It's well-known that on a complex manifold $X$, there is the Riemann Hilbert equivalence
$$\text{regular ...

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### minimizing weighted length of closed curves

Let $\mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $\mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at ...

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### Equivariant resolution of singularities with equivariant centres

From what I understand, given a complex projective variety X inside a compact complex manifold Y, according to Hironaka, there is a sequence of $r$ blowups $Y_i$ of Y along complex submanifolds (...

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### Stokes's Theorem with singularities on projective line

Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...

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### where is the condition class $C^{1}$ used in the definition of current of integration? [closed]

Here is the usual definition of current of integration.
Let $Z \subset M$ be a closed oriented submanifold of $M$ of dimension $p$ and class $C^{1} ; Z$ may have a boundary $\partial Z$. The ...

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### Curves on a Kahler manifold

Let $(X,\omega )$ be a compact Kahler manifold. For any $d>0$ are there only finitely many families of curves $C_i$ such that $C_i\cdot \omega <d$? (More precisely, if $C$ is any curve such that ...

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### Moments of Plücker coordinates on complex Grassmannian and log-concavity

Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...

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### Simple example of non-integrable holomorphic connection

Let $X$ be a complex manifold with complex dimension $d$ and structure sheaf $\mathcal{O}_X$. Let $E$ be a locally free sheaf on $X$. A $holomorphic$ connection on $E$ is a morphism of sheaves
$$\...

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### Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat

I'm having a hard time proving the following:
If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric ...

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### Critical points of polarized endomorphisms of algebraic varieties

My main question is the following:
Let $f: \mathbb{CP}^n \to \mathbb{CP}^n$ be a holomorphic endomorphism of degree $d \ge 2$ of $\mathbb{CP}^n$ .
1. Let $X \subset \mathbb{CP}^n$ be an ...

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### About the embedded resolution

Let $M$ be a Kähler manifold and $V$ a singular hypersurface of $M$. Assume we obtain an embedded resolution $M^{\prime}$ of $V$ in $M$ by finitely many blow-ups along smooth centers.
My question ...

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### Vector field with constant divergence around embedded submanifold

Let $M$ be a smooth $n$-dimensional manifold and $N\subset M$ be a closed embedded submanifold of codimension at least $2$. Furthermore, let $\mu$ be a volume form on $M$.
Question: Does there ...

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### Fun examples relating to Hopf surfaces

A Hopf surface is a compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. I would like to know whether anyone has any of the following ...

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### zero extension of positive currents are always positive

In Demailly's Complex Analytic and Differential Geometry page 139:
He said the trivial (zero) extension of the positive current $T$ (on $X\setminus E$), which denoted by $\tilde T$ is always positive ...

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### Is there Hodge isomorphism between Dolbeault and Harmonic on noncompact manifold

As is well known , Hodge theorem tells us
Let $(X, g)$ be a compact hermitian manifold. Then the canoni.
cal projection $\mathcal{H}_{\bar{\partial}}^{p, q}(X, g) \rightarrow H^{p, q}(X)$ is an ...

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### Variation of the (Chern) curvature with respect to the metric

Let $E\rightarrow X$ be a holomorphic vector bundle, for any Hermitian metric $h$ on $E$ we denote by $F_h$ the curvature of the Chern connection associated to $h$. Fix a metric $h_0$ and consider a ...

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### Derivative of the Bott-Chern forms

The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" ...

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### weak convergence of positive currents vs. $L^1$ convergence of normalized potentials

I have run into the following statement in the literature (e.g. here, p.5, after Theorem 1.1): that weak convergence of positive $(1,1)$-currents on a complex manifold is equivalent to $L^1$ (I ...

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### Closed analytic subvariety of $\mathbb C^n$ not defined by global holomorphic functions

Here is the motivation of this question: $\mathbb C^n$ is already "local" in algebraic category. In other words, algebraic subvarieties of $\mathbb C^n$ are affine, so they are common zero locus of ...

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### Relative Chow's theorem

Suppose you have a smooth quasiprojective complex algebraic variety that is not compact. Suppose you have a complex analytic fiber bundle over the algebraic variety where each fiber is a smooth ...

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### Construction of Kahler Einstein Metric of Poincare Type

I am reading Kobayashi's Kahler-Einstein metric on an open algebraic manifold. In this paper he constructs a Kahler-Einstein of Poincare type on an open manifold X' = X\D, where X is projective and D ...

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### About the current of finite mass

In Demailly's e-book Complex analytic and differential geometry,
chap3-(1.14) Proposition is stated as follows:
Every positive current $T=i^{(n-p)^{2}} \sum T_{I, J} d z_{I} \wedge d \bar{z}_{J}$ ...

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### A differential geometric proof of the Riemann--Roch theorem for lines [duplicate]

I am looking for a differential geometric version of the proof of the Riemann--Roch theorem
for Riemann surfaces, that is, $1$-dimensional compact complex manifolds. The proofs one usually finds are ...

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### Holonomy groups of Hermitian, and hyper-Hermitian, manifolds

An $n$-dimensional complex manifold $M$, endowed with an Hermitian metric $g$, is Kähler if and only if the holonomy group of $g$ is contained in $U(n)$. If $g$ is Hermitian but not Kähler do we ...

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### Linear system corresponding to a holomorphic embedding from compact Riemann surface to projective space is complete

Le $X$ be a compact Riemann surface and $\phi$ be a holomorphic embedding of $X$ into projective space $\mathbb{C}\mathrm{P}^n$ which is induced by $(f_0,\dots , f_n)$. Then there is a linear system ...

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### $S^3$ as a Sasakian Manifold

Reading about Sasakian manifolds one come across two slogans:
A) "A Sasakian manifold is an odd-dimensional analogue of a Kahler manifold."
B) "A Sasakian manifold sits between two Kahler manifolds -...

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### cohomology classes of complex submanifolds

I was wondering if there were restrictions in what the cohomology classes corresponding to complex submanifolds of a complex manifold could be.
For example, say $T^4$ is regarded as a complex ...

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### A relative version of Ehresmann's theorem

Edited: Phil Tosteson suggested Thom's first isotopy lemma, but it does not seem to be in the direction that I'm trying to generalize. Let me reformulate my question again.
Let $N\subset M$ be a pair ...

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### Examples of constant scalar curvature kähler metric that is not kahler einstiein

It is well known that if the first Chern class is proportional to the kähler class given, then every cscK in that class has to be kähler Einstein. So there are two directions to generate examples as ...

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### Kollar-Matsusaka's finiteness theorem from a topological perspective

I am looking to find a reference or proof for the following topological version of Kollar-Matsusaka's theorem, which does not seem to be stated explicitly in the literature.
First I recall the ...

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### Sections of infinite order of elliptic surfaces

Let $X\to \mathbb{P}^1$ be a non-isotrivial elliptic surface over $\mathbb{C}$ with a section and with $X$ a smooth projective connected surface over $\mathbb{C}$. Let $\sigma:\mathbb{P}^1\to X$ be a ...

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### Control the convex combination of two classes on the boundary of the kahler cone

Let $(X,w)$ be a compact kahler manifold, and $[\eta]$ be a class is on the boundary of the kahler cone. The claim is that one can find another class $[\beta]$ also on the boundary of the kahler cone ...

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### Conformal welding of rectifiable curves

In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ together with two conformal isomorphisms $f_1 \colon \mathbb D \to D$ ...

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### Surfaces of general type with $q=1$

Let $X$ be a smooth projective connected surface of general type over $\mathbb{C}$ with $q(X) = 1$, where $q(X) = \mathrm{h}^1(X,\mathcal{O}_X)$.
Let $E$ be the Albanese variety of $X$, and let $X\to ...

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### Organizing mirror pairs

At a maximally vague and naive level, mirror symmetry asks the following question: given a complex manifold $(X, I)$, is there a symplectic manifold $(M, \omega)$ and an equivalence between the ...

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### Some questions about Hitchin's self-duality paper

I am reading this paper (The self-duality equations on a Riemann surface by N. Hitchin), and I don't understand a few things in page 67. In proof of Theorem 2.1 after Equation 2.4, he gives the ...

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### Vector bundle over compact complex manifold which is not holomorphic?

A vector bundle over a complex manifold is said to be holomorphic if its trivialization maps are biholomorphic maps. What is a "natural" example example of a vector bundle over compact complex ...

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### How do I find the portion of a cell/voxel lying within a defined surface?

We have a 3-dimensional grid of voxels (or cells), with individual voxels being of volume $dx\,dy\,dz$ where $dx=dy=dz=1$.
A cone-like surface is defined by some function, $z = f(x, y)$, which in ...

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### Is the $L^2$ or $L^p$ space of global sections of a holomorphic line bundle with respect to a singular metric a Hilbert space?

Let $(X,\omega)$ be a Kahler manifold and $L$ be a holomorphic line bundle over $X$ with a singular metric $e^{-\phi}$. We refer the definition of singular metric (weight) to Demailly 92' Singular ...

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### Chow variety of 1-cycles on abelian surface

It is an easy exercise to show that on a K3 surface, a smooth genus $g$ curve moves in a $g$-dimensional linear system. Nearly the same exercise shows that on an abelian surface, the corresponding ...

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### Equality of the derivative of the exponential map on Kähler manifolds

Let $M$ be a Kähler manifold, $\omega$ its Kähler form and $J$ the complex structure. Moreover, let $V$ be a smooth (or even analytic, I am not sure if this is important) vector field on $M$, $p \in M$...

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### Structure of non-big divisors in an abelian variety

Let $A$ be an abelian variety over $\mathbb{C}$. If $A$ has an effective non-big divisor, then $A$ is not simple. (In a simple abelian variety, every non-zero effective divisor is ample.)
What can ...

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### Vanishing theorems on a non-compact manifold

In complex geometry, various vanishing theorems for cohomology
groups of a hermitian line bundle E over a compact complex manifold X have been found.
My question is
Is there some vanishing ...

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### Chart in $1$-parameter family of Lagrangians in a Kähler manifold

Let $(X,\omega,J)$ be a complex $n$-dimensional Kähler manifold ($\omega$ Kähler form, $J$ complex structure) and $L \subset X$ be a closed real-analytic Lagrangian submanifold. Furthermore, let $L_{t}...

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### Can a real (1,1) form $\phi$ be represented by $u\sqrt{-1}\partial\bar{\partial}u$ on Kahler manifold?

Let M be a 2-dimension (complex dimension) K\"{a}ler maifold and $\phi$ be a real $(1,1)$ form. Is it possible that there exists a function $u$ such that $\phi=u\sqrt{-1}\partial\bar{\partial}u$

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### singular support in the singular case

For any constructible sheaf (or D-module) $\mathcal{F}$ over a smooth variety $X$ over $\mathbb{C}$, there is a notion of singular support $SS(\mathcal{F})$ that lives in the cotangent bundle $T^{*}X$ ...

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### The Yoneda pairing, hypercohomology, and cup product

Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by ...

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### Holomorphic union of sets

Let $\mathbb{D}$ be the unit disk on the plane and let $U,V\subset \mathbb{D}$ be open and such that $U\cup V=\mathbb{D}$.
Is there a holomorphic map $\varphi:\mathbb{D}\times \mathbb{D}\to \mathbb{...