# 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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### Jacobian ideal as primary idea;

Let $R = \mathbb{C}\{x_1, \dots,x_n\}$ be the ring of germs of analytic functions and let $f \in R$ be a homogeneus polynomial of degree $p$ such that $\sqrt{Jac(f)}=\langle x_1, \dots, x_{n-1}\rangle ...

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### Construction of holomorphic line bundles on complex torus

This is an argument for constructing positive line bundles on complex torus. From some knowledge of Abelian varieties, such as Riemann conditions, we know that it is wrong. But I don't know where this ...

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### Is Kähler current class representable by semipositive forms?

A compact complex manifold is called in Fujiki class $\mathcal C$ if it is bimeromorphic to a compact Kähler manifold, or equivalently, if there exists a proper holomorphic bimeromorphic map (i.e. a ...

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### Is the union of Fujiki cones open in $\mathcal H^{1,1}_{\mathbb R}$?

Let $\mathcal X\to B$ be a holomorphic family of compact Kähler manifolds, let $\mathcal K_t$ denote the Kähler cone of the fiber $X_t$, then the union $\cup_{t\in B}\mathcal K_t$ forms an open set in ...

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### Is Aut(X) the group of automorphisms of A which preserve X?

Let $X$ be a smooth closed subvariety of a complex abelian variety $A$. Assume $X$ is of general type and of codimension one with $\omega_X$ ample.
Is the (finite) group $\mathrm{Aut}(X)$ the group ...

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### Holomorphic line bundle trivial outside a divisor

Let $X$ be a complex manifold, $D\subset X$ a divisor, $L$ a holomorphic line bundle over $X$. Suppose that $L$ is trivial on $X\setminus\operatorname{Supp} D$, then is it true that $L\cong \mathscr{O}...

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### Difference between stabilizer and automorphism group of subvariety of an abelian variety

Let $X$ be a smooth closed subvariety of a complex abelian variety $A$. Assume $X$ is of general type and of codimension one with $\omega_X$ ample.
Often, people speak about the stabilizer $\mathrm{...

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### Complex vector bundles on compact complex manifolds

The complex vector bundles on complex projective space $\mathbb{CP}^n$ are explicitly classified for low dimensions. When $n\leq 3$, they are exactly the holomorphic vector bundles; when $n\geq 4$ we ...

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### Interpretation of $\mathcal H^k$

For a holomorphic family $\pi:\mathcal X\to B$ between complex manifolds, the map $\pi$ is a proper holomorphic submersion, $X_t:=\pi^{-1}(t)$, $t\in B$, $X=X_0$, we have the isomorphism $H^k(X,\...

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### Proof of Ehresmann's theorem

In Huybrechts' book Complex geometry: An introduction p.269, Proposition 6.2.2, the author gives a proof of the following theorem
(Ehresmann)
Let $\pi:\mathcal X\to B$ be a proper family of ...

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### Cohen-Macaulyness of Milnor algebra

Denote by $R = \mathbb{C}\{x_1, \dots, x_n\}$ the ring of germs of analytics maps at the origin in $n$ variables and let $f \in R$ such that $Sing(V(f))=V(x_1, \dots, x_{n-1})$ as sets. In addition, ...

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### A very practical 3D geometry problem! [closed]

I really hope you can help me out with a (hopefully basic) very practical problem.
I've bought a two floor house. The first floor is connected to the second with a double flies of stairs that make a U-...

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### Positive integration on P^1

Let $u: \mathbb{P}^1(\mathbb{C}) \longrightarrow \mathbb{R}$ be a smooth function s.t. $u$ is invariant under complex conjugation and $\displaystyle \int_{\mathbb{P}^1(\mathbb{C})}u \; \omega_{\mathrm{...

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### Question about deformation of the metirc on a Riemannian manifold

I'm a bit confused with the deformation of the metric on a given Riemannian manifold $(M,g)$ with a smooth boundary. How can we deform the metric $g$ such that it is a product near $\partial M$, ...

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### Pull back a vector field [closed]

In Voisin's book Hodge theory and complex algebraic geometry, I Section 9.1.2, p.223, the author writes:
Let $\phi:\mathcal X\to B$ be a family fo complex manifolds. The differential $\phi_*$ is a ...

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### Pull-back of factor of automorphy

Let $M=\mathbb C^g/ \Gamma$ be a complex tori and $E$ a be a holomorphic vector bundle of rank $r$ over $M$. Then $E$ is characterised by factor of automorphy, i.e. a holomorphic map $J:\Gamma\times\...

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### Projectivization in the derived category of coherent sheaves

Let $X$ be a compact Kahler manifold. There exists a notion of projectivization of holomorphic vector bundles and coherent sheaves over $X$. Does that concept extend to objects in the derived category ...

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### Non-Kähler Hermitian homogeneous spaces

I am looking for examples of compact homogeneous space endowed with the structure of a non-Kähler Hermitian manifold.

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### Examples of 6-manifolds without an almost complex structure

Question: I am searching for examples for closed (hence orientable ), smooth $6$-manifolds without an almost complex structure.
Finding such an example is equivelant to finding a manifold where the ...

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### Hodge bundle for $\partial\bar\partial$-manifolds

Let $\pi:\mathcal X\to B$ be a holomorphic family of $\partial\bar\partial$-manifolds (compact complex manifolds satisfy $\partial\bar\partial$-lemma, e.g. Kähler manifolds, Fujiki class $\mathcal C$ ...

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### When is bijective map between closed point of varieties a morphism?

Let $f:X\rightarrow Y$ be a bijective map between complex varieties, when will it be a morphism?
I meet this question when working over Fourier–Mukai transforms in algebraic geometry and some papers. ...

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### Infinite-dimensional "algebraic varieties"

This question was formerly posted on MSE but did not receive any answer or comment, so I'm re-asking it here.
Let $H$ be the infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its ...

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### Does the Kähler form $\omega$ satisfy $d^*\omega=0$?

Let $X$ be a compact Kähler manifold with Kähler form $\omega$, then from Kodaira & Spencer's paper on deformations III, p.75, the authors state that the Kähler form satisfies the Laplace equation ...

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### Quantifying the failure of geometric formality in K3 surfaces

It is known that K3 surfaces are never geometrically formal [1]. That is, the wedge product of two harmonic forms on an arbitrary K3 surface is in general not harmonic, or equivalently, the space $\...

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### General fiber and the symmetric product of an ample hypersurface

Let $Sym^m(X)$ be the $m$th symmetric product of a smooth projective variety $X$, $n=\dim(X)$, $Y_1$ an ample hypersurface of $X$, and $CH_0(X)_{hom}$ the Chow groups of $0$-cycles of degree $0$....

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### Blow up and critical points of the projection map

Denote $Z=V(x_1, \dots, x_{n-1}) \subset \mathbb{C}^n$ and let $Bl_Z(\mathbb{C}^n)$ be the blow up of $\mathbb{C}^n$ along $Z$ together with the projection map $\pi \colon Bl_Z(\mathbb{C}^n) \to \...

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### Dimension of the sum of symmetric products of smooth projective varieties [closed]

Let $Sym^{m}(X)$ denote the $d$th symmetric product of the smooth projective variety $X$ with $\dim(X)=n$, and
let $Z$ an irreducible subset of $Sym^{m}(X)$ with $\dim(Z)=mn-K$, (here $K$ is a ...

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### Meaning of torsion points in a Roitman's theorem

I am having some problems to understand the meaning of the following theorem due to Roitmann. I found this theorem in Voisin's book: Hodge Theory and Complex Algebraic Geometry, Volume II, page ...

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### Frölicher spectral sequence of a surface

Asked this on MSE but didn't get much attention.
Let $ S $ be a compact complex surface. Can anyone provide a proof of the fact that the Frölicher spectral sequence of $ S $ degenerates at $ E_1 $?
...

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### Relating the holomorphic Euler characteristic of a family of algebraic varieties to properties of the base and fibers

Let $f : X\rightarrow Y$ be a proper flat morphism (of schemes) with connected fibers over a smooth projective curve $Y$ over $\mathbb{C}$. Let $X_{y_0}$ denote a smooth fiber over $y_0\in Y$.
If $f$ ...

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### Can a holomorphic vector field have an attractor homoclinic loop?

It is well known that a holomorphic vector field $z'=f(z), z\in \mathbb{C}$ does not have any limit cycle.See the last paragraph of this post
Orbits space of real-analytic planar foliations
One can ...

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### parabolic schwarz lemma

Trying to follow the computation in https://arxiv.org/pdf/math/0602150.pdf, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they computed $\Delta \text{tr}_{g}h = g^{i \bar l}...

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### About finite dimensionality of Chow groups of zero cycles

Let $S$ be a connected smooth complex projective surface.
Let $Sym^{d}(S)$, $d\in \mathbb{Z}^+_0$, be the $d$-th symmetric product of $S$ parametrizing $0$-cycles of degree $d$.
Let $Sym^{d,d}(S)=...

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### First Chern form of line subbundle

Let $\pi:E\to X$ be a holomorphic vector bundle over a complex manifold. Denote by $\tilde{E}=\pi^*E\to E$ the pullback of $E$ over itself. There exists a tautological line bundle $L\subset \tilde{E}$ ...

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### Direct image of a sheaf with nowhere vanishing sections

Suppose that $f: X \to Y$ is a morphism of schemes over the complex numbers and $E$ is a vector bundle on $X$ such that all the sections of $E$ are nowhere vanishing sections. Furthermore, assume that ...

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### Representation of fundamental group and flat connections

I read Differential Geometry Of Complex Vector Bundles by Kobayashi, and he says there that a vector bundle $E$ has flat connection is equivalent to $E$ being defined by a representation of $\pi_1$. ...

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### What is known about the moduli of stable rank 3 bundles on the projective plane?

What is known about the moduli space of stable rank $3$ bundles on the projective plane $\mathbb{CP}^2$?
Ideally, there is a concrete complex manifold which is a fine moduli space for such bundles for ...

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### Blow-up of a stratified space

Let $X$ be a smooth projective variety over $\mathbb{C}$, and $D_1, \ldots, D_n$ be a collection of simple normal crossing divisors. The divisors induce a stratification $\mathcal{T}_X$ of $X$.
Let $...

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### Projectivization of a coherent sheaf using resolution by vector bundles

Let $\mathcal{F}\to X$ be a coherent sheaf over a compact Kahler manifold and let $E^{\bullet}\to \mathcal{F}$ be a resolution of $\mathcal{F}$ by holomorphic vector bundles.
Is there a way to ...

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### Gysin homomorphism of an inclusion to Kähler tubular neighborhood

Let $Z\subset U$ be a Kähler tubular neighborhood of a compact manifold $Z$ of codimension $r$.
Consider de Rham complexes of smooth differential forms $\Lambda^{*,*}(Z),\Lambda^{*,*}(U)$, let $\...

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### Does miracle flatness always fail for a non-regular base?

In [1], Huybrechts and Mauri argue that a holomorphic Lagrangian fibration $f: X \to B$ with smooth base $B$ is flat. This is an application of so called miracle flatness [2, Thm 23.1], because ...

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### Understand the Mukai vector

Let $S$ be a K3 surface and $h:=c_1(i^*\mathcal{O}_{\mathbb{P^3}}(1))$, then we can compute that $c_1(S)=0,c_2(S)=6h^2$. Hence
\begin{align}
\sqrt{\text{td}(S)}=1+\frac{c_2(S)}{24}=1+\frac{1}{4}h^2
\...

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### Comparing the first-order theories of different kinds of local rings of a complex variety

Let $X$ be a complex variety containing some point $x$. Then $X$ is naturally a complex-analytic space, and we have an inclusion of rings $\mathbb{C}[X]_x\hookrightarrow\mathbb{C}\{X\}_x\...

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### Stein manifold homotopic to wedge of two Stein manifolds

I am not very conversant with Stein structure on a manifold so this may be a very silly question. Let $X$ and $Y$ be two Stein manifolds of dimension $n$, inside $\mathbb{C}^N$. Take $x\in X$ and $y\...

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### Compactifications of group varieties

Let $V$ be a nonempty, irreducible, smooth projective variety over $\mathbf{C}$.
Is there a smooth projective variety $X$ over $\mathbf{C}$, a surjective map $X\to V$ of varieties over $\mathbf{C}$, ...

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### Milnor hypersurface

I cross-post a question that has not been answered on MSE, see here.
Consider the Milnor hypersurface $H_{ij}$, i.e., the smooth hypersufrace in $\mathbb CP^i \times \mathbb CP^j$ for fix pair of ...

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### A cohomological variant of the second Riemann's extension theorem

Let $X$ be a connected compact complex manifold, $U$ an open subset of $X$ such that the complement of $U$ in $X$ is an analytic subset of codimension at least 2 in $X$. Let $O_X$ (resp. $O_U$) be the ...

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### Finding the (1,1) component of $e^{-\mathbb{A}^2}$ for $\mathbb{A}$ a superconnection

Let $E=E^+\oplus E^-$be a holomorphic superbundle over a compact Kahler manifold, and $v:E^+\oplus E^-$ an odd bundle map. Assume that both $E^+$ and $E^-$ are endowed with Hermitian metrics, and ...

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### 𝔾ₘ extensions vs line bundles over abelian varieties

Given a complex polarized abelian variety $V$, we can define a map $$\operatorname{Ext}^1\left(V, \mathbb{G}_m\right) \to \operatorname{Pic}\left(V\right)$$
by viewing an extension as a $\mathbb{G}_m$-...

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### Do all closed positive currents lift to a resolution?

Consider a normal complex analytic space $X$ and a projective birational resolution $f:Y\rightarrow X$. Let $T$ be a closed positive current of bi-dimension $(p,p)$ on $X$. Is there always a closed ...