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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Source of Proof of a theorem on Area of Pre-image under a complex polynomial

The following fascinating theorem ,attributed to Polya is mentioned in the introduction of the paper "The Areas of Polynomial Images and Pre-Images by Edward Crane" paper link.Could ...
AgnostMystic's user avatar
2 votes
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58 views

Understanding homogeneous projective rational manifolds

Every complex flag manifold is a homogeneous projective rational manifold (HPRM). But are there examples of HPRMs that are not complex flags? Can somebody provide examples of such spaces? For some ...
Bobby-John Wilson's user avatar
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Variation of the metric on Kähler quotient

We can use Kähler quotient to produce a family of Kähler metrics on quotient space. My question is: how do we calculate the variation of these metric? This seems to be a natural question but I can't ...
LYJ's user avatar
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Is inverse image along finite group quotient $t$-exact for the perverse $t$-structure?

Let $q: \mathbb{C}\to \mathbb{C}$ be the quotient by $\mathbb{Z}/n\mathbb{Z},$ i.e. the map taking $z\mapsto z^n$. In the accepter answer to Operations on perverse sheaves on disk the inverse image of ...
Sergey Guminov's user avatar
3 votes
2 answers
288 views

A paper of Borel (in German) on compact homogeneous Kähler manifolds

I am trying to understand the statement of Satz 1 in Über kompakte homogene Kählersche Mannigfaltigkeiten by Borel. Here is the statement in German Satz I: Jede zusammenhängende kompakte homogene ...
Bobby-John Wilson's user avatar
4 votes
0 answers
98 views

A compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group

Is it possible to have a compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group? It seems not to be the case, but a precise argument of reference would be great! Edit: ...
Bobby-John Wilson's user avatar
2 votes
0 answers
94 views

Characterizing the complex structure on a non-compact Riemann surface

A consequence of Torelli's theorem is that a closed Riemann surface $X$ is determined by its period matrix. More precisely, fix $(\alpha_i)$ a basis of $H_1(X, \mathbb{Z})$ and for $J$ some complex ...
Louis Beaufort's user avatar
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107 views

Compact complex manifolds with nef canonical bundle have nonnegative Kodaira dimension

Let $X$ be a compact Kähler manifold with nef canonical bundle. The (Kähler extension of the) abundance conjecture asserts that $K_X$ is semi-ample, and thus $K_X^{\otimes m}$ admits a section for ...
ABBC's user avatar
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1 answer
151 views

Blowup formula for a morphism

Let $f: X\to S$ be a smooth projective morphism between smooth schemes over $\mathbb C$, $i: Z \to X$ a closed subscheme of codimension $c$, also smooth over $S$, and let $g: Y\to S$ be the blowup ...
Aitor Iribar Lopez's user avatar
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1 answer
130 views

Teichmüller theory for open surfaces?

I have a rather straightforward and perhaps somewhat naive question: Is there a Teichmüller theory for open surfaces? My motivation basically is that I would like to find out more about the "...
M.G.'s user avatar
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Existence of meromorphic differential form on curve with given multiplicity of zeroes and poles

Let $m \in \mathbb{Z}^n$ be a partition of $2g-2$. Polishuk showed in his paper "Moduli spaces of curves with effective r-spin structures" (arXiv link) that if all entries of $m$ are ...
Matthias's user avatar
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Identifying the circle bundle of the canonical line bundle $\mathcal{O}(-n-1)$ over a projective space $\mathbb{CP}^n$

It's not hard to see the following fact: the circle bundle of the tautological line bundle $\mathcal{O}(-1)\rightarrow \mathbb{CP}^n$ is $S^{2n+1}$, the unit sphere inside $C^{n+1}.$ I want to see ...
Partha's user avatar
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Blowing up $\mathbb{CP}^2$ nine times and exactness of symplectic form

Consider the (symplectic) blow up $\operatorname{Bl}_k(\mathbb{CP}^2)$ of $\mathbb{CP}^2$ at $k$ points. I have heard that for $k=1,2,\ldots,8$ the size of the balls been blown up can be choosen in ...
kvicente's user avatar
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231 views

Reference for Teichmuller spaces of punctured surfaces

What is a good reference for Teichmuller spaces of punctured surfaces $S_{g,n}$ where $n>0$? I am looking for a reference where there is the correct statement and or proof of say the Bers embedding,...
Chitrabhanu's user avatar
6 votes
1 answer
234 views

Criteria for when Gauss-Manin sheaves are vector bundles

Let $(X,D_X)$ and $(S, D_S)$ be smooth normal crossings pairs over $\mathbb C$; i.e. smooth schemes of finite type over $\mathbb C$ with a normal crossings divisor. If $f:X \to S$ is a proper, flat ...
Aitor Iribar Lopez's user avatar
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Some calculation about Chern connection

The Chern connection is the unique connection satisfying $\nabla^{0,1}=\bar{\partial}$ and $$ \partial_k\langle u, v\rangle=\left\langle\nabla_k u, v\right\rangle+\left\langle u, \nabla_{\bar{k}} v\...
Elio Li's user avatar
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0 answers
274 views

Theories of manifolds w/ extra structure and singularities

Many different objects in mathematics can be described as manifolds with extra structure. Among the most famous examples of these are smooth manifolds, Riemannian manifolds, complex manifolds, and ...
Will Sawin's user avatar
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2 votes
1 answer
159 views

Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves

The following passage is from a thesis I'm reading: Suppose we have a short exact sequence of vector bundles $$0 \to \mathcal{E} \to \mathcal{F}\to \mathcal{G} \to 0.$$ Since these sheaves are ...
Johannes's user avatar
4 votes
1 answer
203 views

Equivariant projective embeddings with optimal dimension

Let $X$ be a complex projective manifold, and $f\in Aut(X)$ an automorphism, which is linearizable, that is, can be extended to an ambient projective space ${\mathbb P}^m$. I am interested to find ...
Misha Verbitsky's user avatar
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0 answers
51 views

To study the elliptic PDE on complex manifold, when can we treat it as the real case?

I wonder when studying the elliptic PDE on complex manifold, especially studying the existence of solutions, when can we directly study the real case, for example, when studying $$\Delta_c u = f(x,u),$...
Elio Li's user avatar
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1 answer
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Number of regions created by r hyper-planes in n-dimensional space [closed]

I found this formula for calculating maximum number of regions created by r hyper-planes in n-dimensional space (n<=r) ...
Mazen Saaed's user avatar
1 vote
0 answers
101 views

Dolbeault class of the curvature of the Chern connection equaling the Atiyah class

The following is a proposition from Complex Geometry by Huybrechts. Proposition $4.3.10$. For the curvature $F_\nabla$ of the Chern connection on an hermitian holomorphic vector bundle $(E,h)$ one ...
Johannes's user avatar
2 votes
0 answers
53 views

A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional

I'm considering the elliptic PDE with complex Laplacian, for example, write $$ \Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot), $$ and $$\Delta_c(u)=f,$$ by [P.Gauduchon, Math.Ann,...
Elio Li's user avatar
  • 729
4 votes
1 answer
194 views

A complex version of the Cahiers topos

Has anyone tried defining a complex version of the Cahiers topos? If we take the definition of $C^\infty$-rings, replace "smooth" with "holomorphic" (of course, one has to take ...
xuq01's user avatar
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2 votes
1 answer
157 views

Gluing local holomorphic connections

On page $180$ of Complex Geometry by Daniel Huybrecths, he defines the so called Atiyah class of a holomorphic vector bundle by the Čech cocycle $$A(E)=\{U_{ij}, \psi^{-1}_j \circ (\psi^{-1}_{ij}d\...
Johannes's user avatar
1 vote
0 answers
40 views

Looking at a frequency reassignment rule as a Möbius transform

Suppose we have some Schwartz function $h$. Denote its Fourier transform $\widehat{h}$. Let $\xi_0$, $a$, $\Delta$ be positive and fixed. I have a function $\Omega: \mathbb{R}\times \mathbb{R}^+ \to \...
mathim1881's user avatar
4 votes
2 answers
421 views

Nearby cycles for stacks

Let $X$ be a variety over $\mathbb C$. Let $f\colon X \to \mathbb{A}^1$ be a regular function. I understand that there is an analytic nearby cycles functor, defined in terms of the exponential map. I ...
user492133's user avatar
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How do I find an algebraic expression for the function $F(ξ, \bar{ξ})$ from this paper?

I am working on understanding the paper "On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg. As part of their definition of equations for a 3D surface, ...
Lawton's user avatar
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6 votes
0 answers
191 views

Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$

In my research I encountered automorphisms of the ring of convergent power series $$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$ which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
red_trumpet's user avatar
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2 votes
0 answers
44 views

Geometric explanation of Fueter-Sce-Qian Theorem and similar situations

In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}...
Giulio Binosi's user avatar
4 votes
1 answer
219 views

Spin$^c$ structures induced by an almost complex structure

Let $M$ be a closed spin$^c$ $4$-manifold with determinant line bundle $L$. If $c_1^2(L)=2\chi(M)+3\tau(M)$, where $\chi$ and $\tau$ denote the Euler characteristic and signature of $M$ respectively, ...
Zhiqiang's user avatar
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1 vote
0 answers
45 views

Elliptic surfaces with monodromy in Borel subgroup

Are there restrictions on the invariants of an elliptic surface $M\overset{\pi}{\longrightarrow} C$ for the monodromy of its homological invariant to be contained in the upper triangular subgroup of $\...
AG14's user avatar
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10 votes
0 answers
256 views

Looking for counterexamples: Are maximal tori in the automorphism groups of smooth complex quasiprojective varieties conjugate?

Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$. It has a group of (algebraic) automorphisms $ \DeclareMathOperator{\Aut}{Aut} \Aut(X)$. Define a torus in $\Aut(X)$ to be a faithful ...
Carlos Esparza's user avatar
1 vote
1 answer
104 views

Nonequidimensional birational Mori contractions

I have been looking for an excplicit example of a birational, divisorial Mori contraction such that the exceptional locus is not equidimensional onto its image. To agree with the setup I like, the ...
p0lydactyl's user avatar
0 votes
0 answers
146 views

“Holomorphic” bump function

I was wondering in what sense can I construct a holomorphic “bump function”? Now, of course we cannot really construct a holomorphic bump function in the usual sense, but I have a much rougher idea in ...
JustSomeGuy's user avatar
4 votes
0 answers
142 views

$\pm 1$-equivariant perverse sheaves on the affine line

Let $G=\mathbb{Z}/\mathbb{2Z}$ act by the map $z\mapsto -z$ on a complex line $\mathbb{C}$. The category $\mathcal{Perv}(\mathbb{C})$ of perverse sheaves smooth along the stratification by the origin ...
Sergey Guminov's user avatar
1 vote
0 answers
189 views

Constructing curves with large tangent space in complex variety

Suppose $M$ is a (singular) complex analytic/algebraic variety. Then for every $p\in M$ there exists a (possibly reducible) curve $C \subset U\subseteq M$ containing $p$ such that $T_pC=T_pM$, where $...
Thomas Kurbach's user avatar
3 votes
0 answers
131 views

Splitting of normal bundle exact sequence and Holomorphic neighbourhood retract

Let $X$ be a compact complex manifold and $Y\subset X$ a complex submanifold of $X$. Consider the two following conditions: The exact sequence $0\to TY\to TX|_{Y}\to N_Y\to 0$, where $TX$, $TY$ ...
SSS's user avatar
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7 votes
0 answers
228 views

Is a smooth projective variety over $\mathbb{C}$ dominated by a Ball?

Suppose that $X$ is a smooth projective variety of dimension $d$ over the complex numbers. Is it true that there is a ball $\Delta_d=\{ z\in \mathbb{C}^d / \lvert z\rvert<1\}$ and a surjective ...
Carletto's user avatar
  • 163
1 vote
1 answer
426 views

Vector bundles on $\mathbb{P}^1$

I am considering an alternative proof of Grothendieck's classification of vector bundles on $\mathbb{P}^1$. Given a vector bundle $E$ on $\mathbb{P}^1$ one can associate a graded module $\Gamma(E)$ ...
Sidana's user avatar
  • 21
2 votes
0 answers
165 views

Splitting of de Rham cohomology for singular spaces

I am currently trying to wrap my head around the following splitting result by Bloom & Herrera (here is a link to the ResearchGate publication) for the de Rham cohomology of (in particular) a ...
Thomas Kurbach's user avatar
2 votes
0 answers
123 views

Hodge coniveaux of Calabi-Yau manifolds

Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
Pène Papin's user avatar
1 vote
1 answer
106 views

Solution to $a=e^t (t-r_1)(t-r_2)$ with Lambert $W$ function, where $r_1, r_2 $ are complex

Lambert $W$ works when $r_1$, and $r_2$ are real. However, I am trying to solve the equation when $r_1$, and $r_2$ are complex numbers.
Hamed Elwarfalli's user avatar
2 votes
0 answers
203 views

Smooth compactification of complex varieties and uniqueness

Since I'm working in differential geometry, for the following I'm strictly interested in the smooth setting over $\mathbb{C}$ and its relation to the setting over $\mathbb{R}$. Here are a few useful ...
Paul Cusson's user avatar
  • 1,735
4 votes
0 answers
210 views

Classification of affine varieties over the affine line whose central fiber is $\mathbb{C}$ and general fiber is $\mathbb{C}^*$

Consider an affine variety $Y$ equipped with a morphism $\pi: Y \rightarrow \mathbb{C}$. The conditions we have are that $\pi^{-1}(0)=\mathbb{C}$, and for any $x$ not equal to zero, $\pi^{-1}(x)=\...
Yunsong WEI's user avatar
3 votes
0 answers
73 views

Let $f: X \to D$ be a proper holomorphic submersion. Does a holomorphic form on $X$, closed on each fiber of $f$, have holomorphic coefficients?

Let $D$ be the unit disc in $\mathbb{C}^n$ and let $f: X \to D$ be a proper surjective holomorphic submersion, which is trivial as a smooth fiber bundle, with connected fiber $F$. We get an induced ...
Vik78's user avatar
  • 487
0 votes
0 answers
116 views

Is any singularity a subgerm of $(\mathbb{C}^n, 0)$?

I am studying singularity theory. I have often come across, in the literature, the sentence which says "let $(X,0) \subset (\mathbb{C}^n,0)$ be a singularity". Here a singularity is a ...
Math1016's user avatar
  • 369
14 votes
2 answers
1k views

Is the Gödel universe Wick rotatable?

Take Wick rotatability being as the way defined in the following article by Helleland and Hervik: Christer Helleland, Sigbjørn Hervik, Wick rotations and real GIT, Journal of Geometry and Physics 123 ...
Bastam Tajik's user avatar
1 vote
0 answers
55 views

Complex geodesic coordinate, local ramified map, and the conic metric

Remark: I have asked this question in MSE, however, I got no responses. This is the reason I come to ask here. I am looking forward to some advices. Thanks in advance Let $X$ be a compact Kaehler ...
Invariance's user avatar
3 votes
1 answer
147 views

Request for non-Einstein positive constant scalar curvature Kähler surfaces

I am curious about concrete examples of compact cscK manifolds in complex dimension two, in particular cscK surfaces with positive scalar curvature. There are of course the Fano (del Pezzo) Kähler-...
Garrett Brown's user avatar

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