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 ...
• 808
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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 ...
1 vote
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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 ...
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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 ...
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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 ...
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: ...
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 ...
1 vote
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 ...
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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 ...
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 "...
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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 ...
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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 ...
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1 vote
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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 ...
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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,...
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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 ...
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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, ...
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1 vote
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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 $TX$, $TY$ ...
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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 ...
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1 vote
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 $\Gamma(E)$ ...
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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 ...
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 ...
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1 vote
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.
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 ...
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