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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On some curves of real values of a rational function

For given parameters $a_{1},\dots,a_{k}\in\mathbb{R}$, define the rational function $\phi:\mathbb{C}\to\mathbb{C}$ as $$\phi(z)=\frac{1}{z}-a_{1}z-a_{2}z^{2}-\dots-a_{k}z^{k}.$$ The domain of its real ...
Twi's user avatar
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10 votes
0 answers
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How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?

Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern abstract algebraic geometry is there in modern complex geometry? What do I mean by complex geometry? ...
Seidarkman's user avatar
8 votes
0 answers
347 views

Symplectic invariance of Hodge numbers?

Let $(X,\omega)$ be a compact symplectic manifold. If $J$ is a $\omega$-compatible complex structure on $X$ then $(X,\omega,J)$ is a compact Kähler manifold and so has Hodge numbers $h^{p,q}$. My ...
user25309's user avatar
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1 vote
0 answers
148 views

Calabi-Yau with nodes

Suppose $X$ is a singular projective irreducible complex variety of dimension 3, and its singular loci are finite number of nodes, and its smooth locus $X_1$ is a Calabi-Yau quasi-projective variety, ...
Wenzhe's user avatar
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4 votes
0 answers
235 views

Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical

Let start with a definition Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So ...
user avatar
1 vote
0 answers
132 views

Poincaré inequality for holomorphic line bundles

Let $M$ be a Riemann surface of genus >1, $g$ be an Hermitian metric on $M$. Let $E$ is a holomorphic negative line bundle over $M$, for example, the holomorhic tangent bundle of $M$. Let $h$ be an ...
Qiongling Li's user avatar
3 votes
0 answers
209 views

local description of $\mathbb{P}^2$-fibrations over $\mathbb{P}^1$

Let $X$ be a rational threefold (over the field of complex numbers) with terminal singularities. It is well-known that $X$ has only finitely many singular points $x_1,x_2, \ldots,x_n$. To be more ...
sabrebooth's user avatar
8 votes
0 answers
229 views

A Hartogs-type criterion for flatness

Let $U$ be a smooth affine connected variety over $\mathbb C$ and let $V\subset U$ be an open whose complement is of codimension at least two. Now, let $Y$ be a smooth quasi-affine connected variety ...
Christophe's user avatar
1 vote
2 answers
202 views

Smooth, irreducible surface with real part containing two projective planes

Let $X$ be a smooth and irreducible projective variety over $\mathbb{R}$ of dimension two. I am looking for an instance of such a variety where two distinct connected components of $X(\mathbb{R})$ are ...
Hans's user avatar
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4 votes
1 answer
351 views

Quantifying the monotonicity property of the hyperbolic metric

Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} &...
Lasse Rempe's user avatar
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8 votes
1 answer
281 views

Complex manifolds with spanning sets of holomorphic tensor fields

This question is an extension of this one. Consider a complex manifold $(M^{2n}, J)$. Fix $1 \leq p \leq n-1$, and suppose that the space of holomorphic sections of $\Lambda^{p,0}$ spans $\Lambda^{p,...
Thisquestionisreallyhard's user avatar
6 votes
1 answer
934 views

Confusion surrounding the Koszul-Malgrange theorem

I recently had the need to appeal to some complex geometry in my research and have been trying to unravel the various relationships surrounding the Koszul-Malgrange theorem. According to nlab, the ...
Daniel Grady's user avatar
7 votes
2 answers
403 views

Are square tiled surfaces dense in the moduli space of translation surfaces?

I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt. At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is dense....
Nuxil's user avatar
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1 vote
1 answer
174 views

Kernel of projection formula

For a closed embedding of compact complex manifolds $$ \iota : Y \hookrightarrow X $$ and any $\alpha \in H^*(X,\mathbb Q)$, we have trivially: $$ \iota^*(\alpha)=0\quad \Rightarrow \quad\iota_*\iota^*...
Nicolas Malebranche's user avatar
5 votes
1 answer
640 views

Complex manifolds with spanning sets of holomorphic vector fields

I want to understand compact complex manifolds $(M^{2n}, J)$ with the following property: there exists a collection $\{X_i\}_{i=1}^L$ of holomorphic vector fields (sections of $(T^{1,0}_{\mathbb C} M)$...
Thisquestionisreallyhard's user avatar
6 votes
1 answer
567 views

Chern-Einstein metrics on complex Hermitian manifolds

Metric on a Riemannian manifold $(M,g)$ is Einstein, if for some function $\lambda\colon M\to \mathbb R$ $$ Ric(g)=\lambda g. $$ It is well know, that such $\lambda$ is, in fact, a constant. The ...
Yury Ustinovskiy's user avatar
-1 votes
1 answer
94 views

Zariski open set in orthogonal grassmanian [closed]

I am confused about the following question. Consider $\mathbb C^4$ endowed with nondegenerate symmetric bilinear form $J:=\left(\begin{matrix}0&0&0&1\\0&0&1&0\\0&1&0&...
user42804's user avatar
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0 votes
0 answers
547 views

A question about invariance of plurigenera

Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So deformation invariance of plurigenera says that the ...
user avatar
7 votes
0 answers
457 views

Kähler quotients of affine varieties and GIT

Let $X\subseteq \Bbb C^n$ be a smooth affine variety and $G=K_{\Bbb C}$ a complex reductive group acting linearly on $\Bbb C^n$ preserving $X$ (where $K$ is a maximal compact subgroup of $G$). Suppose ...
SHP's user avatar
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1 vote
1 answer
299 views

A necessary condition for existence of Ricci flat metric on pair (X,D)

Let $X$ be a complex compact manifold with simple normal crossing divisor $D$. Is the condition $K_X +D = 0$ necessary for the existence of Ricci-flat metric?
Jarek's user avatar
  • 63
8 votes
0 answers
908 views

Fubini-Study form on weighted projective spaces

As it is known, $\mathbb CP^n$ with Fubini-Study symplectic form can be get by the symplectic reduction of $\mathbb C^{n+1}$ with a symplectic form $\sum_{i=0}^n dz_i\wedge d\bar z_i$ by a hamiltonian ...
anna abasheva's user avatar
0 votes
1 answer
520 views

Canonical metric on moduli space of singular Calabi-Yau varieties

Let $\pi:X\to Y$ be a surjective holomorphic map with connected fibers and let fibers are singular Calabi-Yau varieties (i.e. numerical dimension is zero) then is it possible to construct canonical ...
user avatar
7 votes
1 answer
458 views

Embed a bordered Riemann surface into punctured Riemann surfaces?

Let $U$ be a bordered Riemann surface of genus $g$ with $n -1$ punctures and one hole (i.e., the border has one connected component). For any punctured Riemann surface $\Sigma$ of genus $g$ with $n$ ...
user89402's user avatar
9 votes
1 answer
771 views

Is the analytification functor part of a geometric morphism of topoi?

Let $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ be the topos of zariski sheaves on finitely genertaed $\mathbb{C}$-algebras. A complex analytic space for our purpose is a locally ringed space locally ...
Saal Hardali's user avatar
  • 7,549
7 votes
0 answers
201 views

Biholomorphic neighborhoods of the boundary of Stein domains

Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
PVAL's user avatar
  • 773
1 vote
0 answers
431 views

Is the category of mixed Hodge modules bi-filtered?

Let $X$ be a smooth complex algebraic variety and let $MHM(X)$ be the category of mixed Hodge modules on $X$, as defined in (Saito, "Mixed Hodge Modules", 1990), (Peters-Steenbrink, "Mixed Hodge ...
a_g's user avatar
  • 497
12 votes
2 answers
935 views

Bott Chern cohomology via currents

Let $X$ be a compact complex manifold. Is the space of $(p,p)$ $d$-closed currents modulo $\partial\bar{\partial}$-exact ones naturally isomorphic to the Bott-Chern cohomology (made in the same way ...
stefano trapani's user avatar
2 votes
1 answer
335 views

Existence of non-constant solutions for this equations

This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...
Amir Baghban's user avatar
5 votes
1 answer
949 views

On the complexification of a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $...
Amir Baghban's user avatar
1 vote
0 answers
140 views

Equation of the form $\overline{\partial}_{J}f=g$ made holomorphic

In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried ...
Arthur's user avatar
  • 11
2 votes
1 answer
461 views

Local deformations of Lagrangian submanifolds in holomorphic symplectic manifold and their intersections

Let $Y\subset X$ be a Lagrangian submanifold in a holomorphic symplectic manifold $X$. We know that there exists a local moduli space $M$, which parametrizes lagrangian submanifolds in $X$(there are ...
Lya's user avatar
  • 63
3 votes
1 answer
298 views

Do some kind of maximum principle exist on complex manifold?

Consider a holomorphic function $u:C\to C^n$, as $|u|^2 $is sub-harmonic, it satifies a maximum principle. Do some general kind of complex manifold enjoy such property? Say, square of some distance ...
icarus's user avatar
  • 31
1 vote
1 answer
102 views

Triviality of a circle fibration induced by an almost complex structure

Let $E→M$ be a plane bundle endowed with an almost complex structure $J.$ $J$ induces a natural positive definite inner product in the associated bundle $End(E)→M$,denoted by $<,>$. More ...
Mike Cocos's user avatar
-1 votes
1 answer
194 views

Is it easy to see that a cubic surface $V$ in $CP^3$ has no holomorphic 2-forms? [closed]

More specifically, what facts do you need to know to conclude $H^2(V) = H^{1,1}$? In general, are there hypersurfaces in $CP^n$ without holomorphic $k$-forms for some $k$?
mokim's user avatar
  • 107
4 votes
0 answers
348 views

Narasimhan-Simha Hermitian metric vs Weil-Petersson metric

What is relation between Weil-Petersson metric on holomorphic fibre space $f:X\to Y$ of compact complex manifolds $X,Y$ . (let fibres are Calabi-Yau manifolds) And Ricci curvature of Narasimhan-Simha ...
user avatar
1 vote
1 answer
568 views

Is it true that singular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?

It is well known that Kodaira gave an essentially topological classification of the possible singular fibers of elliptic fibrations according to their type: https://en.wikipedia.org/wiki/...
user6319's user avatar
  • 179
2 votes
0 answers
168 views

Question regarding a lemma in Principles of Algebraic Geometry

My question is regarding the lemma on page 81 of the book by Griffiths and Harris. The lemma says the following: A $\bar{\partial}$-closed form $\psi\in Z^{p,q}_{\bar{\partial}}(M)$ is of minimal ...
Alan's user avatar
  • 1,514
4 votes
1 answer
553 views

What is the relation between holomorphic blow-up and symplectic blow-up?

McDuff has shown us exactly how the symplectic blow-up procedure along a symplectic submanifold affects the symplectic structure in the ambient space, i.e., if $\omega$ is the original symplectic form,...
Sinister Cutlass's user avatar
11 votes
1 answer
1k views

Restriction of the Picard group of a surface to a curve

In a paper by Griffiths and Harris on the Noether-Lefschetz theorem, they use the following fact which they don't comment as if it is obvious: For a general (smooth) surface $S$ in $\mathbb{P}^3$ ...
kostya's user avatar
  • 111
18 votes
3 answers
3k views

What are parabolic bundles good for?

The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate ...
Dr. Evil's user avatar
  • 2,641
3 votes
1 answer
483 views

A conjecture from Jean Varouchas on Kahler varieties

Conjecture: Let $\pi: X\to X'$ be a proper flat surjective morphism of complex spaces. If $X$ is Kahler, is $X'$ Kahler? This conjecture when $X$ and $X'$ are smooth solved by Jean Varouchas from ...
user avatar
2 votes
0 answers
524 views

Tian's approach for solving the conjecture of invariance of plurigenera in Kahler setting

Let $f:X\to Y$ be a smooth holomorphic fibre space whose fibres $f^{-1}(y)$ have pseudoeffective canonical bundles. suppose that $$\frac{\partial \omega(t)}{\partial t}=-Ric_{X/Y}(\omega(t))-\omega(...
user avatar
6 votes
0 answers
227 views

Deformation of Complex Spaces

I am trying to learn about deformations of complex spaces from the paper of Palamodov. I am particularly interested in the relative tangent cohomology. Is there any other modern reference to this ...
Chitrabhanu's user avatar
2 votes
0 answers
93 views

Complex Structure Moduli of Elliptic Fibrations

Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration. I know how it is done for the generic Weierstrass ...
Stirling Suleiman's user avatar
1 vote
2 answers
630 views

Contact and CR Examples

What is an example of a manifold such that: (A) It is both a contact manifold and a CR manifold (B) It is a contact manifold but not a CR manifold (C) It is not a contact manifold but not a CR ...
Todd Johnson's user avatar
8 votes
1 answer
507 views

Connectivity of complements of Stein opens

Let $Y$ be an affine open subset of a locally noetherian scheme $X$. Then, $X \setminus Y$ has pure codimension one [EGAIV$_4$, Cor. 21.12.7]. Moreover, if $X$ is proper and of finite type over a ...
Takumi Murayama's user avatar
8 votes
2 answers
520 views

Inequality on Kähler classes

Let $X$ be a compact Kähler manifold of complex dimension $n$, and let $\omega_1, \omega_2$ be Kähler classes on $X$. Denote the Lefschetz operator of a Kähler class $\omega$ by $\Lambda_{\omega}$. ...
Gunnar Þór Magnússon's user avatar
1 vote
0 answers
182 views

horizontal lift along fibres

Let $f:X\setminus D\to Y$ be a smooth family of Kahler manifolds, where $D=\{\sigma=0\}$ is a divisor on $X$. Taking a local coordinate $(s_1,...,s_d)$ of $Y$ and a local coordinate $(z_1,...,z_n)$ of ...
user avatar
4 votes
1 answer
247 views

Smooth algebraic stacks with precisely two $\mathbb C$-objects

In my quest of "understanding" stacks, I recently tried to figure out the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely one $\...
Christian's user avatar
  • 193
5 votes
1 answer
569 views

Is the automorphism group of a Calabi-Yau variety an arithmetic group

Let $X$ be a smooth projective variety over the complex numbers with trivial canonical bundle. Suppose that $X$ is Calabi-Yau. Is the automorphism group of $X$ an arithmetic group? What if $X$ is a ...
Christian's user avatar
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