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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Connectedness of isometry group of closed Kaehler manifolds

Let $(M, g, J)$ be a closed Kaehler manifold. Is there some more-or-less non-tautological condition ensuring that its group of orientation-preserving isometries is connected? Also the same question, ...
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7 votes
2 answers
509 views

Is the symmetric product of an abelian variety a CY variety?

Let $n>1$ be a positive integer and let $A$ be an abelian variety over $\mathbb{C}$. Then the symmetric product $S^n(A)$ is a normal projective variety over $\mathbb{C}$ with Kodaira dimension zero ...
Harry's user avatar
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Symplectic Chern class of holomorphic symplectic manifold

I've posted this question already on math.stackexchage. Unfortunately, it did not receive any answers even though there was a bounty on it. So maybe somebody of you could help me. I apologize if this ...
MrXYZ's user avatar
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Can every non-compact Kähler manifold be realized as the analytification of a smooth variety?

I was just thinking about how we have nice theorems relating compact Kähler manifolds to the algebraic setting, but I was wondering if anything interesting holds in the non-compact case? i.e. Given ...
JJJ's user avatar
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Complex differentials and measured singular foliations

I'm trying to understand the technical basis of singular foliations for pseudo-Anosov diffeomorphisms, and I've hit a bit of a strange calculation I'm having a hard time verifying/unpacking. In Fathi, ...
D. Ford's user avatar
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Constructing new complex manifolds out of old

It is not difficult to build new manifolds out of old in the smooth category, for example taking the direct product or constructing a fiber bundle, taking the level set of a regular value of a smooth ...
Overflowian's user avatar
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2 answers
567 views

Compact complex affine Kähler manifold is a torus

Before giving a motivation let me ask the precise question firstly. By a complex affine manifold I mean a complex manifold $M$ with the property that there exists an holomorphic atlas for which ...
J.E.M.S's user avatar
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Multiple mirrors phenomenon from SYZ and HMS perspective

There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties ...
paul's user avatar
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Differences of $\omega$-plurisubharmonic functions

Let $X$ be a complex manifold, and $\omega$ a Kähler form on $X$. A smooth function $\phi$ on $X$ is $\omega$-plurisubharmonic ($\omega$-psh for short) if the form $\omega+\sqrt{-1}\partial\bar{\...
user135826's user avatar
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Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?

Please let me know whether this question is suitable for Mathoverflow. Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. ...
Zhaoting Wei's user avatar
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Quotient of a smooth projective surface by an involution

Is the quotient of a smooth complex projective surface by an involution projective? Suppose the quotient happens to be smooth; does that change the situation?
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Regular functions vs holomorphic functions

Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves. Is ...
Ari's user avatar
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What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in C^n ( n>1)? [closed]

What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in $\mathbb{C}^n$ ($n>1$) ? I would be pleased if you tell me.
user135645's user avatar
1 vote
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Birational maps mapping ample class to ample class?

I refer to the paper "Normal Subgroups in the Cremona Group". In the last paragraph of the proof of proposition 5.13, the author wrote the following: "Assume now that $h\in \text{Bir}(X)$ preserves ...
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Extending Normal Bundle of a Subvariety of $\mathbb P^n$

This question is related to, but not the same as, an earlier question: Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$ Given a smooth projective variety $X\subset\mathbb P^n$, let $N_X$ be ...
Mohan Swaminathan's user avatar
5 votes
1 answer
482 views

Homogeneous Riemann Surfaces

A Riemann surface $X$ is a connected complex manifold of complex dimension one. A homogeneous space is a manifold with a transitive smooth action of a Lie group. I guess there must be a classification ...
Pierre Dubois's user avatar
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Can we move curves which are members of very ample systems?

Let us take the second degree Hirzebruch surface $\mathbb{F}_2$ which is a holomorphic $\mathbb{CP}^1$ bundle over $\mathbb{CP}^1$ having sections of self intersections $+2$ and $-2$. Let me denote ...
apm's user avatar
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2 answers
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Holomorphic version of Darboux's theorem

I would like to ask if there is a holomorphic version of Darboux's theorem. More concretely, given a holomorphic symplectic manifold $(X, \omega)$ is there a local holomorphic symplectomorphism from $(...
Flavius Aetius's user avatar
4 votes
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107 views

Complexifed Gauge action on determinant line bundle and change of metric

Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $−2πi\Omega$ ...
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Complex manifolds whose Hodge numbers are rigid under small deformations

Let $M$ be a closed complex manifold. Assume that for any family of closed complex manifolds over the unit disk containing $M$ as the central fiber, there exists a sufficiently small neighbourhood of ...
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Local behaviour of the moduli space of almost complex structures (up to conjugation)

Assume $M$ is a closed smooth manifold of real dimension $\geq 4$. What is known about the geometry of the "space" of almost complex structures up to conjugation by diffeomorphisms? There are quotes ...
rori's user avatar
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Higher genus Gromov-Witten invariants and mirror symmetry

As a physicist, my understanding of mirror symmetry is very limited, and perhaps the most "mathematical" literature I have read on mirror symmetry is the book of M. Gross. In the genus-0 Gromov-Witten ...
Wenzhe's user avatar
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3 votes
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Constancy of Hodge numbers in a family of compact complex manifolds

Does there exist a family of compact complex manifolds over unit disk such that the Hodge numbers are not constant in the family? The answer is manifestly positive in complex dimension 1. It is ...
rori's user avatar
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Show Fiber Product of Rational Elliptic Surfaces is Calabi-Yau

In a handful of contexts people study Calabi-Yau threefolds formed by taking the fiber product of two rational elliptic surfaces. I can't find any detailed explanation of why such geometries are ...
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Does there exist a curve which avoids a given countable union of small subsets?

Let $X$ be a projective variety over $\mathbb{C}$. Let $X_1, X_2, \ldots$ be proper closed subsets of $X$. Then $\cup_i X_i \neq X(\mathbb{C})$. However, I am interested in a stronger statement. ...
Harry's user avatar
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1 answer
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Failure of Borel-Schmid quasi-unipotency theorem in non-algebraic case

Let $\mathcal{X} \to \Delta^{\times}$ be a smooth family of compact Kähler manifolds over a punctured disc. When this family is algebraic, the celebrated quasi-unipotency theorem (I think, due to ...
 V. Rogov's user avatar
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Properties of a particular Kummer Surface

Let $Y$ be the abelian variety $\mathbb{C}/\mathbb{Z}[i] \times \mathbb{C}/\mathbb{Z}[i] $ where $\mathbb{Z}[i]$ denote the set of Gaussian integers in $\mathbb{C}$. Let $X$ be the quotient of $Y$ by ...
Soby's user avatar
  • 157
1 vote
1 answer
278 views

Very ample linear systems - intersections with multiplicity >1

On a degree $n$ Hirzebruch surface $F_n$, suppose we have a very ample linear system. It is known that its generic smooth irreducible members give a Lefschetz pencil on $F_n$. Let us take a member, $G$...
apm's user avatar
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1 answer
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Compatible solution of PDE

Let $c=c(z, \bar z)$ be a complex function satisfying $\partial_{z} \bar c=\partial_{\bar z} c$. It follows that there exists a real function $f$ such that $\partial_{\bar z} f=-c$. Would it be ...
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2 answers
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Reference on Complex Geometry

For the preparation of a complex geometry lecture I am looking for a good literature. I already have standard literature like Huybrechts "Complex Geometry. An Introduction" and I am also using it. But ...
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9 votes
0 answers
368 views

Is it possible to glue together complex manifolds?

In the case of Riemannian manifolds, there are ways to take two manifolds and glue them together to get a new Riemannian manifold. For example, taking connected sums in local regions where the two ...
Kim's user avatar
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22 votes
1 answer
854 views

Is being of general type stable under generization

This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families. Definition. An integral projective ...
Ariyan Javanpeykar's user avatar
1 vote
1 answer
137 views

Equalizer of local analytic isomorphisms

Let $a,b : V\to W$ be two morphisms of smooth complex analytic spaces. Assume $a$ and $b$ are local analytic isomorphisms. Does the equalizer $U$ of $a,b$ exist as a smooth complex analytic ...
John P.'s user avatar
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7 votes
0 answers
177 views

Compact Kaehler submanifolds of projectivized Hilbert space

If we take a separable complex Hilbert space $H$, its projective space $PH$ is an infinite-dimensional Kähler manifold in a fairly obvious sense (see below). Suppose $M \subset PH$ is a finite-...
John Baez's user avatar
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2 votes
0 answers
308 views

First Chern Class of Contact Structure which is not Torsion

Let $(M,\xi)$ be a closed connected $3-$dimensional contact manifold with contact structure $\xi$. It is known that the first Chern class $c_{1}(\xi)$ defines an element in $H^{2}(M;\mathbb{Z})$ and ...
Raffael's user avatar
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2 votes
0 answers
104 views

(Singular) metric associated to the higher cohomology

Suppose $X$ is a smooth complex variety and $L$ is a line bundle with a metric $h_L$, then a section $s \in H^0(X, L)$ gives another metric $\tilde h_L:= e^{-\phi}h_L$ where $\phi=\log \|s\|^2_{h_L}$. ...
Li Yutong's user avatar
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5 votes
1 answer
197 views

Does profinite completion commute with mapping spaces?

Does there exist a prime number $p$ and a smooth complex projective variety $X$ such that $F_{\infty p}\mathrm{Map}(B\mathbb{Z}/p\mathbb{Z}, X)$ is not weakly homotopy equivalent to $\mathrm{Map}(B\...
geometer's user avatar
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8 votes
0 answers
217 views

Holomorphic structures on vector bundles over $\mathbb C\mathbb P^2$

It is known that every (topological) complex rank $2$ vector bundle over $\mathbb C\mathbb P^2$ admits holomorphic structures. A proof can be found in the book of Okonek, Spindler, Schneider which is ...
Panagiotis Konstantis's user avatar
8 votes
0 answers
290 views

$c_2$ of Calabi-Yau three-folds

Let $Y$ be a smooth compact Calabi-Yau three-fold (over $\mathbb C$, with $\pi_1(Y)=0$). Is it true that $c_2(Y)$ is Poincare dual to an effective curve? If not, can one construct a counter-example? ...
aglearner's user avatar
  • 14k
0 votes
1 answer
336 views

Counting the number of poles for rational functions in a coordinate ring of a curve

I'm working in an algebra of rational functions in compact Riemann surfaces with arbitrary genus. The idea I'm struggling is how to count the number $n$ of poles for the rational functions defined in ...
Felipe's user avatar
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0 answers
181 views

Complex manifolds as algebro-geometric objects

A result of Artin states that analytification of proper algebraic spaces over $\mathbb{C}$ defines an an equivalence of the category of proper algebraic spaces with the category of Moishezon spaces. ...
geometer's user avatar
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3 votes
0 answers
66 views

Fuchsian groups of singly branched covers

Let $X/\mathbb{C}$ be an algebraic curve with genus $g \geq 2$. Then by the uniformization theorem, with $X(\mathbb{C})$ viewed as a Riemann surface, it can be realized as the quotient $\mathbb{H}/\...
Stanley Yao Xiao's user avatar
6 votes
0 answers
352 views

Complex Riemannian metrics over real manifolds

There is a huge literature on complex manifolds and natural metrics over them, but I was unable to find references about Riemannian complex metric on real manifolds (for which we complexify tangent ...
Phil-W's user avatar
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3 votes
1 answer
49 views

Invariant submanifolds tangent to isotypic subrepresentations

Let $G$ be a Lie group acting on a complex manifold $M$. Let $p$ be an isolated fixed point. Let us look at the representation of $G$ on $T_pM$. Suppose $T_pM = \bigoplus V_i^{\oplus n_i}$ where $V_i$ ...
anna abasheva's user avatar
16 votes
1 answer
1k views

GAGA for stacks

I am curious about stacky generalizations of the following GAGA theorem: If $X, U$ are complex algebraic varieties of finite type, $X$ is proper and $f:X\to U$ is an analytic map then $f$ is ...
Dmitry Vaintrob's user avatar
2 votes
0 answers
91 views

Closed Kaehler--Einstein surfaces are complex ball quotients

Let $X$ be a closed Kaehler manifold of real dimension 4 endowed with a Kaehler--Einstein metric of negative curvature. Is it true that $X$ is isomorphic, as a Kaehler manifold, to a quotient of a ...
geometer's user avatar
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9 votes
0 answers
196 views

Batyrev's theorem in non-algebraic case

Let $X$ and $Y$ be two bimeromorphic closed Kaehler manifolds with trivial real $c_1$. Is it true that $b_n(X)=b_n(Y)$ for $n\geq 0$?
geometer's user avatar
  • 713
2 votes
0 answers
91 views

Automatic plurisubharmonicity for a non-negative function

I feel confused about a point in this very short paper. On the top of page 3, it is claimed that: If $S$ is a totally real submanifold in a compact almost complex manifold $(X,J)$, then any function ...
Hang's user avatar
  • 2,719
1 vote
0 answers
157 views

SL(2,R) invariant which are not SL(2,C) invariants

Consider four points, $\sigma_i$ i=1,2,3,4 on the line $\mathrm{Im}(z) = 0$ in the complex plane $\mathbb{C}$. Does it exist a rational function of these four points which is $\mathrm{SL}(2,\mathbb{R})...
giulio bullsaver's user avatar
14 votes
1 answer
798 views

An almost complex structure on $S^2\times ...\times S^2 / \mathbb{Z_2}$

Consider the product of $2n$ two-spheres $X_n=(S^2)^{2n}$. This manifold admits an orientation preserving involution that preserves the product structure and acts as the (orientation reversing) ...
aglearner's user avatar
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