Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

5
votes
2answers
154 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 ...
5
votes
1answer
144 views

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 ...
1
vote
0answers
39 views

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{\...
2
votes
0answers
172 views

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 $X$. ...
3
votes
0answers
107 views

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?
4
votes
0answers
147 views

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 ...
0
votes
1answer
104 views

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.
1
vote
0answers
91 views

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 ...
2
votes
0answers
65 views

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 ...
4
votes
0answers
269 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 ...
3
votes
1answer
124 views

Can we move curves which are members of very ample systems?

Let us take the second degree Hirzebruch surface F_2 which is a holomorphic CP^1 bundle over CP^1 having sections of self intersections +2 and -2. Let me denote the class of the -2 section by C and ...
3
votes
0answers
121 views

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 $(...
4
votes
0answers
35 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 $−2πi\Omega$ ...
5
votes
0answers
103 views

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 ...
3
votes
0answers
47 views

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 ...
7
votes
0answers
133 views

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 ...
3
votes
1answer
69 views

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 ...
6
votes
2answers
319 views

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 ...
4
votes
1answer
182 views

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. ...
5
votes
1answer
105 views

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 ...
0
votes
0answers
64 views

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 ...
-1
votes
0answers
73 views

Proper projections

Let $D \subset \mathbb{C}^k$ be your favorite complex domain. Suppose we are given a proper holomorphic mapping $f \colon D \to \mathbb{C}^{k+2}$. Let us take $k+1$ generic linear functions $l_i \...
1
vote
1answer
149 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, $G$...
0
votes
1answer
91 views

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 ...
1
vote
2answers
213 views

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 ...
9
votes
0answers
207 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 ...
17
votes
1answer
508 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 ...
1
vote
1answer
76 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 ...
3
votes
0answers
134 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-...
2
votes
0answers
143 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 ...
2
votes
0answers
96 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 $\phi=\log \|s\|^2_{h_L}$. ...
5
votes
1answer
152 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\...
6
votes
0answers
178 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 ...
8
votes
0answers
240 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? ...
0
votes
1answer
156 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 ...
5
votes
0answers
161 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. ...
3
votes
0answers
54 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}/\...
5
votes
0answers
160 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 ...
3
votes
1answer
41 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 $V_i$ ...
14
votes
1answer
868 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 ...
2
votes
0answers
73 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 ...
9
votes
0answers
167 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 $n\geq 0$?
2
votes
0answers
78 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 ...
1
vote
0answers
109 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})...
9
votes
1answer
384 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) ...
1
vote
0answers
72 views

Are there Lorentzian complex manifolds?

Quick and simple... Is it possible to define complex structures on Lorentzian manifolds? If so, Can you point me to some example(s)?
5
votes
1answer
319 views

On limits of manifolds

This question should be fairly elementary. I’d just like to check I’m not missing anything. Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds with transition maps $f_{t,s} : M_t\to M_s$,...
1
vote
1answer
108 views

Mostow rigidity for complex hyperbolic manifolds

A Riemannian manifold $(X,g)$ is hyperbolic if the sectional curvatures are constant and negative. A theorem of Mostow says that these manifolds are determined by their fundamental group. Theorem (...
6
votes
1answer
129 views

How to compute the Kahler potential of a Sasaki metric

The Question Given Hessian manifold $M$, there is a natural Kahler structures on $TM$. Is it possible to write the Kahler potential of these in terms of the Hessian potential? Background To ...
1
vote
1answer
137 views

Hodge numbers of compact Ricci-flat Kaehler manifold

Assume that $M$ is a closed connected Ricci-flat Kaehler manifold $M$ of complex dimension $n\geq 3$ with $h^{2,0}(M)=0$. Is is possible that $h^{n, 0}(M)\neq 1$ $h^{p, 0}(M)\neq 0$ for some $0< p&...