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Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

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Is a domain of a holomorphic flow pseudoconvex?

Let $Z$ be a holomorphic vector field on $\mathbb{C}^n$. I would like to know whether (it seems that it is) the domain $D_\phi \subset \mathbb{C} \times \mathbb{C}^n$ of a maximal flow $\phi: D_\phi \...
25
votes
2answers
726 views

References for Riemann surfaces

I know this question has been asked before on MO and MSE (here, here, here, here) but the answers that were given were only partially helpful to me, and I suspect that I am not the only one. I am ...
5
votes
0answers
137 views

Diffeomorphism type of Ricci-flat four manifolds

Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows: A) Is there a classification of the possible homeomorphism types of ...
3
votes
1answer
97 views

The ample cone of a surface with an algebraic $\mathbb C^*$ action

Let $X$ be a compact complex protective surface that admits a nontirvial algebraic $\mathbb C^*$-action. It seems to me, that the ample cone of $X$ is polyhedral with finite number of faces. I wonder ...
6
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1answer
231 views

Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$

Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.) Is it true that for any ...
3
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0answers
113 views

Cohomology of a tubular neiborhood of submanifold vs cohomology of the formal neighborhood

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $\hat Z$ be the formal neighborhood of $Z$. Let $U$ be an open neighborhood of $Z$. Is ...
5
votes
1answer
404 views

Are there enough meromorphic functions on a compact analytic manifold?

Let $X$ be a compact complex analytic manifold, $D\subset X$ an irreducible smooth divisor, given as zeroes of a global meromorphic function $f\in {\mathfrak M} (X)$. Are there enough other ...
8
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0answers
223 views

Cohomology of complex manifold vs cohomology of its complex submanifold

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that $$H^i(Z, A|_Z)=0 \mbox{ for any } ...
3
votes
1answer
202 views

Does a projective variety have only finitely many associated Hilbert polynomials?

Let $X$ be a projective variety over $\mathbb{C}$. If $L$ is an ample line bundle, then $h_L$ denotes the Hilbert polynomial. Is it true that, if $L$ and $L'$ are ample line bundles which are ...
5
votes
2answers
108 views

Lelong numbers and integrability of psh functions

Let $\varphi$ be a plurisubharmonic function in the unit ball $B_1\subset \mathbb{C}^n$ with $\varphi\le 0$. Suppose that the Lelong number $\nu(\varphi,0)<k$ for some $k>0$. Does it follow that ...
4
votes
1answer
304 views

Kahler manifolds and algebraic varieties

Let $X$ be a smooth complete algebraic variety over $\mathbb{C}$. Can it happen that the underlying complex manifold is not Kahler? If yes, are there explicit examples? If not - how to prove this?
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1answer
145 views

Is the stack of stable curves with no rational component algebraic?

Let $g\geq 2$ be an integer and let $\overline{\mathcal{M}}_g$ be the (smooth proper Deligne-Mumford) algebraic stack of stable curves of genus $g$. Let $\mathcal{M}_g^{nr}$ be the substack of ...
4
votes
0answers
34 views

numerically approximating the conformal map between two curvilinear triangles to high precision

Here is a triangular region $T$ whose two curved edges are complicated analytic curves that I know only numerically, but can compute to any desired precision: And here is a simpler region $H$ whose ...
5
votes
0answers
212 views

Have complex manifolds with dual number structure on the holomorphic tangent bundle been studied?

If $M$ denotes a $2n$-dimensional real smooth manifold, then $M$ together with $J\in\operatorname{End}(TM)$ with $J^2 =\mathrm{id}$ (also called almost product structure on $M$) have been surveyed in ...
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0answers
126 views

Analytic Aspects of Rational Maps

I would like some help finding references for a analytic treatment of rational maps between compact complex manifolds (that is holomorphic maps defined away from a codimension at least 2 subvariety). ...
1
vote
1answer
49 views

Minimal complex surfaces with pseudo-effective canonical bundles

A complex line bundle $L$ over a complex surface $X$ is said to be pseudo-effective if it admits a (possibly singular) Hermitian metric $h$ whose curvature is positive semi-definite in the sense of ...
4
votes
2answers
215 views

Question on PDEs which are related to certain geometric problems (e.g. Calabi conjecture, Gauduchon conjecture)

There are interesting symmetric functions $P_k$ arising from differential geometry and PDEs, where $P_k$ is given by \begin{equation} \begin{aligned} P_k(\lambda) = \prod_{1\leq i_{1}<\cdots < ...
19
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1answer
590 views

Hodge decomposition and degeneration of the spectral sequence

I am teaching a course on Hodge theory and I realised that I don't understand something basic. Let first $X$ be a compact Kahler manifold. Let $H^{p,q}(X)=H^q(X,\Omega^p_X)$ where $\Omega^p_X$ is the ...
2
votes
2answers
354 views

Conformal mappings that preserve angles and areas but not perimeters?

Conformal mappings from $U$ to $V$, both subsets of $\mathbb{C}$, locally preserve angles. But, in general, such mappings neither preserve areas nor preserve perimeters. Q. Are there examples of ...
4
votes
1answer
154 views

Hodge decomposition of the symmetric product of a curve

Let X be a smooth projective connected curve over $\mathbb{C}$ and let $n>1$ be an integer. Let $Y= Sym^n_X$ be the $n$-th symmetric product of $X$. Is there, for every $i$, a nice formula ...
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0answers
386 views

What are hyperkähler metrics used for?

It seems that a lot of effort has been devoted to endow holomorphic-symplectic manifolds with hyperkähler metrics. It started with Calabi [4] with $T^*\mathbb{CP}^n$. Other examples include coadjoint ...
3
votes
1answer
117 views

Sections of tangent bundle on hypersurface

If $X\subset \mathbb{P}^n$ is a smooth hypersurface (more generally a complete intersection) of dimension at least 2, and if $K_X+\mathscr{O}_X(-1)\geq 0$, why is it true that $H^0(T_X(1))=0$? (...
11
votes
1answer
199 views

On non-representability of certain hom schemes

Let $k$ be an algebraically closed field. It is well-known that the isom-sheaf Isom$(\mathbb{A}^1_k,\mathbb{A}^1_k)$ is not representable by an algebraic space. (To be clear, the functor Isom$(\mathbb{...
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votes
0answers
96 views

Unobstructedness of nodal holomorphic curve in symplectic manifold

Suppose $(X,\omega)$ is a compact symplectic manifold and $J$ is an $\omega$-compatible almost complex structure on $X$ (the symplectic structure seems to be irrelevant for this question actually). ...
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vote
2answers
272 views

Is there an example to show the Hodge decomposition fails on non-compact case

The theorem of Hodge decomposition is on the compact Kahler manifold, is it generally true for the non-compact kahler manifold or are there examples to show the failure? Here is my Hodge ...
2
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0answers
106 views

Equivariant proper modifications of $\mathbb{C}^{n}$

Let $f:X\to Y$ be a proper surjective holomorphic map between two $n$-dimensional connected complex manifolds $X$ and $Y$. $X$ is called a proper modification of $Y$ if there are nowhere dense compact ...
3
votes
1answer
151 views

Homotopy of paths at the boundary

Denote by $\Gamma$ a hypersurface in $\mathbb{C}^2$, i.e. the zero locus of a polynomial of two complex variables. Denote by $X$ the complement of $\Gamma$ in $\mathbb{C}^2$. I am trying to define a ...
9
votes
1answer
327 views

DGLA controlling deformation of holomorphic curves

Suppose $C$ is a compact Riemann surface and $X$ is a compact Kähler manifold. Suppose $f:C\to X$ is a stable holomorphic map. Then, the deformations of $f$ are controlled by the complex $L^\bullet = ...
3
votes
1answer
188 views

Proper modifications of $\mathbb{C}^{n}$

Let $f:X\to Y$ be a proper surjective holomorphic map between two $n$-dimensional connected complex manifolds $X$ and $Y$. $X$ is called a proper modification of $Y$ if there are nowhere dense compact ...
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33 views

An effective torus action on compact Kähler manifold satisfying some property

Let $M$ be a compact connected Kähler manifold of real dimension $n$. Assume that $M$ is not diffeomorphic to a product of a smooth projective toric variety and a torus. Let $k$ be a positive ...
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vote
2answers
149 views

Can we extend a logarithmic form to some appropriate compactification?

Given some simple normal crossings divisor $D$ on a complex manifold $X$, which is not assumed to be compact. Given a form $\omega\in H^0(X,\Omega_X^1(\log D))$, when is it possible to find a ...
4
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1answer
195 views

Complex Structure on Manifold of Maps

Suppose $M$ is a compact smooth manifold and $V$ is a compact complex manifold. I want to show that the spaces $C^{k,\alpha}(M,V)$ and $W^{k,p}(M,V)$ (the latter for $kp>\dim M$) are complex ...
5
votes
2answers
236 views

Complex Analytic Structure on Moduli Space of Stable Maps

Suppose $(X,\omega,J)$ is a compact Kähler manifold, and $\beta\in H_2(X,\mathbb Z)$ is given. Then, we can form the space $\overline{\mathcal M}:=\overline{\mathcal M}_{0,0}(X,\beta)$ of stable maps $...
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0answers
54 views

Deformation theory of holomorphic vector bundles in Donaldson-Kronheimer

There is the Proposition 6.4.3 in Donaldson-Kronheimer as follows: Proposition (6.4.3) (i) There is a holomorphic map $\psi$ from a neighborhood of $0$ in $H^1(\operatorname{End} \mathscr{E})...
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1answer
120 views

Does the holomorphic curvature determine the connection?

Let $M$ be a simply connected complex manifold (of dimension greater than one), $L$ a line bundle, and $\nabla$ a connection on $L$ with possibly singularities along a divisor $D$. We define the ...
2
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1answer
181 views

non-existence of global coordinates

Assume we have a smooth manifold, $M$, of dimension $n$. (An example of interest is the case when $M$ is a compact and orientable Riemann surface of genus $g$, but the question is intended to be broad....
3
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1answer
262 views

Chow theorem in $\mathbb{C}^2$

I have the following question the answer to which I cannot find in the literature (but it must have been studied): Suppose that $M\subset\mathbb{C}^2$ is a real surface which may locally be written ...
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0answers
44 views

Differential operator of globally unbounded order on connected complex manifold?

Let $X$ be a connected complex manifold. Consider the module $\mathcal{D}_X$. I recall hearing somewhere that one has to be careful with regards to differential operators that are not globally of ...
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0answers
79 views

Universal cover of ladderly puntured complex plane

The twice punctured complex plane $\mathbb{C}-\{0,1\}$ has as its universal cover the upper half plane via elliptic modular function. I am looking for the constructions of the covering map from the ...
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0answers
292 views

Is Serre duality related to Pontryagin duality?

I am wondering if there is some relationship between Serre duality and Pontryagin duality for compact complex manifolds. In this case Serre duality reduces to the commutativity of Hodge-star operator ...
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0answers
37 views

Is singular Cauchy operator bounded in Morrey spaces?

The singular Cauchy operator is defined by $$S_\Gamma :f \to \int_\Gamma \frac{f(\xi)}{\xi-z} d\xi , z\in \Gamma.$$ Is this operator bounded in Morrey spaces and weighted Morrey spaces? i.e. is there ...
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0answers
93 views

Can the base of an elliptically fibered Calabi-Yau threefold be an Enriques surface?

For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$. If ...
6
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0answers
79 views

Good reduction of abelian varieties over valuation rings via coverings

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$, and let $A$ be an abelian variety over $K$. Suppose that there is a smooth proper scheme $\mathcal{X}$ over $\mathcal{O}_K$ whose ...
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votes
2answers
316 views

Do abelian varieties have Neron models over arbitrary valuation rings?

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$. Let $A$ be an abelian variety over $K$. Does $A$ have a Neron model? If $\mathcal{O}_K$ is a discrete valuation ring, then this is ...
7
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0answers
152 views

Explicit descriptions of a flop

I want to know how to describe explicitly the flop of the following flop contraction. Because the construction is so natural and simple, I was wondering such descriptions should already exist in the ...
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0answers
67 views

What is the unitary $1$-parameter group generated by a vector field on a manifold?

If $M$ is a (compact, Riemannian) manifold, and one complexifies its tangent bundle, is it possible to give a meaning to the "unitary group"-like expression $t \mapsto \exp (\mathrm i t X)$ when $X$ ...
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vote
1answer
86 views

Chain rotation of a point

Let $n$ be a positive integer number and $P$ be a point in a plane. Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ points in the plane, we take modulo $m$ for $A_j$ (it is mean $A_{m+i}=A_{i}$ for $i=1, 2, \...
3
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1answer
164 views

Formal complex manifold without dd^c

Is there an example of compact complex manifold, which is formal, but does not admit complex structure satisfying $dd^c$-lemma?
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0answers
124 views

What is the Jarlskog invariant, conceptually?

Let $U$ be a $3\times 3$ unitary matrix, and call $(u_{ij})$ its coefficients. For $i,j,k,\ell$ in $\{1,2,3\}$ with $i\neq j$ and $k\neq\ell$, consider the quantity: $$J_{ij,k\ell} := \operatorname{...
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votes
4answers
260 views

Lattices of PU(n,1) with large abelianization

I am interested in properly discontinuous cocompact subgroups of the group $PU(n,1)$ of automorphisms of the complex hyperbolic space $H^n_{\mathbb{C}}$, says for $n=2,3$. Is there such a lattice $G$ ...