Questions tagged [complex-manifolds]

For questions about or involving complex manifolds.

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Manifolds whose tangent spaces have a special behavior

Consider an $n$-dimensional complex manifold $M\subset\mathbb{C}^N$ and let $$f:\mathcal{U}\subset\mathbb{C}^n\rightarrow \mathcal{V}\subset M\subset\mathbb{C}^N$$ be a local parametrization of $M$. ...
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  • 91
1 vote
0 answers
80 views

Maximal analytic continuation

I'm currently reading about the concept of a “maximal analytic continuation” from Forster's book Lectures on Riemann Surfaces (see Section 7). There are a bunch of definitions to unpack before I can ...
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  • 111
1 vote
0 answers
131 views

Isometries of the complex projective space for the Fubini Study metric

$\DeclareMathOperator\SU{SU}$I am trying to understand a geometric proof in our mathematical quantum mechanics lecture regarding Wigner's theorem in finite dimensions. We have already shown that it ...
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3 votes
0 answers
68 views

Complete intersections in complex manifolds

Let $X$ be a complex manifold of dimension $n$ and $Y\subset X$ a closed submanifold of codimension $k$. a) Say that $Y$ is a complete intersection if the ideal $I(Y)\subset \mathcal O(X)$ of global ...
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0 votes
0 answers
77 views

Phase (argument) of a complex top form

Let $\xi\in\Omega^{n,n}(X,\mathbb{C})$ be a volume form on a complex manifold $X$ of dimension $n$. For any other non vanishing top form $\eta\in\Omega^{n,n}(X,\mathbb{C}) $, there exists a function $...
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  • 765
5 votes
0 answers
262 views

Viewing an algebraic subset through hyperplane sections

Suppose $n \ge 3$ and $X$ is a path-connected subset of $\Bbb{CP}^n$ under the manifold topology. If for every complex hyperplane $H \subset \Bbb{CP}^n$, $H \cap X$ is a degree $k$ algebraic ...
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  • 685
2 votes
1 answer
80 views

Complement of complex submanifolds of codimension $\ge1$ is connected

Let $X,Y$ be complex manifolds of $\dim X=n$, $\dim Y=m>1$, $U\subset X$ open and $g\colon U\to Y$ holomorphic embedding. Then $g(U)$ is a submanifold of codimension $m-n\ge1$. It seems clear that $...
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  • 751
2 votes
0 answers
171 views

Riemann-Roch theorem for higher-dimensional complex manifolds

Does an analogue of the Riemann-Roch theorem hold for higher-dimensional complex manifolds? (Hirzebruch-Riemann-Roch theorem is for algebraic manifolds, but not for general complex manifolds, right?)
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8 votes
1 answer
308 views

Do we know any examples of complex surfaces where we have explicit knowledge of the Chern–Weil functions?

Let $X$ be a compact complex surface (smooth). Let $\gamma_1, \gamma_2$ denote the Chern–Weil functions. That is, if $\omega$ is a Kähler form on $X$ with volume form $\omega^2$, then $\gamma_1, \...
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7 votes
1 answer
244 views

Automorphism group of compact almost complex manifold

Does the automorphism group of a compact almost complex manifold carry a (canonical) Lie group structure? Part 3 of Theorem 4.1 in *"The automorphism group of a homogeneous almost complex ...
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  • 241
6 votes
0 answers
92 views

Kahler property and finite covering

Let $(M,\omega)$ be a compact symplectic manifold and $\pi:\tilde M\to M$ a finite covering. Clearly $(\tilde M,\pi^*\omega)$ is a compact symplectic manifold. Suppose we know that $(\tilde M,\pi^*\...
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  • 753
15 votes
1 answer
499 views

Proving algebraicity of compact Riemann surfaces without Chow's theorem

I am trying to write a report for a complex analysis class where I prove Riemann-Roch and apply it to prove algebraicity of compact Riemann surfaces. While writing this, I found that Riemann-Roch ...
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  • 253
5 votes
1 answer
187 views

Example of usual Laplacian does not respect bidegree for general hermitian manifolds

We know that the Kähler identity $\Delta=2\Delta_{\partial}=2\Delta_{\bar{\partial}}$ on a Kähler manifold $(X,g)$ implies that the usual Laplacian $\Delta:=dd^*+d^*d$ respects the bidegree, i.e. for ...
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1 vote
0 answers
81 views

Holomorphic mapping on a manifold approximating a constant map

Let $X,Y$ be complex manifold, $Y$ Stein. It sounds quite reasonable to formulate the following claim: given $y_0\in Y$, for every $\epsilon>0$ and $M\subset X$ compact, there exists an holomorphic ...
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  • 751
7 votes
0 answers
111 views

holomorphic functions on Stein manifolds reaching maxima in a given point on the boundary (Shilov boundary)

Let $M$ be a Stein manifold with smooth, strictly pseudoconvex boundary, and $x$ a point on its boundary. Is there a holomorphic function $f$ on $M$, smooth on the boundary, with strict maximum of $|f|...
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1 vote
1 answer
218 views

Interesting examples of direct image bundles

Let $\pi : W \rightarrow Y$ be a holomorphic fibration of complex manifolds. Let $L\rightarrow W$ be a holomorphic line bundle on its total space and denote by $$E^k_q := R^q \pi_*L^k$$ the direct ...
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2 votes
1 answer
144 views

A Riemann surface is automatically paracompact

[A question I remember from many years ago.] Definition A Riemann surface is a connected complex manifold $X$ of complex dimension one. This means that $X$ is a connected Hausdorff space that is ...
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13 votes
1 answer
912 views

Artin vanishing for Stein manifolds and restriction maps

In the setting of complex Stein manifolds $X$ of complex dimension $d$, the theorem of Andreotti--Frankel implies the vanishing of the singular cohomology group $H^i(X,\mathbb Z)=0$ for $i>d$. With ...
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4 votes
0 answers
121 views

How to judge whether an orbifold is good

My own case comes from dynamic system on compact complex manifolds. To be precise, let $M$ be a compact complex 3-dimentional manifold, $W^c$ a holomorphic foliation of M with 1-dimentional uniformly ...
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0 votes
0 answers
87 views

Complex manifold structure on a real manifold of real dimension 2n

A real manifold of dimension $2n $ sometimes can be given a complex structure. If I start with a $3-$dimensinal CW complex $X$ which can be embedded in $\mathbb{R}^6$ and get a neighbourhood of $X,$ ...
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  • 851
4 votes
1 answer
142 views

Family of Dolbeault operators on complex vector bundles over $\mathbb{CP}^1$

Let $\pi \colon E \rightarrow \mathbb{CP}^1$ be a complex vector bundle. It is a well-known fact that a Dolbeault operator on $\pi\colon E \rightarrow \mathbb{CP}^1$ gives a holomorphic structure on $...
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  • 369
4 votes
0 answers
163 views

Approximation of a holomorphic function vanishing at a submanifold by polynomials

Suppose $M$ is a complex affine algebraic manifold in ${\mathbb C}^n$ (I mean, a set of common zeroes of a system of polynomials on ${\mathbb C}^n$, which is at the same time a smooth manifold). ...
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1 vote
1 answer
142 views

Motivation for the definition of $L^p$ norm for quadratic and Beltrami differentials

According to Riemann surfaces, dynamics and geometry by C. McMullen (Course notes), the definition for a quadratic differential $\phi$ on a Riemann surface $X$ is given by $$ \|\phi\|_p = \left(\...
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  • 1,457
3 votes
0 answers
90 views

Which non-compact quaternion-Kähler spaces are Kähler?

The list of quaternion-Kähler compact symmetric spaces can be found here. I am curious to know which of the non-compact versions of these spaces are Kähler. If the answer is known also for non-...
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  • 31
1 vote
0 answers
89 views

Quasi-plurisubharmonic function with polynomial decay

Let $(M, \omega)$ be a compact Kähler manifold. An $\omega$-quasi-plurisubharmonic function on $M$ is an upper semi-continuous function $\varphi : M \to \mathbb{R} \cup \{ - \infty \}$ such that $\...
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2 votes
1 answer
208 views

Hyperkahler and symplectic complex geometry: reference?

I would need some references regarding symplectic and hyperkahler (complex) geometry. My background is mostly from algebraic geometry and I know a little bit the basics on Kahler manifolds. I would be ...
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3 votes
1 answer
254 views

Finite self-maps exist on rigid CY3s

Let $X$ be a smooth projective rigid Calabi-Yau threefold. Question. Does there exist a finite map $X\to X$ of degree $>1$?
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  • 31
7 votes
0 answers
210 views

Is there a reasonable definition of an octonionic manifold?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\End{End}$ Todorov and Dubois-Violette have recently shown how to understand the structural gauge group of the standard model via octonions. Q. Is there ...
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5 votes
1 answer
938 views

An almost complex structure on the real $n$-sphere $S^n$

If $R\mathrel{:=}\mathbb{R}[x_1,\dotsc,x_{n+1}]/(x_1^2+\dotsb+x_{n+1}^2-1)$ and $S^n\mathrel{:=}\operatorname{Spec}(R)$ is the real $n$-spere, a classical result of Borel and Serre says that the only ...
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2 votes
1 answer
104 views

Quasiconformal map from a subset of $\mathbb{C}$ to a polytope

Question. Does a quasiconformal map exist between a subset of $\mathbb{C}$ (such as a unit disc or rectangle) and a polytope? Here, I take a polytope to be a two-dimensional surface that could be ...
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  • 475
0 votes
0 answers
83 views

Change in Connection on a complex Line bundle

Let's say $M$ is a compact Kähler manifold and $L$ is a complex line bundle on $M$. Now let's say $A$ be a connection or equivalently a hermitian metric on $L$. Hence one can have the operators $\bar\...
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  • 485
3 votes
1 answer
228 views

Irreducibility of the base and of the general fiber

Let $f:X\rightarrow Y$ be a morphism of scheme over $\mathbb{C}$. Assume that $Y$ and the the general fiber $F_y = f^{-1}(y)$ of $f$ are irreducible. Does there exists an irreducible component $X'$ of ...
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3 votes
0 answers
339 views

Consequence of the failure of Nagata's conjecture

A modern version of the Nagata's conjecture says that $$ L_{N,t}:=f_{N}^{*}(-K_{\mathbb{P}^{2}})-t\sum_{j=1}^{N}E_{j} $$ is Ample for any $t<\frac{3}{\sqrt{N}}$, where $f_{N}:Y_{N}\to \mathbb{P}^{2}...
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  • 71
2 votes
0 answers
186 views

(1,1)-form that does not come from a divisor

Let $M$ be projective complex manifold. The Lefschetz (1,1)-theorem says that the cycle map $$ \text{cl}:\operatorname{Pic}(M) \to \text{Hod}^1(M) $$ is surjective. Question. Is there an interesting ...
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9 votes
1 answer
427 views

Status of a conjecture of Hirzebruch

I was reading a paper from 1994 which claimed that the following statement was a conjecture of Hirzerbruch: If a complex surface X is homeomorphic to either $S^2 \times S^2$ or $\mathbb{C}P^2 \# \...
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  • 663
11 votes
1 answer
417 views

Holomorphic Urysohn Lemma

Let $M,N$ be two disjoint closed holomorphic submanifolds of $\mathbb{C}^n$. Is there a holomorphic map $f:\mathbb{C}^n\to \mathbb{C}$ with $f(M)=0,\;f(N)=1$.
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1 vote
1 answer
113 views

Real part of the Ward correspondence

I am currently very confused about the real side of the Ward correspondence. Recall that the Ward correspondence gives a one-to-one correspondence between: $M$-trivial holomorphic bundles $E$ on $Z$, ...
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  • 133
3 votes
1 answer
226 views

Equivalent definitions of normality for complex algebraic varieties

In Kollár's article The structure of algebraic threefolds: an introduction to Mori's program he gives the following definition of a normal variety: Definition 5.4. Let $V \subset \mathbb{C}^n$ be an ...
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1 vote
1 answer
142 views

Deform a complex structure fixing marked points

Let $\Sigma$ be a closed orientable surface of genus $g$ with $m$ marked points $x=\{x_1, \ldots, x_m\}$ and $j_0$ denote a complex structure on $\Sigma$. Take a neighborhood $U$ of the isomorphism ...
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  • 369
4 votes
1 answer
370 views

(Contradiction) All symplectic manifolds are holomorphic

I’m studying symplectic manifolds and almost complex structures. This lead to two propositions: Proposition 1 (from da Silva’s Lectures on Symplectic Geometry): If $J_0$ and $J_1$ are almost complex ...
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2 votes
0 answers
46 views

complex eigenspace sub-bundle of holomorphic tangent bundle

Let $X$ be a compact K"ahler manifold and let $\varphi: TX\rightarrow TX$ be a self-adjoint endomorphism of the holomorphic tangent bundle, with the property that at each point, the eigenvalues ...
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2 votes
0 answers
50 views

Regarding projective manifolds with decomposable real tangent bundle

Let $X$ be a complex projective manifold. Suppose its real tangent bundle $T_{\mathbb{R}}X$ splits as a direct sum of two sub-bundles of even rank. Does this give any useful information about the ...
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2 votes
0 answers
72 views

Finding $\Omega$ such that the 1-form $\Omega^2 \omega$ is $L^2$ orthogonal to conformal killing vector fields on $S^2$

Consider the space $\mathcal{A}$ of functions $\Omega$ such that $\Omega^2 \gamma_0$ is isometric to the round sphere, where $\gamma_0$ is the round sphere. (so $\Omega^2 \gamma_0$ is of constant ...
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  • 683
4 votes
1 answer
329 views

Exotic $\mathbb{R}^4$ with a complex structure?

Is there an exotic $\mathbb{R}^4$ admitting an integrable almost complex structure?
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  • 1,353
0 votes
0 answers
107 views

Fundamental theorem for real submanifolds into complex space forms

It is a well-known result that Gauss, Codazzi and Ricci equations are necessary and suficiently conditions to guarantee the existence of an isometric immersion of a given $n$-dimensional real ...
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  • 193
32 votes
1 answer
1k views

About a claim by Gromov on proper holomorphic maps

At p. 223 of his paper [G03], Mikhail Gromov makes the following claim: Let $X$, $Y$ be two complex manifolds (not necessarily compact or Kähler) of the same dimension and having the same even Betti ...
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10 votes
1 answer
656 views

Is every surjective holomorphic self-map on a compact complex manifold finite-to-one?

I have already asked this question on stack exchange, but I didn’t get any answer. Let $X$ be a compact connected complex manifold. Let $f:X \to X$ be a surjective holomorphic map. Is it true that $f$...
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-2 votes
1 answer
153 views

$f$ is the complexification of a map if $f$ commutes with almost complex structure and standard conjugation. What if we had anti-commutation instead? [closed]

I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures ...
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3 votes
0 answers
190 views

Is there a compact complex surface $X$ with $c_2(X)=7+6n$ and $c_1^2(X)=17+18n$?

As stated in [1], most pairs of positive integers $c_1^2$, $c_2$ satisfying $c_1^2+c_2=0$ $\mod 12$, the BMY inequality and the Noether inequality are actually Chern numbers of compact complex ...
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  • 7,982
4 votes
1 answer
385 views

Complexification of realification: Why compute eigenvalues? [closed]

I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures ...
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