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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1answer
307 views
What is the “complex third derivative”?
Background
I am including this information about real higher order derivatives because it does not seem to be common knowledge. I also include a review of the complex Hessian.
If $f:\mathbb{R}^n ...
4
votes
2answers
326 views
Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$
This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better.
I believe it is a theorem of Grauert that any holomorphic vector ...
1
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0answers
55 views
About the integral of the plurisubharmonic functions on the whole manifold
Let $(M,\omega)$ be a Kahler manifold with positive first Chern class. $\omega\in c_1(M)$ is a Kahler current. $D$ is a smooth simple divisor in $-K_M$. $s$ is a determining section of $D$. The ...
3
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0answers
87 views
Moment map in the singular case
The moment map is defined on the symplectic manifold $(M,\omega)$, or particularly, $(M,\omega)$ is Kahler. While $\omega$ is smooth or differential enough, the definition is obvious to understand, in ...
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0answers
115 views
Moment map of CP^1 as rational normal curve
I am little confused about some basic symplectic geometry about Hamiltonian actions on sphere. I appreciate your comments.
Consider sphere $S^2 = \mathbb{C}P^1$ with its standard symplectic (Kaehler) ...
3
votes
1answer
106 views
3d-analog of “every 2d oriented manifold is complex”
Is there an analog of the statement of "every 2d oriented surface is a complex manifold"?
I saw a theorem in Blair's book, that "every 3d contact metric manifold is a strongly pseudo convex CR ...
3
votes
1answer
150 views
Smooth mixed hodge modules - representations of fundamental group?
I do not know much about mixed Hodge modules. I would like to ask: Let $X$ be a smooth connected algebraic complex variety, with a chosen point. Could one describe smooth mixed Hodge modules on $X$ as ...
2
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1answer
192 views
Cotangent bundle of coadjoint orbit is stein manifold?
Let me first define stein manifolds and coadjoint orbits.
A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold:
$X$ is holomorphically convex, ...
2
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1answer
340 views
Topological degree and polynomial degree
Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ...
0
votes
1answer
134 views
Kahler structure on holomorphic principal bundles
Let $G$ be a compact complex Lie group and $M$ be a compact Kähler manifold.
Does there exist any example of a holomorphic principal $G$-bundle over $M$ admitting Kähler structures?
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1answer
115 views
Arithmetic property of a surface of general type
In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general ...
2
votes
2answers
207 views
Etale covers of a hyperelliptic curve
Let $X$ be a hyperelliptic curve of genus at least two.
Let $Y\to X$ be a finite etale morphism with $Y$ connected.Then $Y$ is a smooth projective connected curve.
Is $Y$ hyperelliptic?
More ...
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1answer
135 views
A family of examples of (Brody) hyperbolic surfaces
Let $P(x)$ and $Q(x)$ be two general polynomial of the same degree $d$ (d can be arbitrary large). Consider the surface $S : z^2 = P(x) Q(y)$ in the projective space $\mathbb{P}^3$. I want to prove ...
2
votes
1answer
75 views
Uniform estimate for the Cauchy-Riemann equations on a hyperbolic Riemann surface
I have been trying to find the answer to this question in the literature, but have not succeeded. The question is as follows.
Suppose $X$ is a Riemann surface that admits a green's function (i.e. ...
2
votes
1answer
127 views
Atiyah classes of holomorphic vector bundles with trivial Chern classes
Let $X$ be compact Kahler and $E \to X$ a holomorphic vector bundle. Then $E$ has an Atiyah class, $At(E)$, valued in the sheaf cohomology $H^1(\Omega_X \otimes \operatorname{End} E)$. Suppose the ...
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vote
0answers
76 views
Translation of “Über kompakte homogene Kählersche Mannigfaltigkeiten”
Has anyone translated Borel and Remmert's 1962 paper titled:
Über kompakte homogene Kählersche Mannigfaltigkeiten?
2
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1answer
106 views
What are the known examples of compact complex $n$-folds that do not have a strongly Gauduchon metric for $n >2$?
Gauduchon showed that every conformal hermitian structure on a compact complex $n$-fold contains an hermitian metric such that the associated 1,1-form $\omega$ satisfies $\partial ...
3
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0answers
216 views
Hirzebruch's ICM talk
Is there an English translation of Hirzebruch's 1958 ICM lecture note:
KOMPLEXE MANNIGFALTIGKEITEN
Thank you very much!
3
votes
0answers
107 views
Is there any advantage to knowing that Gauss-Manin is Hermitian flat?
Let $S$ be a complex manifold and let $p : E \to S$ be a holomorphic vector bundle. Is there any advantage to knowing that $E$ carries a flat Hermitian metric $h$, ie a smooth Hermitian metric with ...
1
vote
0answers
82 views
algebraic varieties whose fundamental group is subgroup separable wrt subvariety subgroups
Call an algebraic variety $\pi_1$-subgroup separable iff,
for every $Y\subseteq X$ a closed subvariety and $\hat Y\xrightarrow{i} Y$ a normalisation of $Y$,
and subgroup $\Gamma=Im(\pi_1(\hat ...
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0answers
64 views
filling by holomorphic disks method
Can you give me a reference for the proof of the filling by holomorphic disks method, besides Bishop's original paper?
2
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1answer
128 views
mixed Hodge structure on Cohomology with compact support
Let $X$ be a smooth quasi-projective variety of dimension $n$ over the complex numbers (let us assume it is the complement of a normal crossings divisor in a smooth projective variety).
Then how does ...
10
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0answers
197 views
What is the holomorphic sectional curvature?
Let $X$ be an $n$-dimensional complex manifold, let $\omega$ be a Kahler metric on $X$ and let $R$ be the $(4,0)$ curvature tensor of $\omega$. We can simplify the tensor $R$ in different ways, two of ...
4
votes
1answer
84 views
Weyl group action on complexified Iwasawa decomposition
Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [SW02]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. ...
1
vote
2answers
222 views
The Schottky group and the fundamental group of a compact Riemann surface
I am quoting the following description from a paper,
"...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a ...
1
vote
1answer
138 views
A question about the first eigenvalue for two Kahler metrics
While reading the paper of Gang Tian, "Kähler-Einstein metrics with positive scalar curvature". In the proof of Theorem 1.6, he pointed that if two Kahler metrics $\omega $ and $\omega'$ satisfies ...
8
votes
4answers
241 views
Automorphism group of flag manifolds?
If $F=F(n_1, \ldots, n_k)$ is the (complex) flag variety associated to the partition $(n_1, \ldots, n_k)$, what is the automorphism group of $F$? Here I mean holomorphic and/or variety automorphisms.
...
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votes
0answers
67 views
moduli space of equivariant holomorphic embeddings into the quintic
I'd like to understand better the following problem, whether it is mathematically well-posed, trivial, etc.
Fix a non-negative integer $g$ and consider
the space
...
1
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1answer
232 views
A question about complex polarization
Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar P ...
2
votes
1answer
164 views
almost holomorphic line bundles
Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...
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vote
1answer
70 views
Do Hermitian metrics also split on the Riemann sphere?
Maybe this is well known, but i could not find a pointer to some literature:
Let us assume $E$ is a rank n vector bundle on the Riemann sphere $\mathbb{C}\mathbb{P}^1$. We know that ...
3
votes
1answer
178 views
Newlander-Nirenberg in dimension 2
What is the easiest (and what is the most elementary) way of proving
Newlander-Nirenberg theorem for Riemannian surfaces? I was able to reduce
it to existence of non-trivial harmonic functions ...
5
votes
2answers
263 views
Making Hironaka's theorem explicit for hypersurfaces
Given a smooth hypersurface $H$ in $\mathbb{C}^n$, a theorem of Hironaka promises that one can find a strict normal crossings compactification $\bar{H}$ inside of a projective variety $X$. For me, ...
3
votes
1answer
135 views
$\mathbb{P}^1$-fibrations over $\mathbb{P}^2$ that are not rational
We know that if $X$ is a smooth complex projective variety and we assume that there is a dominant morphism $f : X \to Y$ with $Y$ and the general fibers of $f$ rationally connected. Then $X$ itself is ...
1
vote
0answers
78 views
Algebraic set given by sequence of polynomials
When working on some problem, I have end up with a following situation. Suppose
$P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...
5
votes
1answer
127 views
The proof of Belyi theorem by Lando and Zvonkin
I'm sorry for asking such a specific question, but i have trouble understanding one detail in the proof of Belyi's theorem in the book "Graphs on surfaces and their applications" by Lando and Zvonkin" ...
7
votes
1answer
264 views
Log forms and Tate classes
Let $X$ be a smooth finite type variety over $\mathbb{C}$. Suppose that $\theta$ is a closed algebraic $1$-form whose cohomology class is weight $2$.
Can we always express $\theta$ as
$$\theta = ...
1
vote
0answers
66 views
Boundary of fibers of submersions
Let $M$ be a smooth manifold with boundary (say of dimension $m$) and let $N$ be a smooth manifold with no boundary (say of dimension $n$ with $m\geq n$). We have the following classical result:
...
9
votes
2answers
506 views
Vector bundles on vector bundles
Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base?
In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X ...
-1
votes
1answer
125 views
Order of the zero of a meromorphic function under the action of $Gal(\mathbb{C},\mathbb{Q})$
Let's take $X$ a Riemann surface as an algebraic curve in $\mathbb{P}^n$.
The group $Gal(\mathbb{C},\mathbb{Q})$ (automorphisms of $\mathbb{C}$ which act as the identity on $\mathbb{Q}$) acts on $X$ ...
5
votes
1answer
271 views
Root space decomposition
What is the root system for the special unitary lie algebra $\mathfrak{su}(p, q)$. Remind that these are matrices of the form
$\left(
\begin{array}{cc}
X & Y \\
\overline{Y}^t & Z ...
0
votes
1answer
51 views
Quasiconformal extensions of diffeomorphisms
Let $\gamma:\mathbb R\to\mathbb R$ be an increasing diffeomorphism. Then it is well known that there exist quasiconformal mappings of the upper half plane which extends $\gamma$. One way to construct ...
2
votes
0answers
109 views
counterexample to the Chern number inequality on Fano manifold-II
Several days ago I asked in
counterexample to the Chern number inequality on Fano manifold
that whether there exists an $n$-dimensional Fano manifold such that it does not satisfy Yau's Chern number ...
2
votes
1answer
190 views
fearful of defining equivalent germs for non isolated singularities
Two power series $G(x_1, \ldots, x_n)$ and $F(x_1, \ldots, x_n)$ are equivalent over $\mathbb{C}$ if there is an automorphism of the ring
$\mathbb{C}[[x_1, \ldots, x_n]]$ given by $x_1 \to \phi(x_1, ...
1
vote
1answer
106 views
Bakry-Emery Laplacian and Hodge Decomposition
I have a question about the Hodge Decomposition theorem. Let $(X,\omega)$ be a compact Fano Kaeler manifold, we know the Hodge theorem works very well with respect to the $\bar\partial-$Laplacian ...
8
votes
1answer
238 views
counterexample to the Chern number inequality on Fano manifold
We know that if an $n$-dimensional Fano manifold admits a Kahler-Einstein metric, it satisfies the following Chern number inequality
$$nc_1^n\leq 2(n+1)c_2c_1^{n-2}.$$
My question is whether there ...
1
vote
1answer
124 views
Algebraic Hodge decomposition of CM abelian varieties
On p. 205 of Katz's paper entitled "p-adic L-functions for CM fields" Katz says that
"Shimura's algebraicity theorem, in our context, is an easy consequence of the fact that Hodge decomposition of ...
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0answers
69 views
Groups and triangle-square complexes
I would like to know what kind of groups (and/or their group presentation) acting geometrically on CAT(0) curved piecewise Euclidean triangle-square complexes.
Thanks
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votes
0answers
52 views
Singularities of the Remmert reduction of a holomorphic convex manifold
Let $X$ be a holomorphically convex manifold, namely, for any infinite discrete subset of $X$ there exists a holomorphic function on $X$ which is unbounded on this set, then a theorem of Remmert says ...
3
votes
1answer
137 views
Higher Weierstrass points on curves of genus 3
So this question is directly related to a comment made by David Mumford in his
Lecture 1 given at U. Michigan in 1974 entitled: What is a curve and how explicitly can we describe them ?
Mumford ...
