Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.
0
votes
0answers
34 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
votes
0answers
22 views
Contour deformation and inversion [on hold]
I am working on a research problem and would like to know peoples views about what is contour deformation and its inversion. It will be appreciable if one can explain it with the help of an example.
1
vote
1answer
188 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
145 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 ...
1
vote
1answer
61 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
0answers
79 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
251 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
116 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
64 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
111 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
248 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
60 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:
...
8
votes
2answers
469 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
108 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
256 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
46 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
100 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
129 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
votes
0answers
126 views
Does the Cech cohomology of punctured complex tori vanish? [closed]
I am trying to understand the paper "Osservazioni sui nuclei di Bochner-Martinelli" of Paolo Zappa.
Let $\Gamma$ be an integer lattice in $\mathbb{C}^n$. $\Gamma$-invariant and $ d^{''}$-closed in ...
1
vote
1answer
98 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
230 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
115 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 ...
0
votes
0answers
59 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
2
votes
0answers
44 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
132 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 ...
3
votes
2answers
164 views
Connectedness of a section of an algebraic bundle
Let $X$ be a complex projective variety, $E$ be a rang $n$ bundle with $n<dim X$ and $s$ be a (holomorphic) section of $E$.
There is a relatively straightforward criterium to check if the space ...
3
votes
0answers
232 views
holomorphic embeddings of the sphere into the quintic in degree 2
Is there an explicit way of
classifying (with regard to their compatibiliy with $\Omega_+$ or $\Omega_-,$ see below)
the various families of
equivariant holomorphic embeddings from $\mathbb{CP}^1$ to ...
1
vote
0answers
64 views
Example of n-parameter family of real-analytic diffeomorphisms acting on $S^3$, constant on the Hopf fibres
I am trying to construct an n-parameter family of measure preserving real analytic diffeomorphisms on $S^3$ which preserves the $S^1$ fibres of the Hopf fibration but acts transitively on the image ...
0
votes
1answer
67 views
geodesic convexity of regular sets for metrics with cone singularities
Consider $\mathbb{C}^2 = (r,\theta,\rho,\varphi)$ with the following cone metric in polar coordinates
\begin{equation*}
ds^2= (dr^2 + \alpha^2 r^2 d\theta^2) + (d\rho^2 + \beta^2\rho^2 d\varphi^2)
...
3
votes
1answer
132 views
The de Rham complex of a quaternion-Kahler manifold
As we all know, for a complex manifold $M$, its de Rham complex admits a decomposition into a double complex called the Dolbeault complex. If $M$ also admits a Kahler metric, then we get the wonderful ...
1
vote
1answer
179 views
Kahler-Einstein metrics on Toric manifolds are Torus-invariant?
let $(M,\omega)$ be a Kahler-Einstein toric manifold of complex dimension $m$. By toric manifold i mean a manifold that has an open dense subset $X$ biholomorphic to an algebraic torus ...
3
votes
3answers
282 views
Finiteness of De Rham cohomology of smooth quasi-projective varieties
Let $U$ be a smooth quasi-projective variety over $\mathbf{C}$. Let $U^{\infty}$
be $U$ but thought of as a smooth manifold.
Q1: Is there a simple proof (so it should avoid Hironaka's ...
0
votes
1answer
60 views
Resolution of the indeterminacy locus of a rational map away from an irreducible component
Suppose I have a rational map between two smooth, complex, projective varieties (or a meromorphic map between two compact, complex, manifolds)X and Y. Ordinarily one eliminates the indeterminacy locus ...
1
vote
1answer
126 views
Kähler Identities: from the untwisted to the twisted case
For any Kähler manifold $M$, we have the well known Kähler identities
\begin{align*}
[L,\partial^*] = i\overline{\partial}, & & [L,\overline{\partial}^*]=-i\partial, & & ...
3
votes
2answers
188 views
Integrable compatible complex structures
In reading Gauduchon's notes (cannot link them, anyway it is standard material) I ran into the following construction.
Let $(M, \omega_0)$ be a compact symplectic manifold which is fixed.
An almost ...
2
votes
1answer
265 views
Definition of a complex space
In the definition of a complex space (in the sense of Grauert), one defines a model space (one to which we require a complex space to be locally isomorphic) to be the support of the quotient sheaf ...
0
votes
0answers
103 views
When the leaves of a distribution are compact
Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
...
3
votes
0answers
117 views
Local holomorphic equations for symplectic divisors
If $(X,\omega)$ is a symplectic manifold and $J\colon TX \to TX$ is an almost complex structure, we know that $X$ admits a complex structure if and only if the Nijenhuis tensor $N_J(\cdot,\cdot)$ of ...
1
vote
1answer
103 views
the group of all biholomorphic group automorphisms of complex tori
My background is complex geometry, but when I confront complex tori, I feel it is easier to consider it as a compact connected complex Lie group although I just know the definition of Lie group.
Let ...
2
votes
0answers
153 views
Explicit form for hermitian structure $h$ with respect to $\omega$
Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...
3
votes
0answers
128 views
Hilbert's syzygy theorem in the analytic setting
If $X$ is a projective variety then Hilbert's syzygy theorem says that any coherent sheaf of $\mathcal O_X$ modules has a finite global resolution by locally free modules.
By GAGA, I believe this ...
1
vote
1answer
79 views
Non-equivariant vector bundles over complex projective $N$-space
From Grothendieck's lemma, we know that all holomorphic vector bundles over the complex projective line are direct sums of line bundles, and so, are $SU(2)$-equivariant.
I wonder, do there exist ...
2
votes
0answers
45 views
Formal adjoint of covariant derivative on endomorphism bundle
Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$, equipped with an integrable, unitary connection $D=D'+D''$. Let $\beta\in\Lambda^{p,q}(\mathrm{End}\,E)$ be ...
0
votes
1answer
171 views
Birkhoff decomposition vanishing of the Chern numbers
Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...
3
votes
0answers
138 views
Vector bundle connection over complex manifold vs. over underlying real manifold
Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$ of rank $r$. Denote by $(X^{\mathbb{R}},g^{\mathbb{R}})$ the underlying Riemannian manifold of $(X,g)$.
...
1
vote
1answer
105 views
lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology
Start from a connected closed Riemann surface $\Sigma_g,$
obtained as the (symmetric) covering of an open and/or unoriented surface
$\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
0
votes
1answer
152 views
A problem related to connectivity of analytic functions
Let $f(z)\in A(\mathbb D)$, where $A(\mathbb D)$ is the space of analytic functions on the open unit disk $\mathbb D$ and continuous on $\overline{\mathbb D}$.
Question: Is the connectivity of ...
6
votes
2answers
438 views
Trying to Understand Lefschetz Pencils
All:
I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) below better, though I would appreciate insights on condition i), and in general. Please forgive if the presentation ...
2
votes
0answers
31 views
Explicit local expression for Bers embedding in genus 2
Let $\mathcal T_{g,n}$ be the Teichmüller space of genus g compact Riemann surfaces with $n$ marked points. According to Riemann, this is a complex manifold of complex dimension 3g-3+n.
Bers ...
3
votes
1answer
289 views
top chern class
Let $E\to M$ be a holomorphic vector bundle ($M$ is compact) and assume $s\colon M\to E$ is a non-trivial section transverse to the zero section.
Is it be possible that $s^{-1}(0)\neq \emptyset$, ...
