Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
Questions tagged [complex-geometry]
3134 questions
39
votes
2 answers
Do compact complex manifolds fall into countably many families?
Do there exist countably many proper holomorphic submersions of complex manifolds $\mathcal{X}_n \to B_n$ such that every compact complex manifold appears as a fiber in at least one of the families?
(Conventions: Assume that each $B_n$ is…
Bjorn Poonen
- 23,617
17
votes
1 answer
Are all holomorphic vector bundles on a contractible complex manifold trivial?
It is true that over a contractible manifold all differentiable vector bundles are trivial. However the method of proof does not apply in the holomorphic category.
It is also true that a contractible one dimensional complex manifold has no…
Emre
- 813
12
votes
1 answer
Parameterization of complex analytic subvarieties
Let $V$ be an analytic subvariety of some open set of $\mathbb{C}^n$ (intersection of finitely many zero-locii of analytic functions). Hironaka's desingularization theorem provides a parameterization of $V$, that is: there exists a proper, onto map…
Loïc Teyssier
- 5,289
12
votes
2 answers
Bott Chern cohomology via currents
Let $X$ be a compact complex manifold. Is the space of $(p,p)$ $d$-closed currents modulo $\partial\bar{\partial}$-exact ones naturally isomorphic to the Bott-Chern cohomology (made in the same way with smooth forms)? I can prove this statement for…
stefano trapani
- 121
8
votes
2 answers
How to count the total zeros of a complex polynomial outside a closed curve?
Set up
Suppose $\gamma$ a simple closed curve, oriented in a counterclockwise direction. $f(z)$ is a complex polynomial
$$
f(z)=a_nz^{n}+a_{n-1}z^{n-1}+\cdots+a_0.
$$
We already know that the integral
$$
N=\frac{1}{2\pi…
Guoqing
- 431
8
votes
1 answer
Complex manifolds with spanning sets of holomorphic tensor fields
This question is an extension of this one. Consider a complex manifold $(M^{2n}, J)$. Fix $1 \leq p \leq n-1$, and suppose that the space of holomorphic sections of $\Lambda^{p,0}$ spans $\Lambda^{p,0}_x$ for all $x \in M$. (The referenced…
8
votes
1 answer
Intersections in almost complex manifolds
Question: Suppose $(M,J)$ is an almost complex manifold, and $X$ and $Y$ are two almost complex submanifolds (i.e. $J(T(X)) \subset T(X)$ and $J(T(Y)) \subset T(Y)$). Then must $X \cap Y$ also be an almost complex submanifold?
Clarification: If $X$…
user2028
7
votes
3 answers
Did Kahler say "a long list of miracles occur"?
I've been reading Moroianu's Kahler geometry notes and found a unattributed quote that says that if the Kahler properties hold, then
"a long list of miracles occur"
I am guessing that this quote belongs to Kahler himself, but I can't back this up.…
Jean Delinez
- 3,339
7
votes
1 answer
Existence of closed manifolds with more than 3 linearly independent complex structures?
A Riemannian manifold is hyperkähler, if there are three complex structures $I,J,K$, which are all compatible with the Riemannian metric (i.e., $(v,Iw)$ defines a symplectic form and similarly for $J$ and $K$). Furthermore, we also need the complex…
Doris Hein
- 73
6
votes
1 answer
Status of global spherical shell conjecture for minimal complex surfaces?
A class VII surface is a compact complex surface $M$ such that $b_1(M)=1$ and $kd(M)=-\infty$. Class VII surfaces with vanishing second Betti number have been classified by Bogomolov (and are either Hopf surfaces or Inoue surfaces).
The situation…
user74900
6
votes
1 answer
Almost complex manifold fibered by holomorphic sub-manifolds
Suppose $p:(M,J)\rightarrow (N,I)$ is a submersion between smooth manifolds M and N such that:
$(M,J)$ is an almost-complex manifold.
$(N,I)$ is a complex manifold where $I$ is the integrable almost-complex structure.
$p$ is almost-holomorphic,…
Andy Sanders
- 2,890
6
votes
3 answers
Dolbeault Cohomology of $\mathbb{P}^1$
So from the $\overline{\partial}$-Poincare lemma, there is a short exact sequence of sheaves on $X = \mathbb{P}^1$
$$0 \to \Omega \to A^{1,0} \to Z^{1,1} \to 0$$
where $\Omega$ is the sheaf of holomophic 1-forms, $A^{1,0}$ is the sheaf of…
solbap
- 3,938
5
votes
1 answer
Dual Lefschetz Operator and Contraction with the Fundamental Form
Let $M$ be a Kahler manifold, with metric $g$, fundamental form $\omega$, and dual Lefschetz operator $\Lambda$. Now $\Lambda$, and contraction with $\omega$, both map the two forms $\Omega^2(M)$ to $0$-forms, ie smooth functions. Are they equal? I…
Ago Szekeres
- 389
5
votes
1 answer
Quillen metric definition
I am confused as to why a regularised determinant is used in the definition of Quillen's metric on the determinant line bundle defined over the space of $\bar{\partial}$ operators on a (hermitian) vector bundle over a compact Riemann surface. I…
Vamsi
- 3,323
5
votes
0 answers
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 0 such that $h^{p, q}(M_t)=h^{p, q}(M)$ for all…
user74900