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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6
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2answers
844 views

What is the role of projective spaces in GAGA?

The GAGA theorem is a celebrated elaboration of the idea that complex analytic and complex algebraic geometry are equivalent, at least for smooth projective varieties/manifolds. I am aware why this ...
8
votes
0answers
323 views

rings of modular functions on the upper half plane

Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index. Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index. Let $M_0(\Gamma_i)$ denote the ring of modular ...
4
votes
1answer
189 views

What's the minimal embedding of orthogonal Grassmannian?

Suppose $X$ is the orthogonal Grassmanian. We know the Plücker embedding does not span the whole background $\mathbb{CP}^n,$ just a subspace $\mathbb{CP}^m.$ My question: is there an ...
6
votes
1answer
177 views

Loci in the moduli space of K3 surfaces associated to lattices

The moduli space of K3 surfaces forms a 20-dimensional family with countably many 19-dimensional components $M_d$ corresponding to the polarized K3s $(X,L)$ with $L^2=d$. The moduli space $M_d$ has a ...
3
votes
1answer
74 views

compact almost complex submanifolds of complex Lie groups

I find the following Corollary 1.21: Question: does there exist any complex Lie groups $G$ such that there are some compact almost complex submanifolds (for example, $\mathbb{C}P^m$) of $G$? I want ...
4
votes
1answer
381 views

vanishing theorem in algebraic geometry

This is a general question: As we know there are a lot of vanishing theorems like Fujita vanishing, kodaira Nakano vanishing, vanishing for big nef line bundle, Kollár vanishing, etc. Those ...
8
votes
3answers
336 views

Trivialisation of vector bundles on Stein spaces

Does every vector bundle on a Stein space have a finite local trivialisation? Definitions: Stein space means either a complex analytic Stein space or a nonarchimedean Stein space in the sense of ...
5
votes
1answer
254 views

When is the tangent bundle of a manifold naturally a complex manifold?

It is well-known that the cotangent bundle of a manifold is naturally a symplectic manifold. Inspired by mirror symmetry, when is the tangent bundle $TM$ of a manifold $M$ naturally a complex manifold?...
2
votes
0answers
42 views

cayley transformation of bounded symmetric domain

Can anyone write down the biholomorphic map between classical bounded symmetric domains(defiend by matrixs) with their siegel upperhalf plane models. I know if it's type 2, i.e $I-Z\bar{Z}^{t}>0$ ...
4
votes
1answer
88 views

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-...
2
votes
0answers
123 views

Examples of holomorphic Killing vector fields on compact Kahler manifolds

I'm looking for concrete examples of compact Kahler manifolds that admit global holomorphic Killing vector fields. The only examples I can think of so far are quite trivial: (i) CP^N with the Fubini ...
2
votes
0answers
121 views

Do complex schemes locally deformation retract onto closed subschemes in the analytic topology?

Let $X$ be a scheme of finite type over $\mathbb{C}$ and let $Z \hookrightarrow X$ be a closed subscheme. Consider the associated closed inclusion $Z_{an} \hookrightarrow X_{an}$ between their ...
7
votes
3answers
309 views

Are compact, complex, affinely flat manifolds geodesically complete?

Let $M$ be a real, even dimensional, compact manifold endowed with a symplectic form $\omega$ and a flat, torsionless connection $\nabla$ compatible with $\omega$, that is $$\nabla \omega=0.$$ Under ...
2
votes
0answers
108 views

Is there an algorithm to compute the intersection of tautological classes on the moduli space of genus one curves?

Let $\overline{M}_{1,1}(\mathbb{P}^2, d) $ be the moduli space of degree $d$ genus one curves on $\mathbb{P}^2$ with one marked point. Let $L\longrightarrow \overline{M}_{1,1}(\mathbb{P}^2, d) $ ...
2
votes
0answers
66 views

kahler einstein metric for exceptional compact type hermitian symmetric space

Can anyone write down the kahler einstein metric for exceptional compact type hermitian symmetric spaces($\frac{E_6}{SO(10)*SO(2)}$ and $\frac{E_7}{E_6*SO(2)}$). I can find the bergmann kernel for ...
2
votes
0answers
36 views

Why generalized vectors can be written locally as sum of vectors and 1-forms?

I would like to understand better this point. In generalized complex geometry the generalized bundle $E$ is defined as a non-trivial fibration of the cotangent bundle $T^*M$ over the tangent bundle $...
1
vote
0answers
80 views

If there exists an immersion, then does a neighbourhood of a singular rational curve contain a genuine cuspidal point?

Let $X$ be a compact complex surface and $u_1, u_2: \mathbb{P}^1 \longrightarrow X$ be rational curves that are not multiply covered that represents a class $\beta \in H_2(X, \mathbb{Z})$. Suppose $...
3
votes
0answers
152 views

Possible Betti numbers of smooth complex varieties

Given a smooth projective complex variety $X$ of dimension $n$, there are various restrictions on its sequence of Betti numbers $b_0, b_1, ..., b_{2n}$. Of course, $b_0=b_{2n}=1$ and $b_i=b_{2n-i}$ by ...
0
votes
1answer
103 views

Unique Equivariant Symplectic Structure for the Full Flag Manifold of $SU(3)$?

I was looking at the following interesting question about the number of equivariant almost complex structures on the full flag manifold of $SU(3)$, and I began to wonder how many equivariant ...
6
votes
1answer
301 views

Complex Geometry Consequesnces of Serre's Kahler-Zeta Function

Serre's famous paper Analogues K\"ahl\'eriens de Certaines Conjectures de Weil proves an analogue of the Weil conjectures for compact K\"ahler manifolds. It would go on to inspire the line of attack ...
2
votes
0answers
106 views

Deformations of blow ups of $\mathbb{C}^{m}$

let $\mathbb{C}^{m}$ be the complex $m$-space with the standard complex structure and let $$P:=\left\{p_{1},\ldots,p_{N} \right\}\subset \mathbb{C}^{m}$$ a finite set of points. Now we blow up ...
2
votes
0answers
96 views

Can a class be represented by both a $(p,q)$ form and a $(p',q')$ form?

Suppose $X$ is a complex manifold. If $X$ is Kahler, the cohomology groups decompose into subgroups represented by $(p,q)$ forms. If $X$ is not Kahler, I think the decomposition may not hold? Is ...
4
votes
2answers
265 views

Riemannian metric of hyperbolic plane

I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector space....
42
votes
0answers
696 views

Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here. $\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an ...
4
votes
0answers
141 views

Lie algebra of holomorphic vector fields

It's well known that the holomorphic vector fields on a complex manifold form a Lie algebra. In simplest situations, this Lie algebra can be described explicitly. For example, take $X=\mathbb{P}^n$, ...
5
votes
0answers
144 views

Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2

Does every compact complex manifold of complex dimension greater than or equal two possess a nontrivial rank 2 holomorphic vector bundle?
3
votes
2answers
134 views

Has anyone developed a technique to generate a polytope given (possibly redundant) inequality constraints? [closed]

I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm just trying to find the vertices for a feasible region, given a set of inequality ...
2
votes
1answer
94 views

Higher dimensional analogue of Ahlfors covering surface theory

It is well known that Ahlfors covering surface theory in one dimensional is very powerful in dealing with many problems. I wonder whether there exists some generalization of this theory into higher ...
4
votes
1answer
172 views

Equivariant Almost Complex Structures on the Full Flag Manifolds

On complex projective space ${\bf CP}^m$, there exists a unique $SU(m+1)$-equivariant almost-complex structure. What happens for the case of the full flag manifold of $SU(m+1)$, which is to say the ...
0
votes
1answer
236 views

Torsion in the (co-)homology of a smooth projective variety - what is known in general?

There are lots of ways in which the complex singular (co-)homology of a smooth projective variety over $\mathbb{C}$ is "special" among complex manifolds - the hard Lefschetz theorem, the Hodge ...
2
votes
0answers
55 views

Group of real analytic isometries of $g$-fold product of the Poincare upper half plane

Let $\mathfrak{h}^g$ be the cartesian product of $g$ copies of the Poincare upper half plane. We endow $\mathfrak{h}^g$ with the usual Poincare metric given in local coordinates by $ds^2=\sum_{i=1}^g ...
7
votes
2answers
486 views

Generalising the Penrose Twistor Fibration

As is well known, there exists a fibration $\mathbb{CP}^3 \to S^4$, of the four sphere by complex projective $3$-space, called the Penrose twistor fibration. Does this fibration admit a "canonical" ...
3
votes
1answer
147 views

Stokes-like Theorem for Dolbeault Operator

I have a simple question regarding complex geometry: is there an analog for the Stokes Theorem for the Dolbeault Operator $\bar{\partial}$? For instance, suppose that $M$ is a closed complex manifold ...
0
votes
0answers
50 views

What is an induced Abel Jacobi map?

Let $X$ be a compact Kahler threefold, $C\subset X$ be a smooth curve, $\tau\colon\tilde{X}\to X$ be the blow up of $X$ along $C$, $j\colon E\to \tilde{X}$ be rthe exceptional divisor, $\tau_E$ be the ...
3
votes
2answers
363 views

Three and a half basic questions on the Weil restriction of scalars

(This is reposted from mathstackexchange, where it received no answer so far.) I am currently trying to get familiar with the Weil Restriction functor. For a finite field extension $L|K$ it ...
7
votes
1answer
262 views

Does there exist an algebraic space with large fundamental group but no finite etale covers by schemes

Does there exits a smooth proper algebraic space $X$ over $\mathbb C$ with "large" fundamental group such that no finite etale cover of $X$ is a scheme? By "large" fundamental group I mean that $X$ ...
1
vote
1answer
120 views

extension of holomorphic mappings

In 1971 Phillip.A.Griffith proved that Let $B_n^\ast=\{z\in\mathbb{C}^n:0<\|z\|\le 1\}$ be the punctured ball in $\mathbb{C}^n$,and $f:B_n^\ast\rightarrow M$ a holomorphic mapping into a compact K\...
3
votes
1answer
139 views

Polynomials with Unique Critical Value

My question is extremely simple to state: I am looking for a characterization of multivariate complex polynomials $f$ such that $f(Sing(f))=\{0\}$. My motivation is that I recently read somewhere that ...
2
votes
0answers
45 views

Uniqueness of Riemann Constant Vector Solution

Let $X$ be a compact, genus $g$ Riemann surface (given as the desingularization and compactification of a plane algebraic curve), $J(X)$ its Jacobian, and $A : X \to J(X)$ the Abel map $$A(P) = \left(...
14
votes
3answers
2k views

Intuition behind the Kodaira Vanishing Theorem?

As the question suggests, what is the intuition behind the Kodaira Vanishing Theorem? The Kodaira Vanishing Theorem says that the cohomology groups $H^q(M, L \otimes K_M)$ vanish for $q \ge 1$ when $L$...
4
votes
0answers
151 views

Explicit metrics on non-compact Calabi-Yau threefolds

I would like to know which explicit metrics on non-compact Calabi-Yau (CY) threefolds are known. For instance, an important class of such spaces can be constructed algebraically, including local $\...
1
vote
2answers
158 views

Transformations that leave the Plucker embedding of G(2,4) invariant

I am interested in a group of transformations that leave the Plucker embedding of complex Grassmannian $G(2,4)$ into $CP^5$ given by $\lambda_{12}\lambda_{34}-\lambda_{13}\lambda_{24}+\lambda_{14}\...
5
votes
1answer
118 views

Metrics on Teichmüller spaces

I know that Teichmüller $\mathcal{T}_g$ spaces support different metrics. One of them is the Bergman metric; which is a particular case of the Bergman metric on any domain of holomorphy. On the other ...
3
votes
0answers
87 views

Toponogov comparison theorem for complex manifold

I would like to know some reference for the Toponogov comparison theorem for complex manifolds, in particular for complex manifolds with bounded holomorphic sectional curvature. As far as I know, the ...
0
votes
1answer
118 views

How does $H_1$ change after projection

Suppose $X\subset \mathbb{CP}^N$ is a $n$ dimensional projective manifold (and complex dimension $n>1$), take a general projection $p\colon X\to\mathbb{CP}^{n+1}$. Suppose $H_1(X)$ is nontravial. ...
1
vote
1answer
238 views

Two limit cycles which lie on the same leaf

Edit 1: For a related discussion see this MSE post I apologize in advance, if this question is obvious: 1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two ...
4
votes
1answer
194 views

Infinitesimal deformations of fake projective planes (or ball quotients)

This question is related to the answer I gave to this MO question. What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial. By ...
6
votes
1answer
417 views

Analytic Chern classes

I have two questions on Chern classes, following Huybrechts' Complex Geometry. Are the analytic Chern forms just the elementary symmetric polynomials of the eigenvalues of the curvature? I googled ...
1
vote
1answer
157 views

Orientation form on the blow up of a Kaehler manifold

Let $(X,\omega)$ be a complex Kaehler manifold of (complex) dimension $d$, and let $Y\subset X$ a complex submanifold of dimension $k$. Evidently $[\omega]^d\in H^{2d}(X,{\mathbb{R}})$ is always ...
5
votes
1answer
242 views

How to get a polygon from a translation surface $(X,\omega)$

Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence $\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces ...