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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7
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1answer
401 views

Complex geometry text/research introduction for the analyst

To give some background, I am mainly an analyst trained in harmonic/functional and do work on geometric pde's and spectral multipliers. Of late, I am trying to learn more about (research level) ...
3
votes
1answer
229 views

Second betti number of compact analytic spaces

Let $V$ be a proper singular complex algebraic variety, possibly nonprojective ($dim(V)=n>0$). I would like to know: 1) if its second Betti number is non zero, 2) same question but now $V$ is a ...
2
votes
0answers
150 views

What is the relation between Beilinson's conjectures and Standard conjectures of algebraic cycles?

Do Standard conjectures on the K-theory of varieties over finite field have implications in the motivic cohomology of Z where exist the correct formalism of Beilinson's conjectures? What is the ...
2
votes
1answer
196 views

How does one show that slope stability of a vector bundle is an open condition with respect to the polarisation?

I would like a source for the following result, which I expect to be true (probably well known): Let $X$ be a complex projective variety, $L$ an ample line bundle and $E$ a slope stable vector bundle ...
0
votes
1answer
82 views

A semi-ampleness criterion for homogeneous bundles on homogeneous spaces?

Let $X$ be a (compact) homogeneous space and $V$ be a homogeneous vector bundle on $X$ of rank $n$, and such that $\operatorname{dim}X\ge n$. Suppose $V$ has a section $s$, whose zeros $s=0$ form a ...
3
votes
1answer
146 views

Flat family with special fiber $\mathbb{C}\mathbb{P}^1$

Let $C=Spec \mathbb{C}[t]/(t^{n+1})$. Let $X$ be an algebraic (or complex analytic) scheme, flat over $C$ with the structure morphism $f\colon X\to C$. Assume that the special fiber is isomorphic to ...
4
votes
1answer
174 views

Holomorphic Foliations having transverse sections

In the introduction to the paper "On the Geometry of Holomorphic Flows and Foliations Having Transverse Sections" by Ito and Scardua, one reads the following "a holomorphic codimension one foliation ...
1
vote
1answer
287 views

When flatness of a morphism implies smoothness?

EDIT: Let $f\colon X\to C$ be a flat proper morphism of complex algebraic (or analytic) varieties. Assume the special fiber over a point $p\in C$ is smooth. Is it true that there exists a ...
5
votes
0answers
74 views

Density of rational functions in open Stein

I repost here, after I tried here. Lately I have been wondering on this problem: if $U \subset \mathbb C^n$ is an open Stein and I denote by $\mathcal R(U)$ the set of rational functions on $\mathbb ...
0
votes
1answer
167 views

is grassmannian rational connected or not [closed]

I wan to know if Grassmannians are rational connected? Any reference describe how to tell if a variety is rational connected or not?
3
votes
1answer
151 views

Can someone tell me properties of Douady space?

I want to know the parallel properties of Douady space with respect to Hilbert scheme. For example I want to know what is the irreducible component of Douady space, what if I consider a family of ...
0
votes
0answers
82 views

A question related to the Grauert semi-continuity theorem

Let $f\colon X\to Y$ be a proper holomorphic map of complex analytic manifolds. Assume $f$ to be submersive for simplicity, but probably it is not important. Let $\mathcal{F}$ be a coherent sheaf on ...
2
votes
1answer
213 views

Generalizations of de Franchis and function field Mordell

The classical de Franchis theorem, as generalized by S. Kobayashi and T. Ochiai ("Meromorphic mappings onto compact complex spaces of general type," Inventiones, 1975), states that if $X$ is a complex ...
8
votes
1answer
149 views

Non projective hyperbolic compact complex space

A famous conjecture by Kobayashi (perhaps slightly revisited subsequently) states that every compact hyperbolic Kähler manifold $X$ has ample canonical bundle. This implies in particular that $X$ is ...
1
vote
1answer
237 views

A generalization of the Grauert direct image theorem

EDIT: Let $f\colon X\to Y$ be proper holomorphic submersive map of complex analytic manifolds. Let $\mathcal{F}$ be the sheaf of holomorphic sections of a holomorphic vector bundle over $X$. Assume ...
2
votes
1answer
207 views

The stability of vector bundle with trivial Chern classes is independent of ample divisor, a direct proof?

Let $X$ be a smooth projective variety over $\mathbb{C}$. For an ample divisor H, we can define the slop of vector bundle with respect to $H$, then we can define stablilty of vector bundle with ...
4
votes
1answer
181 views

Analytic representatives for Kahler classes

If we are given compact complex manifold $X$ and a Kahler class $[\omega]$, can we always find a positive definite representative $\omega \in [\omega]$ that is real analytic?
-1
votes
1answer
172 views

what is the first cohomology group of structure sheaf of grassmannian [closed]

I want to know if the first cohomology group of structure sheaf of grassmannian vanishes.
8
votes
1answer
313 views

Hodge numbers of diffeomorphic complete intersections

Is there an example of two complex projective complete intersections that are diffeomorphic but have different Hodge numbers? Edit: as written by Daniel Loughran in the comments below, complete ...
4
votes
1answer
418 views

What is the automorphism of a grassmannian?

I want to know what is the holomorphic automorphism of a grassmannian. Can someone tell me this?
2
votes
1answer
97 views

Projectively flat Hermitian curvature proportional to Kähler form?

Is there a classification of the holomorphic Hermitian vector bundles $\pi:E\rightarrow M$, over a given complex Hermitian manifold, which are projectively flat and the curvature is proportional to ...
5
votes
1answer
279 views

A geometric characterization of smooth points of a complex algebraic variety

Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric. Fix an arbitrary point $p\in X$. Let ...
0
votes
0answers
89 views

complementary bundle for a divisor

For a divisor in a complex manifold, what is known about a complementary bundle to the divisor in the manifold (either for the tangent or the cotangent bundle). Is there a description in terms of ...
2
votes
1answer
120 views

Show properness of Ahlfors map

If have got a $k$-fold connected surface $G$, which is bounded by n distinct, non-intersecting Jordan-curves. By Ahlfors it is known that there exists a unique function $\phi$ which maps G to the unit ...
6
votes
1answer
531 views

Kodaira-Spencer theory of deformation done right

I thought in asking this question on Math StackExchange, but by my experience I don' t think anyone will notice me. Recently, I started studying deformation of complex manifolds in the sense of ...
4
votes
1answer
153 views

Simple maps: Flat versus locally trivial

In deformation of complex analytic spaces, one usually considers an analytic proper simple surjective map $\varpi: \mathscr{M} \twoheadrightarrow \mathscr{P}$ as an analytic family. However the term ...
0
votes
1answer
92 views

Condition for infinite dimensional complex manifold to be Kähler by pullback form

For a finite dimensional Kähler manifold $M$, there is the condition that if $N\xrightarrow{f}M$ is a holomorphic map with maximal rank at each point, then $N$ is Kähler with the pullback form induced ...
4
votes
1answer
232 views

Non Kähler blow-up of a Kähler manifold

Is it possible to find a complete, non compact Kahler manifold $(X,\omega)$ with a closed, connected, non compact complex submanifold $Y\subset X$ of codimension at least 2 such that the blow-up of ...
2
votes
2answers
307 views

Is the Nijenhuis tensor an obstruction to the existence of non constant pseudo-holomorphic maps?

Let $(N,J_N)$ and $(M, J_M)$ be two compact almost complex manifolds with non integrable almost complex structures (i.e., the Nijenhuis tensor is non zero for both $J_N$ and $J_M$). Does it imply ...
18
votes
2answers
3k views
+400

The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see this related post and also the following post. Added : According to their ...
10
votes
1answer
275 views

Extending holomorphic functions

Suppose $K\subset \mathbb{C}^n$ is a compact subset and $f:\mathbb{C}^n\setminus K\to \mathbb{C}$ is a holomorphic function. Then, provided $n>1$, $f$ extends to a holomorphic function defined on ...
1
vote
0answers
145 views

Derived category of product of complex manifolds

Let $X$ and $Y$ be a compact complex manifold. Is it possible to describe the derived category $D^b(X\times Y)$ of coherent sheaves in terms of $D^b(X)$ and $D^b(Y)$? I am particularly interested in ...
6
votes
1answer
150 views

Is the total space of a family of normal varieties a normal variety?

Let $f:X \rightarrow C$ be a flat morphism from a complex variety $X$ to a smooth curve $C$. If any fiber $X_{t}=f^{-1}(t)$ is a reduced normal projective variety, is the total space $X$ a normal ...
0
votes
0answers
124 views

classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto ...
4
votes
1answer
176 views

Level set of convex and plurisubharmonic functions (Ricci curvature and other curvature conditions)

Let $\phi$ be a stricly plurisubharmonic function on a domain in ${\Bbb C}^n$, and $S=\phi^{-1}(c)$ its level set. Consider $S$ as a Riemannian manifold equipped with a metric induced by $dd^c\phi$. I ...
3
votes
2answers
232 views

Families of Fano varieties over non-hyperbolic curves

Let $C$ be a non-hyperbolic (smooth quasi-projective connected complex algebraic) curve. That is, $C$ is isomorphic to $\mathbb P^1, \mathbb A^1, \mathbb G_m$, or an elliptic curve. Let $f:X\to C$ be ...
2
votes
0answers
94 views

Chern-Weil theory for degenerated metric

If $\omega$ is a Kähler metric on a compact complex manifold $X$, the standard Chern-Weil theory says that the Chern classes $c_{i}(M)$ can be represented by forms involving the curvature of ...
7
votes
1answer
459 views

Geometric interpretation for fourth coeficient of the polynomial for Hirzebruch–Riemann–Roch theorem

Hey guys :) I have a question on a theorem which is known as Tian-Yau-Zelditch theorem now. If there is some reference for following question, please let me know Let $(M,\omega)$ be a compact ...
1
vote
1answer
117 views

characterization of structure group

Somebody tell me that: For a bundle(maybe polystable) over algebraic manifold, take a symmetric power of the bundle and tensor with its determinant line bundle to some power. Assume that the ...
2
votes
0answers
367 views

What implications would a solution of the *Standard Conjectures* have on the *Hodge Conjecture*?

I'm new to the field, so I just would like to know what implications would have a solution of the Standard Conjectures on the Hodge Conjecture. I read somewhere they are related in some way, but I ...
1
vote
1answer
177 views

examples of Kähler manifolds with trivial Hodge numbers and first Chern classes

Yesterday I asked the following question to which abx has given a positive answer. examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes But I suddenly realized ...
1
vote
1answer
310 views

A question about Quantized closed Kaehler manifolds

Let $(M,\omega)$ be a Quantized closed Kaehler manifold then by Koderia embedding theorem , $M$ must be algebraicly projective i.e, we have the embedding $$\phi: (M,\omega)\to  (\mathbb CP^N, ...
3
votes
1answer
264 views

examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes

To my limited knowledge, many compact Kähler manifolds have trivial odd Betti numbers. For instance, flag manifolds $G/P$,where $G$ is a semisimple complex Lie group and $P$ a parabolic subgroup, and ...
1
vote
0answers
57 views

some intuition about the degree of a map

Consider a map $$ f: \Sigma \to X/\sigma,$$ where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution, $\sigma:X\to X$ is an antiholomorphic involution of some ...
6
votes
1answer
201 views

Chern-Weil Theory for $p_1$

I'm studying Riemannian manifolds that admits a almost-complex manifold, thus $$3\tau+2\chi=c_1^2,$$ where $\tau$ is the signature, $\chi$ is the euler characteristic and $c_1$ is the first Chern ...
0
votes
1answer
79 views

Meromorphic extension of local defining equations of a complex submanifold

let $M$ be a smooth compact complex manifold of dimension $m$ and $N\subset M$ a smooth complex submanifold of dimension $1\leq n \leq m-2$. Covering $N$ with well chosen open sets of $M$ we can ...
4
votes
1answer
71 views

Minimal projective space containing projective variety independent of base field

In this question I ask whether ambient spaces descend to models of varieties. Let $k\subset K$ be a non-trivial extension of algebraically closed fields, e.g., $\overline{\mathbb Q}\subset \mathbb ...
5
votes
2answers
515 views

Are “ample” and “positive” line bundle the same concept?

A line bundle is ample if some power of it is very ample. A line bundle is positive if the chern class in $H^2(X,\mathbb{Z})$ is represented by a Kahler metric in $H^{1,1}(X,\mathbb{Z})$.(Regarded as ...
2
votes
1answer
142 views

Bound for the Picard number of a 3-fold

Are there some upper bounds for the Picard number of a non-singular threefold? We know that in the surface case, we have $h^{1,1}=10\chi-c_1^2+2q$. Hence the picard number $\leq 10\chi-c_1^2+2q$. Is ...
0
votes
0answers
40 views

Expansion of the squared distance function from a submanifold

let $(M,g)$ be a compact Kahler manifold of dimension $m$ with the Kahler metric $g$ real-analytic. Let $N\subset M$ be a complex submanifold of dimension $0<k<m$. Let $$d_{g}(M,\cdot): ...