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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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 ...
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152 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 ...
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582 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 ...
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182 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 ...
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1answer
106 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 ...
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274 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 ...
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2answers
376 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 ...
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The error in Petrovski and Landis' proof of the 16th Hilbert problem

Edit:For a recent progress on the Hilbert 16th problem see the following note which consider an infinite dimensional nature for this apparently 2 dimensional amazing problem . Best wishes for ...
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309 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 ...
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158 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 ...
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157 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 ...
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137 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 ...
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196 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 ...
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2answers
240 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 ...
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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 ...
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474 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 ...
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1answer
118 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 ...
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606 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 ...
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185 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 ...
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1answer
316 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, ...
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1answer
278 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 ...
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68 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 ...
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1answer
231 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 ...
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84 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 ...
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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 ...
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670 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 ...
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1answer
145 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 ...
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43 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): ...
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98 views

Ampleness of Hodge bundles over complex curves

Let $C$ be a smooth, proper and connected curve over the complex numbers $\bf C$. Let ${\cal G}\to C$ be a smooth group scheme over $C$ and let $\epsilon_{\cal G}:C\to{\cal G}$ be its zero-section. ...
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54 views

Characterisation of convergence in Deligne-Mumford compactifiaction

1) Is there a (simple) way to characterise the convergence of complex curves toward a stable curve in term of hyperbolic metrics for the Deligne Mumford compactification of the moduli space of complex ...
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74 views

Deformations of a pair of compact, complex manifolds

This is somewhat related to the question found at What is the DGLA controlling the deformation theory of a complex submanifold?, though not exactly the same, so I hope it's not duplicating too much. ...
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297 views

Must a canonical line bundle be associated to a cartier divisor?

Suppose $X$ is a complex manifold, we have the map $H^0(X,K_X^*/O_X^*)\to H^1(X,O_X^*)$, is the canonical line bundle $\wedge^n{\Omega}$ always in the image of the map?
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298 views

A Generalized De Rham cohomology

Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version. Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by ...
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227 views

Is the number of minimal models finite

Let $X$ be a variety of general type. Assume that $\dim X = 3$. In https://eudml.org/doc/164223 it is proven that $X$ has only finitely many minimal models (i.e., only $\mathbb Q$-factorial terminal ...
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148 views

Degeneration of varieties ---Mumford reduction---minimal model, what's the relationship?

Every variety here is over $\mathbb{C}$. Let $f: X \rightarrow C $ be a flat proper surjective morphism from a quasi-projective variety $X$ to a quasi-projective smooth curve. Let $p\in C$. We ...
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170 views

When is the analytification of a variety homeomorphic to $\mathbb C$

Let $X$ be the analytification of a variety $V$ such that $X$ is homeomorphic to $\mathbb R^2$. What can we say about $X$? Can $V$ be singular?
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89 views

How far do conjugated Mobius transforms move points?

I start with an automorphism $f$ of the complex unit disc $S^1 = \{ z \in \mathbb{C} : |z| \leq 1\}$. I assume that such a map is given by a Mobius transform, namely $$ f(z) = \frac{z - ...
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223 views

Lattice polarized K3 surfaces

I've recently encountered the definition of a lattice polarized K3 surface. What is the idea behind the definition? Surely, there's something deeper to it than merely being a natural generalization of ...
3
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1answer
168 views

Deformation of Hitchin-Simpson correspondence

Let $X$ be a compact Riemann surface, Hitchin-Simpson correspondence over $X$ says that irreducible representations of $\pi_1(X)$ one-to-one correspondend to stable Higgs bundles with vanishing Chern ...
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1answer
789 views

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 ...
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197 views

Generalizing a result of Paul Andi Nagy

I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact $$ ...
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1answer
141 views

Harmonic function osculating a given subharmonic function

Let $\Omega\subseteq\mathbb C$ be an open set and let $\phi:\Omega\to\mathbb R_{\geq 0}$ be an exhausting (i.e. proper) smooth subharmonic function. Fix $p\in\Omega$. Does there exist a harmonic ...
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2answers
919 views

A question about Marsden-Weinstein reduction theory

Let $G$ be a compact Lie group and $\frak g$ be its Lie algebra. Then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J \colon M\to \frak g^*$ be its moment map then the ...
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1answer
229 views

The behaviour of holomorphic mapping of curves

Given a polynomial, (or rational function, transcendental entire (meromorphic) function) $f$, and a smooth closed Jordan curve $\gamma$, can we give a complete description of the image of $f(\gamma)$ ...
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183 views

Is it possible to find $h$ hermitian metric such that $Isom_{h}(X) \cong Aut(X)$?

I suspect this is true for some class of analytic manifolds (Riemann surfaces maybe), but my knowledge in differential geometry is very poor, so I could not conclude it. For complex manifolds, is it ...
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2answers
681 views

Spicing up Riemann surfaces course (revised)

I am a master's student planning to write a master's thesis on Riemann surfaces. I plan to study Forster's Lectures on Riemann surfaces. What side topics could one study to spice up the thesis? I am ...
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407 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 ...
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381 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 ...
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120 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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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) ...