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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Kahler Einstein metric with minimal singularities

Let $X$ be a Kahler variety with snc divisor $D$ such that $K_X+D$ is ample. then there is a Kahler metric $\omega_E$ such that $Ric(\omega_E)=-\omega_E$ on $E=X\setminus D$, then $h=\frac{1}{\omega_E^...
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21 views

Finding the analytic Zariski decomposition singular hermitian metric on a relative line bundle

Let $f:X\to S$ be a proper surjective projective morphism between complex manifolds with connected fibers and let $D$ be an effective $\mathbb Q$-divisor on $X$ such that $$S^°=\{s\in S| f\text{ is ...
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25 views

Gauss Bonnett on a flat surface with border

I'm reading "Euler Characteristics of Teichmuller Curves in genus two" by Matt Bainbridge and there's a point I don't understand in the proof of theorem 5.5. Maybe you can help me clarify it. Let $Y,...
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25 views

Singular canonical hermitian metric

Let $M$ be a complex manifold , take $$K_{M,m}:=\sup \{|\sigma|^{\frac{2}{m}}; \sigma\in H^0(M,\mathcal O_M(mK_M)), |\int_M(\sigma\wedge\bar\sigma)^{\frac{1}{m}}|\leq 1\}$$ Let $$K_{M,\infty}:=\lim\...
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1answer
139 views

Kähler classes for surfaces of general type with $c_1^2=3c_2$

Given a smooth, compact complex surface with ample canonical bundle satisfying $c_1^2=3c_2$, is it true that every Kahler class is a multiple of $c_1$? This seems to be the case for fake projective ...
8
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125 views

Smooth quotients of algebraic spaces that are varieties away from codimension $\ge 2$ subset

This is a question about when a smooth complex algebraic space that is very close to being an algebraic variety is actually an algebraic variety. General question: Let $X$ be a smooth separated ...
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82 views

Riemann surface defined by a Beltrami differential

Let $R$ be a Riemann surface, $\omega$ and $\tau$ respectively a holomorphic and an antiholomorphic 1-form on $R$. Locally $\omega=fdz$ and $\tau=gd\overline{z}$ with $\partial_{\overline{z}}f=0$ and $...
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2answers
185 views

cohomology of configuration space of punctured variety

Given a smooth projective variety $X$ of dimension $l$, we denote with $F(X,n)$ the configuration space of points $$ F(X,n):=\{(x_{1}, \dots, x_{n})\in X^{n}\: : \: x_{i}\neq x_{j}\text{ for each }i,j ...
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96 views

Unexpected isomorphisms between “unrelated fields”

I read in the post Why worry about the Axiom of Choice ? that the existence of isomorphisms between $\overline{\mathbb{Q}_p}$, $p$ any prime, and $\mathbb{C}$, makes some worry about the Axiom of ...
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46 views

Thom form of holomorphic bundle over Kaehler manifolds/orbifolds

Consider a holomorphic vector bundle $\pi:E\rightarrow X$ of complex rank $m$ over a Kaehler manifold $X$. Can we find a Thom form $\Theta$ of $E$ such that as a form on the complex manifold $E$, it ...
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1answer
163 views

Intersection of two curves is not Cohen Macaulay

Let be $R=\mathbb{C} \lbrace x,y,z \rbrace$ the formal series ring and let $f_{1},f_{2},f_{3} \in R$ be nonzero elements of $R$. (a) Consider the varieties $M:=V(f_{1},f_{2})$ and $N:=V(f_{2},f_{3})$ ...
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1answer
118 views

Pseudoeffective line bundle

Let $L$ be a pseudoeffective line bundle on a complex manifold $X$ then is there a singular hermitian metric of $L$ which its curvature is not semi-positive?
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67 views

Schwarz–Ahlfors–Pick theorem for hyperbolic pair of pants

Let $\mathcal{P}$ be a hyperbolic pair of pants with geodesic boundary and $\Sigma_3$ be the hyperbolic thrice punctures sphere. I want to construct a conformal map $f:\mathcal{P}\rightarrow \Sigma_3$ ...
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1answer
76 views

Hyperplane generic to a given arrangement

At the moment, I am reading the paper "on the connectivity of the realization spaces of line arrangements" of Nazir and Yoshinaga. I would like to extend their Lemma 3.2 to higher dimension. However, ...
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105 views

Ellipticity of Bott-Chern Laplacian

I want to prove that Bott-Chern Laplacian $$\tilde{\Delta}_{BC}^{p,q}=(\partial\bar\partial)(\partial\bar\partial)^*+(\partial\bar\partial)^*(\partial\bar\partial)+(\bar\partial^*\partial)(\bar\...
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2answers
232 views

Alternative construction of the first Chern class map

Let $X$ be a compact Kähler manifold. Consider the exponential sequence $0 \to \mathbf Z \to \mathscr O_X \to \mathscr O_X^* \to 0$. The boundary map gives a map $H^1(X, \mathscr O_X^*) \to H^2(X, \...
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1answer
163 views

A very general complex torus is simple

Let us parametrize the set of lattices inside $\mathbb{C}^g$ with the open dense subset $U = \text{GL}_{2g}(\mathbb{R})$ of $\mathbb{R}^{4g^2}$. Does there exist a countable family $(Z_n)_{n \in \...
4
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1answer
216 views

Hodge decomposition for Bott-Chern cohomology

$\DeclareMathOperator{im}{im}$ I want to prove that Bott-Chern cohomology group $H^{p,q}_{BC}=\frac{\ker\partial\cap \ker\bar{\partial}}{\im\partial\bar{\partial}}$ has finite dimension via Hodge ...
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1answer
91 views

Reference request: Existence of plurisubharmonic potential for positive (1,1)-current on Stein (affine) manifold

I am looking for a reference for the following fact, which I believe should be true. Let $X$ be a Stein manifold (or smooth affine variety over $\mathbb{C}$). If $\omega$ is a positive closed $(1,...
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1answer
229 views

Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure?

Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure? Every paper or article I looked at, talk only about singular laminations on $\mathbb{CP}^2$. I was wondering why. If you know ...
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1answer
185 views

Question on Weil-Petersson metric on Teichmuller space

I'm reading Ahlfors' original articles about Weil-Petersson metric: "Some remarks on Teichmüller's space of Riemann surfaces" and "Curvature properties of Teichmüller's space". The tangent space at ...
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1answer
49 views

On the limit set of eigenvalues of banded Toeplitz Hessenberg matrices

Let $T_{n}(b)$ be the $n\times n$ Toeplitz matrix determined by the symbol $$ b(z)=\frac{1}{z}+\sum_{j=0}^{k}a_{j}z^{j} $$ where $k\in\mathbb{N}$ and $a_{0},\dots,a_{k}\in\mathbb{R}$, $a_{k}\neq0$. ...
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1answer
191 views

Question on period map, Gauss-Manin connection and complex coordinates of $\mathcal{H}^1(k)$

Let $\mathcal{L}_g$ be the space of abelian differentials on Riemann surfaces of genus $g\ge 2$ and $\mathcal{TH}_g:=\mathcal{L}_g/Diff_0^+(S_g)$ be the Teichmuller space of abelian differentials on ...
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98 views

On dimension of the moduli space of abelian differentials on Riemann surfaces

I fear I'm missing something important here, so forgive me if my question is stupid. Consider $\mathcal{M}_g$ the moduli space of Riemann surfaces of genus $g>2$ and $\mathcal{H}_g$ the moduli ...
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80 views

Action of a lattice on abelian varieties

Let $\pi\colon Y\to\mathbb{P}_\mathbb{C}^1$ be a ramified cover of degree two of $\mathbb{P}_{\mathbb{C}}^1$ such that $Y$ is smooth. I fix a point $x$ on $\mathbb{P}^1$, over which the cover is étale,...
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2answers
163 views

Kähler forms arising as the curvature form of a singular metric on a line bundle

The Fubini-Study metric on complex projective space $\mathbb{P}^n$ is a smooth metric $h = e^{-\phi}$ on the line bundle $\mathcal{O}(1)$ and it is a standard calculation to check that its curvature ...
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0answers
98 views

How many compact complex 3-folds with $b^1 = b^2=h^{1,2}=0$?

Are there any compact complex 3-folds with Betti numbers, $b^1 = b^2 = 0 $ and Hodge number, $h^{1,2}=0$? If yes, then how plentiful are they?
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462 views

Spin^c structures on manifolds with almost complex structure

Let $M$ be a smooth even-dimensional manifold. Is it true that for each almost-complex structure $J$ on $M$ there exists a canonical spin$^c$ structure $S_J$ associated to $J$ ? (I've read this ...
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1answer
261 views

Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?

Is there a closed subscheme $D$ in $\mathbb P^2_{\mathbb C}$ pure of codimension one such that, for all algebraic varieties $X$ over $\mathbb C$, any analytic map $$ \phi: X(\mathbb C) \to \mathbb P^...
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1answer
153 views

Normal bundle of a fiber of the family of curves

If we have the family of complex curves $f:X\rightarrow Y$, over a complex smooth curve $Y$ , we consider a fiber $C=f^{-1}(y)$ and its tangent bundle $T_{C}$. We know that $df: f^{*}{T_{Y}}_{|C}\...
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99 views

Decompositon of the Euler class in the ideal generated by Weyl-invariant polynomials

Let $G$ be a complex reductive Lie group, $B$ be a Borel subgroup, $T\subset B$ be a maximal torus, $W$ be the Weyl group. Then the space $X:=G/B$ is a complex manifold of dimension $n$, denote by $\...
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1answer
124 views

Density of non-algebraic leaves in the characteristic foliation

Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of $D$, the restriction of $\omega$ to $D$ has one-...
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2answers
508 views

List of Applications of the $\partial\overline{\partial}$-lemma

Quoting from Huybrecht's book Complex Geometry on the $\partial\overline{\partial}$-lemma for Kaehler manifolds: Although it looks like a rather innocent technical statement, it is crucial for ...
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1answer
404 views

Holomorphic vector bundles over $\mathbb{C}^{n}\setminus 0$

Is it true that every holomorphic vector bundle over $\mathbb{C}^{n}\setminus 0$ is trivial? If not true, how can one construct a counterexample? And just a small note here (wrong): For $n\leq 2$, ...
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1answer
324 views

How to tell if it's a Moishezon morphism

Suppose that $f \colon X\rightarrow S$ is a proper morphism of reduced and irreducible complex spaces and $f$ is a smooth deformation in the sense of Kodaira and Spencer. If we know each fiber $X_s$, ...
6
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1answer
355 views

The Hypercomplex Structure of $SU(3)$

(A) In this really stylish answer it is shown that one can define a family of complex structures $J_{\lambda}$ on the Lie group SU(3), dependent on the parameter $\lambda \in {\mathbb C}\backslash {\...
6
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1answer
293 views

Is polynomial convexity a topological invariant?

Is the property of being polynomially convex a topological invariant? In other words, let $M$ and $N$ be two homeomorphic, compact subsets of $n$-dimensional complex Euclidean space, and assume in ...
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80 views

Open Period Integrals of Elliptically Fibered K3 surfaces

Let M be the period domain for elliptic K3 surfaces $(X,\Omega)$ with a holomorphic two-form. Denote the fiber class $f$. Then $$M=\{\Omega\in f^\perp\otimes \mathbb{C}\,:\, \Omega\cdot \Omega=0, \,\...
4
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1answer
171 views

Hyper-Kaehler Strucutre for Compact Lie Groups?

We know from the classy work of Joyce that "any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus". The quote comes from the Wikipedia page. I am asking if it ...
3
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1answer
185 views

Deformation of $\mathbb{P}^1 \times \mathbb{P}^1$

I learnt that $\mathbb{P}^1 \times \mathbb{P}^1$ is rigid, but can be deformed to a non-rigid Hirzebruch surface $S$. Suppose $\pi: M \to B$ is such deformation such that $\mathbb{P}^1 \times \mathbb{...
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1answer
74 views

Find the Picard Fuchs operator of a four parameter fundamental period

In my research, we have constructed a Calabi-Yau as a hypersurface of a toric variety and we could compute the fundamental period of the complex moduli, which is a series in four parameters. Say \...
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1answer
217 views

Normal bundle to fibers of a rational morphism

Let $f:X\dashrightarrow C$ be a rational fibration from a 3-dimensional variety $X$ to a curve $C$ such that generic fiber is smooth and different fibers intrsect in smooth curves. Take $S$ to be a ...
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1answer
248 views

Algebraicity and non-algebraicity of leaves of the characteristic foliation

Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of $D$, the restriction of $\omega$ to $D$ has one-...
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47 views

Hypercomplex, hyperKahler, or quaternion-Kähler from Joins/Connected Sums

I am looking for examples of (compact) hypercomplex, hyperKahler, or quaternion-Kähler manifolds which can be constructed as joins/connected sums of manifolds which do are not hypercomplex, ...
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2answers
249 views

Chern classes and singular hermitian metrics on vector bundles

Let $L$ be a holomorphic line bundle on a complex manifold $X$, and assume it is equipped with a singular hermitian metric $h$ with local weight $\varphi$. Then, one can show that the de Rham class of ...
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1answer
241 views

Relative tangent bundle and trivilization, tautological foliation

Let $T_{X}\rightarrow X$ be the tangent bundle over a complex manifold $X.$ Let $\pi:PT_{X}\rightarrow X$ be a projectivization of that bundle. Let $L$ be the tautological line bundle of $PT_{X}.$ ...
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128 views

Kodaira fibration and moduli space of Riemann surfaces

Here we mean Kodaira fibration $f: X \rightarrow C$ where $f$ is a holomorphic submersion with maximal rank everywhere, but not a complex fiber bundle map. Such a surface has been constructed by ...
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48 views

Hyper-Complex Connected Sums of Grassmannians?

As we all know, the Grassmannians are Kaehler manifolds. Is there anyway to take connected sums of Grassmannians and produce a examples of hypercomplex manifolds, hyper-Kaehler or Calabi--Yau exmples ...
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69 views

On K-theory of blow-ups of compact complex manifolds

Is there a long exact sequence for the K-theory of (coherent sheaves on) blow-ups of compact complex manifolds? Does it split? What can one say on (possibly, singular) complex analytic spaces here? ...
5
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152 views

When a proper smooth fibration is isotrivial?

Let $\pi : X \to Y$ be a proper smooth morphism between complex quasi-projective varieties. Assume there exists a connected and Zariski dense analytic subvariety $Z$ of $Y$ such that for any two ...