Tagged Questions

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Complex structures on Riemann surfaces

This is cross posted from math.SE: http://math.stackexchange.com/q/876432/9 Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq (H^0(M;K^2))^*$. Considering $\alpha$ as a map ...
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A topological criterion for connectedness of a semi-ample divisor

I have a half page long proof of the following statement, and I would like to know if this is (a corollary of) a well known statement. Maybe there is a reference or a three lines proof? Statement. ...
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Basic questions on the Hilbert scheme/ Douady space

Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More ...
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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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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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Almost Complex Structure approach to Deformation of Compact Complex Manifolds

I don't know much about the deformation of compact complex manifolds, I've only read chapter 6 of Huybrechts' book Complex Geometry: An Introduction. There are two parts to this chapter. The second ...
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motivating examples of family of Hodge structure

Let $\phi: \chi \to B$ be a proper holomophic submersion with smooth fiber $X$. Then, we get local systems $R^i \phi_* \mathbb C$ and corresponding flat connection $(B, \bigtriangledown)$ In this ...
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Algebraic De Rham cup product versus Betti cup product

Let $X$ be a smooth projective variety over $\mathbf{C}$ of complex dimension $2n$. Let $C_1,C_2\subseteq X$ be two closed subvarieties of complex dimension $n$. Then we get two Betti homology ...
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Moderate growth functions on complex algebraic varieties.

Does anybody know a reference that discusses moderate growth (continuous or measurable) functions on complex algebraic varieties? I'm interested in such a discussion in the generality of varieties ...
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Why the letter “p” for genus?

Does anybody know why the genus (arithmetic or geometric) of a curve was historically denoted by $p$ ($p_a$ and $p_g$)? What does the letter "$p$" stand for? Any references would be greatly ...
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Complex manifold which is algebraic away from codimension \ge 2

If $X$ is a complex manifold and $Z$ is a closed subset of codimension $\ge 2$ such that $X-Z$ has an algebraic structure, then is $X$ algebraic; i.e. a scheme? If not is $X$ an algebraic space? ...
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Curvature and Symmetry on Kähler manifolds

Hi there, Suppose $X$ is a Kähler manifold that has an analytic isometry $S$, with $S^k = \operatorname{Id}$ ($k \in \Bbb N$). In a situation like this (maybe with additional assumptions on $X$) can ...
636 views

Monge Ampere equations

I am a graduate student trying to understand complex Monge-Ampere equations(mostly on complex manifolds with or without boundary, but also in C^n), but I can't put my hand on any monograph/textbook ...
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Composition of circle inversions

I would like to understand the map of $\mathbb{C}$ to $\mathbb{C}$ that results by iterating inversion in a unit circle. Let $f(z)$ for $z \in \mathbb{C}$ invert $z$ in a unit circle centered on ...
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Calculating a curvature tensor by polarization

I'm reading some articles by Siu and Nannicini on the Weil-Petersson metric associated to families of compact Kahler-Einstein manifolds. In each article Siu and Nannicini construct a Weil-Petersson ...
Given a hermitian holomorphic vector bundle (E, h) on a complex manifold-with-a-metric (X,g), then consider the following (natural) construction of a metric on the total space of $\mathbb{P}(E)$ : ...
In their book "Compact complex surfaces",W,P.Barth,K.Hulek,C.A.M.Peters and Van De Ven refer to the following theorem: Let $X$ be a compact complex space and $L$ a holomorphic line bundle on $X$.Then ...