Questions tagged [complex-geometry]
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
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Is the associated G/B fibration to a G-torsor projective?
Let $X$ be a smooth projective variety over $\mathbb{C}$, $G$ a reductive group, and $P \to X$ a $G$-torsor. Let $B \subset G$ be a Borel subgroup. Is the associated $G/B$ fibre bundle $$ Y=G/B \...
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Inequality of Lübke
The following famous inequality by Lübke is given in Differential Geometry of Complex Vector Bundles by S. Kobayashi
Let $(E, h)$ be an Hermitian vector bundle of rank $r$ over a compact Hermitian ...
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Ramified covering map between analytic sets
Suppose $ B_{n}(0, 1) $ is the open ball of radius 1 in $n$-dim complex space $\mathbb{C}^{n}$ and $B_{m}(0,1)$ is the open ball of radius 1 in $m$-dim complex space $\mathbb{C}^{m}$. Let $V$ be a $n$-...
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Compute the Euler class of tautological $C$-bundle over $CP^1$
$\DeclareMathOperator\SO{SO}$This might be an old question. But since I have not found an explicit answer to this question, I put the question here.
The background is that we need to use a similar ...
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Hodge numbers of a complement
Let $Y\subset X$ be an analytic subvariety of codimension $d$ of a smooth compact complex variety $X$. Denote $U = X\setminus Y$. The relative cohomology exact sequence implies that
$$
H^i(X) \to H^i(...
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Holomorphic fibre bundles over noncompact Riemann surfaces
Some days ago I came across the paper "Holomorphic fiber bundles over Riemann surfaces", by H. Rohrl.
At the beginning of Section 1, the following theorem is quoted:
Theorem. Every fiber ...
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Moduli space of curves away from singular subsets
Recently, I'm interested in the moduli spaces of curves in a (possibly noncompact) complex orbifold (resp. symplectic and almost complex) away from singularities.
More specifically, I'm interested in ...
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Prefactor $2\pi i$ for Tate-Hodge structure
A rather basic question. What was the original reason to consider the underlying $\mathbb{Z}$-module of the - as canonical object regarded - Tate-Hodge structure $\mathbb{Z}(1)$ to be given as $2 \pi ...
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Subset of a complex manifold whose intersection with every holomorphic curve is analytic
The Osgood's lemma in complex analysis says that if $f \colon \mathbb{C}^n \to \mathbb{C}$ a continuous function such that $f|_{L}$ is holomorphic for every complex line $L \subset \mathbb{C}^n$, ...
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Curvature and Hermitian-Einstein conditions
The following is from a set of lecture notes I'm following and I have had some difficulties understanding it.
Let us discuss a few equivalent formulations of the Hermite-Einstein condition ($\Lambda_\...
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Gabriel's theorem for complex analytic spaces
Let $X,Y$ be noetherian schemes over $\mathbb{C}$.
Then, it is known that
$$
\text{Coh}(X) \simeq \text{Coh}(Y) \Rightarrow X \simeq Y,
$$
by P. Gabriel(1962).
Are there some results in the case of ...
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A closed $(1,1)$-form $\eta$ is harmonic if and only if $\Lambda\eta = \text{constant}$
Let $X$ be a Kähler manifold and consider the dual operator $\Lambda$ of the Lefschetz operator $L$. Is the following a true statement?
A closed $(1,1)$-form $\eta$ is harmonic if and only if $\...
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Understanding the Hodge filtration
Let $X$ be a smooth quasiprojective scheme defined over $\mathbb{C}$, and let $\Omega^{\bullet}_X$ denote its cotangent complex, explicitly, we have:
$\Omega^{\bullet}_X:=\mathcal{O}_X\longrightarrow \...
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Reference request. Looking for a specific compact complex manifold
For my research I need to construct a compact complex manifold with quartic ramification loci. By quartic ramification loci I mean that $L_1,L_2,L_3$ are complex algebraic varieties of degree four and ...
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Mean curvature as a contraction
I'm going over some of Kobayashi's work on complex vector bundles and trying to state some of the notions in a more familiar language to me.
The set up is the following. We have a hermitian vector ...
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