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Results tagged with complex-geometry
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user 105103
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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A result in Zheng's complex differential geometry book
In Section 9.5 of Fangyang Zheng's Complex Differential Geometry Book, he proves the following:
Lemma 9.25. Let $(M^2,h)$ be a Kähler surface and $p \in M$. Suppose $M$ has negative holomorphic sectio …
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Can every Hodge structure be polarized?
I suspect this is very elementary, but it is not stated anywhere. A Hodge structure of weight $k$ consists of a finite rank lattice $H_{\mathbb{Z}}$ together with a decomposition of its complexificati …
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What do we necessarily need for the image of a domain of holomorphy to be a domain of holomo...
I posted this on Math.Stack.Exchange with no luck, so I thought it would be perhaps better suited for this site.
We recall that a domain of holomorphy is a domain in $\mathbb{C}^n$ that is holomorph …
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Fibrations in complex geometry
Let $X^n$ be a compact Kähler manifold with $K_X$ semi-ample, i.e., a sufficiently high power of $K_X$ is basepoint free. The associated pluricanonical system $| K_X^{\ell} |$ furnishes a birational m …
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Show that these Kähler forms are cohomologous
Let $Y$ be a closed Kähler manifold with $c_1(Y)=0$ in $H^2(Y,\mathbb{R})$. Let $\omega$ be a Ricci-flat Kähler form on $\mathbb{C}^m \times Y$ such that $$A^{-1} (\omega_{\mathbb{C}^m} + \omega_Y) \l …
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Request for Acta Math Sinica 1984 paper
The mathscinet reference for the paper I am after is here:
MR807424 53C55 (32H99)
Chen, Zhi Hua; Yang, Hong Cang Estimation of the upper bound on the Levi form of the distance function on Hermitian ma …
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Cohomology of the base of an elliptic fibre space
Work over $\mathbb{C}$. Let $\Phi : X \to S$ be an elliptic fiber space, where $X$ is a smooth projective threefold with $H^1(\mathcal{O}_X)=H^2(\mathcal{O}_X)=0$, and $S$ is a smooth projective surfa …
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Examples of compact non-Kähler complex manifolds with Kodaira dimension zero
Let $X$ be a minimal compact non-Kähler complex manifold. Suppose that Kodaira dimension $\kappa(X)=0$.
Is there a known example where the canonical bundle is not holomorphically torsion?
For minima …
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Balanced manifolds and the $dd^c$-lemma
Let $X$ be a compact complex manifold. A Hermitian metric $\omega$ is balanced if $d\omega^{n-1}=0$, where $n=\dim_{\mathbf{C}} X$. By a theorem of Alessandrini-Basanelli, this class of Hermitian mani …
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Practically calculating the domain of a power series for function of several complex variables
For simplicity, let us consider a function $f$ holomorphic on a domain $D \subseteq \mathbb{C}^2$. We may therefore write $f$ as a sum of power series $$f(z) = \sum_{\nu_1 \nu_2 =0}^{\infty} c_{\nu_1 …
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What is the meaning of the monodromy theorem in Hodge theory?
Let $f : X^m \to Y^n$ be an algebraic fiber space (between projective manifolds) whose discriminant locus is denoted by $E$. Let $U$ be a polydisk in $\mathbb{C}^n$ (with coordinates $(y_1, ..., y_n)$ …
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Universal deformation space of a cuspidal plane cubic curve
Does anyone have a reference for the universal deformation space of a cuspidal plane cubic curve? Specifically, a reference that discusses its discriminant locus -- Apparently it has a cuspidal discri …
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Exponential Sequence of Sheaves
Let $(X, \mathcal{O}_X)$ be a complex analytic space in the sense of Grauert, i.e., a $\mathbb{C}$-analytic ringed space which is locally isomorphic to a local model. We may assume that $X$ is a compl …
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Suppose that two cohomologous forms agree on every restriction. Do they agree?
Let $\eta$, $\omega$ be two $(1,1)$-forms on $\mathbb{C}^m \times Y$, where $Y$ is a compact Kahler manifold with vanishing first Chern class, i.e., a Calabi-Yau manifold. Suppose that for all $z \in …
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Condition for Integrability of an Almost Complex Structure
The following question concerns a remark made in the paper:
Lebrun, C., Complete Ricci-flat Kähler metrics on $\mathbb{C}^n$ need not be flat, Proceedings of Symposia in Pure Mathematics, Volume 5 …
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