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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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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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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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7
votes
1
answer
519
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Do non-projective K3 surfaces have rational curves?
Define a compact Kähler surface $X$ to be a K3 surface if $X$ is simply connected, $K_X \simeq \mathcal{O}_X$, and $h^{0,1}=0$. If $X$ is projective, then a theorem typically attributed to Bogomolov a …
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110
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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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Representing homotopy classes of Kähler manifolds by harmonic maps
Let $(M,g_M)$ be a compact Kähler manifold with negative bisectional curvature. Let $\alpha : (S,g_S) \to M$ be a continuous map from a compact Riemannian manifold $(S,g_S)$.
Is $\alpha$ homotopic to …
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1
vote
2
answers
262
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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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2
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175
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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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2
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Does every non-compact hyperbolic manifold admit compact complex submanifolds?
Let $(X,\omega)$ be a complete Kähler manifold with a metric of negative holomorphic sectional curvature. Does $X$ admit a proper, positive-dimensional, compact complex submanifold?
In general, it is …
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5
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Examples of compact Kähler manifolds whose Bochner curvature tensor has constant norm?
The Bochner curvature tensor is the Kähler analog of the Weyl curvature tensor in the curvature decomposition of a Kähler, discovered by Bochner in 1949. The article on Bochner-Kähler metrics by Rober …
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4
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Is there a compact complex manifold with $b_1(X)=b_2(X)=b_3(X)=b_4(X)=0$?
As the title suggests, I have the following question:
Is there a compact complex manifold with $b_1(X)=b_2(X)=b_3(X)=b_4(X)=0$?
Clarification:
Denote by $b_k$ the $k$th Betti number of a compact com …
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4
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273
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How many ways are there to characterise $\mathbb{P}^n$?
Let $\mathbb{P}^n$ denote the complex projective space of dimension $n$. In many respects, this is the model of (positivity in) complex geometry. There are some well-known characterisations of $\mathb …
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6
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513
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What is the geometric meaning of $H^2(X, \mathscr{O}_X)$?
Let $X$ be a compact complex manifold with structure sheaf $\mathscr{O}_X$ (the sheaf of holomorphic functions on $X$).
What is the geometric meaning (if any) of $H^2(X, \mathscr{O}_X)$?
In the comm …
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answer
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Reference for the Hodge diamond of the Iwasawa threefold
Let $X = G/\Gamma$ denote the Iwasawa threefold, where
$$G = \left\{\begin{pmatrix} 1 & z_1 & z_3\\ 0 & 1 & z_2\\ 0 & 0 & 1\end{pmatrix} : z_1, z_2, z_3 \in \mathbb{C} \right\},$$
and $\Gamma$ is the …
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6
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Is there a relation on Hodge numbers, weaker than $h^{2,0}=0$, that implies a compact Kähler...
The Kodaira embedding theorem yields as a corollary that a compact Kähler manifold $X$ with $h^{2,0} =0$ is projective.
Is there a weaker relation on Hodge numbers that implies that a compact Kähler m …
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2
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0
answers
203
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Yau proof of $K_X>0$ using a non-smooth metric which restricts to a metric of negative holom...
In this lecture of Yau's on the Existence of complete Kähler-Einstein metrics with negative scalar curvature he mentions the following, I quote:
Negative holomorphic sectional curvature is a rather i …
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