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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.
8
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861
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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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7
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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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6
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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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6
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0
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206
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Concerning the omnipresence of hyper-elliptic curves in the construction of examples
Vague rambling: I hate asking these types of questions, but I feel that I would benefit immensely from hearing some discussion of the use of hyperelliptic curves in constructing certain examples. What …
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6
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306
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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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6
votes
1
answer
625
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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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6
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329
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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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5
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1
answer
356
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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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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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Fun examples relating to Hopf surfaces
A Hopf surface is a compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. I would like to know whether anyone has any of the following e …
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4
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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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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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0
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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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4
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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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3
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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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