Questions tagged [complex-manifolds]
For questions about or involving complex manifolds.
377 questions
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Picard-Fuchs equation, Schwarzian derivative and Bers embedding of Teichmuller Space
Let $X$ be the elliptic curve
$$y^2=x^3-g_2x+g_3$$
The $j$ invariant of $X$ is
$$j=\frac{g_2^3}{g_2^3-27g_3^2}$$
I came across the formula of Dedekind
$S(\tau)(j)=\frac{1-\frac{1}{2^2}}{(1-j)^2}+\frac{...
- 515
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The Hantzsche-Wendt manifold
Does the direct product of the Hantzsche–Wendt manifold and a circle admit a complex structure?
- 21
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Holomorphic fiber bundles with holomorphic and non-holomorphic sections
I am looking for examples of holomorphic fiber bundles $\pi\colon E \rightarrow B$ with the following properties:
Both $E$ and $B$ are connected complex manifolds.
There exists a holomorphic section $...
- 1,438
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Why Isotopic Markings Define the Same Point in Teichmüller Space
Let $S_g$ be a compact orientable surface of genus $g \geq 2$. The Teichmüller space $\mathcal{T}(S_g)$ is defined as the set of equivalence classes of pairs $(X, f)$, where:
$X$ is a Riemann surface,...
- 111
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Is there any Galois correspondence about fiber bundle (vector bundle)?
I have learned Galois correspondence about universal covering space over a topological manifold $M$. Since a covering space over $M$ can be viewed as a fiber space over $M$ with discrete fibers, and a ...
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Invariant sets of foliations
Definition: Let $\mathcal{F}$ be a regular holomorphic foliation on a complex manifold $M$. We say that a subset $ A \subseteq M $ is invariant for $\mathcal{F}$ if $A = \bigcup\limits_{x \in A} L_x $...
- 601
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Suppose that $S^6$ admits a complex structure, is there an almost complex submanifold of $S^6$?
Suppose that $S^6$ admits a complex structure, is there necessarily an almost complex submanifold of $S^6$ of dimension $2$ or $4$?
By almost complex submanifold, I mean that the tangent space of the ...
- 101
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Kummer surfaces and foliated K3 surfaces
In "FOLIATIONS ON COMPLEX PROJECTIVE SURFACE", M. Brunella proves (at the end) that K3 surfaces $M$ with algebraic dimension zero and admitting a holomorphic foliation are obtained as a &...
- 171
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Are bundles with continuous metric and positive curvature big?
Let $L$ be a holomorphic line bundle on a compact complex manifold $M$, and $h$ a continuous Hermitian metric on $L$ such that its curvature current $dd^c \log h$ is positive and strictly positive ...
- 9,459
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Stably complex representations
Let $\Gamma$ be a finite group and suppose $V$ is a finite-dimensional real $\Gamma$-representation. If there exists a finite-dimensional complex $\Gamma$-representation $W$ such that $V \oplus W$ is ...
- 945
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Extension of Harvey-Lawson's theorem to general pseudoconvex hypersurfaces
Harvey-Lawson have this remarkable theorem (which can be seen here):
Theorem: Let $X$ be a strongly pseudoconvex CR manifold of dimension $2n −1$, $n \geq 2$. If $X$ is contained in the boundary of a ...
- 181
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Projective dimension of analytic coherent sheaves
Let $X$ be a compact complex manifold. Let $\mathrm{Coh}(O_X)$ be the category of coherent $O_X$-modules. Let $F,G$ be coherent sheaves on $X$. Let $\mathrm{Ext}^i_{\mathrm{Coh}(O_X)}(F,G)$ be the Ext ...
- 837
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Quasi-coherent gaseous sheaves and affineness
I am trying to get a feeling for quasi-coherent sheaves in complex geometry, in the manner of Clausen-Scholze. I find myself a little unclear how to expect such to behave.
Apologies in advance, I ...
- 241
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Almost complex manifold with at least one almost complex chart is automatically complex?
Let $(M,J)$ be an almost complex manifold. Further, let $(U,\varphi)$ (with $U \neq \emptyset$) be an almost complex chart, i.e. $\varphi: U \to V$ is an almost complex map where $V\subseteq\mathbb{C}^...
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A somewhat simple looking question in complex geometry: Levi Civita vs. Chern connection
I am currently reading the following article https://arxiv.org/pdf/2409.04382
I am interested in eq. (4.19). It seems to suggest that on any complex manifold $X$ with Hermitian metric $g$ and ...
