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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Lie algebra cohomology and Lie groups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}\DeclareMathOperator\Sp{Sp}$Let $G$ be a complex Lie group and $\mathfrak{g}$ its Lie algebra (which is over $\mathbb{C}$). My first question is:
(...
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Cup-product ($\cup$) vs. cross-product ($\times$) on the space of graph homology
I'm currently reading Nieper-Wißkirchen's Chern numbers and Rozansky-Witten invariants of compact Hyper-Kähler manifolds. He introduces the space of graph homology $\mathcal B$ as the free $K[\circ]$-...
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Different Hodge numbers arising from different holomorphic structures?
Does anyone have an example or know any references for a complex manifold $M$ with two different holomorphic structures that give rise to different Hodge numbers?
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Hyperplane section argument on Kähler manifold
On a projective variety, there exist very ample line bundles and hyperplane sections. It is a useful trick to take a general hyperplane section to reduce a problem to a lower dimension. However, on ...
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Negative definite of exceptional curve in higher dimension
One direction of the Grauert's contractibility theorem shows
Let $f:S\rightarrow T$ be a surjective holomophic map where $S$ is a compact holomorphic surface. If $C$ is a reduced connected effective ...
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Enriques-Kodaira classification of minimal resolution of surface with quotient singularities
Let $X$ be a normal projective complex surface with at worst quotient singularities. Let $\bar{X}\to X$ be the minimal resolution. Further assume that $b_2(X)=1$ and $b_1(X)=b_3(X)=0$. Since quotient ...
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$H^2(X,T_X)=0$ implies the Frölicher spectral sequence degenerates at $E_1$?
Let $X$ be a compact complex manifold, if $X$ satisfies $H^2(X,T_X)=0$, then it is well-known that the Kuranishi space of $X$ is smooth by Kodaira and Spencer's deformation theory. On the other hand, ...
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Analogue of Bochner's formula for compact Kähler manifolds
Let $X$ be a compact Kähler manifold and $(E,h)$ a Hermitian vector bundle over $X$. Suppose that $\nabla$ is a Hermitian-Einstein connection on $E$, that is $$i\Lambda F_\nabla = \lambda\text{id}_E.$$...
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How to "eliminate" the log pole of a logarithmic $(p,q)$-form？
Let $X$ be a compact complex manifold, and $D=\sum_{i=1}^{r} D_i$ be a simple normal crossing divisor on $X$. Let $\alpha$ be a logarithmic $(p,q)$-form, namely, on an open subset $U$, we can write
$$\...
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Branched covering maps between Riemann surfaces
What is an example of a branched covering map between Riemann surfaces of infinite degree? i.e. something like a branched version of the exponential map $exp: \mathbb{C} \to \mathbb{C}^*$.
Thanks!
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Isometry group of the Fubini-Study metric on complex projective spaces
Let $(\mathbb CP^n,g_{FS})$ be the complex projective space equipped with the standard Fubini-Study metric.
What is the Riemannian isometry group of $(\mathbb CP^n,g_{FS})$? It seems to me that its ...
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Locality of Kähler-Ricci flow
Let $(M,I, \omega)$ be a compact Kähler manifold with $c_1(M)=0$. Denote by $\operatorname{Ric}^{1,1}(\omega)$ the Ricci (1,1)-form, that is, the curvature of the canonical bundle. It is known ("...
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The discriminant of a holomorphic vector bundle
Let $M$ be a complex manifold and $E$ a holomorphic vector bundle over $M$. The discriminant $\Delta(E)$ of $E$ is then defined to be $$\Delta(E)=c_2(\text{End}(E))=2rc_1(E)-(r-1)c^2_1(E).$$
This ...
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Seeking for bridges to connect K-stability and GIT-stability
We consider the variety $\Sigma_{m}$ := {($p$, $X$) : $X$ is a degree $n$ + 1 hypersurface over $\mathbb{C}$ with mult$_{p}(X) \geq m$} $\subseteq$ $\mathbb{P}$$^{n}$ $\times$ $\mathbb{P}$$^{N}$, ...
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Birational deformations of holomorphic symplectic manifolds
Let $X$ and $X'$ be birational holomorphic symplectic manifolds. Then the birational morphism between them identifies $H^2(X)$ with $H^2(Y)$. The period space of $X$ is defined to be a subset of $\...
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