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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Monge–Ampère operator
I'm studying the article of Bedford–Taylor "Fine topology, Šilov boundary…" but I don't
understand the proof of the following proposition.
Let $u$, $v$ be plurisubharmonic functions defined ...
CommunityBot
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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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Converses to Cartan's Theorem B
Here is a phrasing of some Cartan Theorem B statements:
Consider the following conditions:
$X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible ...
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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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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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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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Relation of some Euclidean geometry theorems and more conjecture generalizations
In this topic I want to share relation of the Pythagorean theorem, the Stewart theorem and the British Flag theorem, the Apollonius' theorem, the Ptolemy's theorem and the Feuerbach-Luchterhand. Since ...
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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
$$\...
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Analogue of Bochner's formula for compact Kähler manifolds
Let $X$ be a compact Kähler manifold and $(E,h)$ a Hermitian vector bundle over $X$. Suppose that $\nabla$ is a Hermitian-Einstein connection on $E$, that is $$i\Lambda F_\nabla = \lambda\text{id}_E.$$...
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Obstruction to the existence of global resolution of coherent sheaf
It is well known that any coherent sheaf on a complex manifold (or more generally on some complex spaces) admits locally a resolution with locally free sheaves. It is also well known that for non-...
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Kähler manifold with Ricci-flat Kähler form
hallo,
I have the following problem: Let $X$ be a $n-$dim Kähler manifold with Ricci-flat Kähler form $\omega$. There is a known fact that then there exists a holomorphic (n,0)-form $\Omega$ such ...
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