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Results tagged with complex-geometry
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user 42804
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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463
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self intersection of a curve in a surface
Suppose $S$ is a compact complex surface, $C\subset S$ is a one dimensional irreducible subvariety (a curve). Suppose further, there exists a family of biholomorphism of $S$ nearby the identity map. …
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120
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dimension of singular set of torsion free sheaves over a unit disc
Suppose $D\subset\mathbb C$ is a unit disc and $\mathcal F$ is a torsion free analytic coherent sheaf over $D$. Define $S(\mathcal F)=\{x\in D|\mathcal F_{x}\, is \,not\, locally\, free\}$. Is $S(\m …
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110
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Can we find a torus on $K^3$ surface
Suppose in $P^3$ we have $K3$ surface defined by $x^4+y^4+z^4+w^4=0$ can we find a complex subvariety that is a torus?
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1
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132
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generic irreduciblity
Suppose we have a proper morphism $f:X\rightarrow Y$ and $0\in Y$. If the fiber $f^{-1}(0)$ is irreducible and reduced, is the set $\{y\in Y|f^{-1}(y) \text{ is irreducible and reduced}\}$ open?
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1
answer
600
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what is the first cohomology group of structure sheaf of grassmannian [closed]
I want to know if the first cohomology group of structure sheaf of grassmannian vanishes.
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0
answers
49
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moduli space of curves under prescribed tangency conditons
We consider an irreducible component of the Hilbert Scheme of curves in $\mathbb P^2$. Denote it as $\mathcal D.$ We fix a line $L$ and a point $A\in L.$ Denote $\mathcal D_0$ as the subscheme of $\m …
- 1,091
5
votes
2
answers
985
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What's the minimal embedding of orthogonal Grassmannian?
Suppose $X$ is the orthogonal Grassmanian. We know the Plücker embedding does not span the whole background $\mathbb{CP}^n,$ just a subspace $\mathbb{CP}^m.$
My question: is there an expressio …
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1
answer
321
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functoriality of hilbert scheme
suppose $f:X\rightarrow Y$ is a morphism between two schemes over scheme $S.$ Do we have the morphism between their hilbert schemes, i.e. is there a natural morphism $Hilb(X/S)\rightarrow Hilb(Y/S)$ o …
- 1,091
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1
answer
595
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is grassmannian rational connected or not [closed]
I wan to know if Grassmannians are rational connected? Any reference describe how to tell if a variety is rational connected or not?
- 1,091
2
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102
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kahler einstein metric for exceptional compact type hermitian symmetric space
Can anyone write down the kahler einstein metric for exceptional compact type hermitian symmetric spaces($\frac{E_6}{SO(10)*SO(2)}$ and $\frac{E_7}{E_6*SO(2)}$). I can find the bergmann kernel for th …
- 1,091
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1
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139
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Is there some lattice not rigid
I heard that in complex hyperbolic space setting for example CH2, there is some deformation of lattice nontrivial. What confused me is it seems contradicting Mostow Rigidity. Could someone explain t …
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3
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answers
114
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cayley transformation of bounded symmetric domain
Can anyone write down the biholomorphic map between classical bounded symmetric domains(defiend by matrixs) with their siegel upperhalf plane models. I know if it's type 2, i.e $I-Z\bar{Z}^{t}>0$ whe …
- 1,091
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2
answers
505
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Is there an Oka-Grauert principle for homogeneous spaces?
Suppose we have a fibration over the punctured disc (i.e., a deformation of complex manifolds) such that each fiber is a homogeneous space. Is the total space a product of a fiber with the punctured d …
- 1,091
4
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1
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227
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can we write down the holomorphic vector fields on the compact hermitian symmetric spaces ex...
Can we write down the holomorphic vector fields on the compact hermitian symmetric spaces explicitly? Do you have any idea of which paper has disscussed this topic? For example, what is the dimensio …
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answer
1k
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What are the automorphisms of a Grassmannian?
I want to know what are the holomorphic automorphisms of a Grassmannian. Can someone tell me this?
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263
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Is a G-invariant metric always Kähler-Einstein?
Suppose there is a Hermitian symmetric space of compact type $X$. It is realized in the following way: $X\hookrightarrow\mathbb{P}^N$ and equipped with the induced Fubini-Study metric $g$.
What's m …
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1
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175
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What is the Fano index for Hermitian symmetric spaces of compact type?
As we know Hermitian Symmetric spaces of compact type are all Fano picard number one, we can talk about his Fano index. Suppose $X$ is one of those Hermitian symmetric spaces, $L$ is the generator of …
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5
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1
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523
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How to tell if it's a Moishezon morphism
Suppose that $f \colon X\rightarrow S$ is a proper morphism of reduced and irreducible complex spaces and $f$ is a smooth deformation in the sense of Kodaira and Spencer. If we know each fiber $X_s$, …
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5
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1
answer
579
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Can someone tell me properties of Douady space?
I want to know the parallel properties of Douady space with respect to Hilbert scheme. For example I want to know what is the irreducible component of Douady space, what if I consider a family of fixe …
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201
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classification of homogenous complex manifolds
Suppose $X$ is a complex manifold (doesn't assume it's Kahler), and it's holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ?
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-1
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1
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92
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Zariski open set in orthogonal grassmanian [closed]
I am confused about the following question.
Consider $\mathbb C^4$ endowed with nondegenerate symmetric bilinear form $J:=\left(\begin{matrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{matrix}\right)$. L …
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5
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520
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a question on Hodge and Atiyah's paper "integrals of the second kind on an algebraic variety"
I have a question on Hodge and Atiyah's paper "Integrals of the second kind on an algebraic variety". It is about the exact sequence below formula (14) and above formula (15) on page 71:
$$H_{2n-q}(S …
- 1,091
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Classification of principal monodromy elements
Let $(X,0)$ be a germ of normal analytic space with an isolated singularity at $0$, and let $Y:=X\backslash\{0\}$. Suppose $Y$ has a complex-hyperbolic metric which is complete at $0$. Burns-Mazzeo pr …
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