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Results tagged with complex-geometry
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user 16183
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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2
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637
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Examples of pluripolar sets
I have a very basic question on pluripolar sets. First remind their definition.
Let $\Omega\subset \mathbb{C}^n$ be a domain. A subset $E\subset \Omega$ is called pluripolar if there exists a plurisub …
- 21.8k
6
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193
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Betti numbers of non-projective compact algebraic varieties
Are there examples of complex smooth compact algebraic varieties (necessarily non-projective) with vanishing second Betti number?
If not, are there such examples whose Betti numbers do not satisfy th …
- 21.8k
3
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204
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Cohomology rings of complex varieties and combinatorics
It is a classical fact that the cohomology ring (with complex coefficients) of a complex smooth projective manifold is a bigraded algebra satisfying (1) Poincare duality; (2) hard Lefschetz theorem; ( …
- 21.8k
11
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743
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Gluing Riemann surfaces
Let $X$ be a compact Riemann surface with boundary $\partial X$. Assume (for simplicity only) that $\partial X$ has a single connected component. Let us fix an orientation preserving diffeomorphism $\ …
- 21.8k
0
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113
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Kahlerness of the projectivized cotangent bundle [duplicate]
Let $X$ be a smooth not necessarily compact complex manifold which admits a Kahler metric. Is it true that its projectivized cotangent bundle also admits a Kahler metric? If not, are there sufficient …
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6
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315
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Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$
Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.)
Is it true that for any o …
- 21.8k
0
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144
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Limit of a sequence of smooth varieties in Hilbert scheme
Let $\{Z_i\}_{i=1}^\infty$ be a sequence of smooth irreducible $k$-dimensional submanifolds of $\mathbb{C}\mathbb{P}^n$ which converges to a closed subscheme $Z$ in the sense of the Hilbert scheme of …
- 21.8k
3
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Cohomology of a tubular neiborhood of submanifold vs cohomology of the formal neighborhood
Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $\hat Z$ be the formal neighborhood of $Z$. Let $U$ be an open neighborhood of $Z$.
I …
- 21.8k
2
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1
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311
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Flatness of a morphism of complex analytic spaces
Let $f\colon X\to D$ be a morphism of a complex analytic space $X$ into the 1-dimensional disk $D$. Assume for simplicity that $X$ has a single irreducible component which may not be reduced.
Questi …
- 21.8k
3
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A question on the proof of Bedford-Taylor theorem in Demailly's book
I am trying to understand a proof of the Bedford-Taylor theorem on the weak convergence of Monge-Ampere operators of decreasing sequences of plurisubharmonic functions.
I am reading a proof in the bo …
- 21.8k
6
votes
1
answer
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A property of nearby cycles functor
Let $f\colon X\to Y$ be a flat morphism of irreducible projective algebraic varieties over $\mathbb{C}$ (or any other algebraically closed field of characteristic 0). Assume that $Y$ is smooth, and th …
- 21.8k
6
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A topological property of flat morphisms
Let $f\colon X\to Y$ be a faithfully flat morphism of smooth projective varieties over $\mathbb{C}$. Assume that the generic fiber of $f$ is smooth. Then there exists a non-empty Zariski open subset $ …
- 21.8k
8
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430
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Different notions of convergence of complex subvarieties
Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges …
- 21.8k
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Top integer homology of compact analytic variety
Let $V$ be a compact connected complex analytic subvariety (possibly singular) of a complex smooth manifold. Let $n$ denote its complex dimension.
Is it true that $H_{2n}(V,\mathbb{Z})\simeq \mathbb{Z …
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19
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Classification of complex structures on $\mathbb{R}^{2n}$
Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a reaso …
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