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Results tagged with complex-geometry
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user 99826
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.
2
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1
answer
134
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The kernel of $H^{\bullet,\bullet}_{\bar\partial}(X)\to H_A^{\bullet,\bullet}(X)$
In Angella and Tomassini's paper p.75, there is an exact sequence:
$\cdots\to B^{\bullet,\bullet}\to H_{\bar\partial}^{\bullet,\bullet}\to H_A^{\bullet,\bullet}\to \cdots$
where $B^{\bullet,\bullet} …
- 471
1
vote
0
answers
87
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Is any holomorphic $p$-form on a Fujiki class $\mathcal C$ manifold closed?
In Fujiki's 78 paper, p.227, Corollary 1.7. (1), there is
Let $X$ be a compact complex manifold in Fujiki class $\mathcal C$, then every holomorphic $p$-form on $X$ is closed.
In Kodaira & Morrow's …
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1
vote
1
answer
394
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Motivation behind spectral sequences
It is well known that spectral sequence is very important in algebraic geometry and complex geometry, but its definition seems very unnatural. For example, in Voisin's book Hodge theory and complex al …
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3
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1
answer
117
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Manifold $X$ in Fujiki class $\mathcal C$ with $c_1(X)=0$ admits arbitrarily small deformati...
It is known that any Calabi-Yau manifold $X$, i.e. compact Kähler manifold with $c_1(X)=0$, has arbitrarily small deformations which are algebraic (see, for example, Buchdahl's paper, Proposition 5).
…
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1
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0
answers
42
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Characterize manifolds in Fujiki class $\mathcal C$ by smooth forms
Let $X$ be a compact complex manifold, we say $X$ is in Fujiki class $\mathcal C$ if it is bimeromorphic to a compact Kähler manifold, or equivalently, if there exists a proper holomorphic bimeromorph …
- 471
5
votes
1
answer
508
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Deformation invariance of Chern classes
Let $\pi:\mathcal X\to B$ be a deformation of a compact complex manifold $X=\pi^{-1}(0)$, then for any $t\in B$, the first Chern class $c_1(X)=c_1(X_t)$?
I know the Chern class of a manifold depends o …
- 471
1
vote
0
answers
341
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Lefschetz theorem on (1,1) classes for a compact complex surface
In Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius' book Compact complex surfaces. Second edition p.142, there is a Lefschetz theorem on (1,1) classes for compact surface:
The …
- 471
2
votes
0
answers
189
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Hodge bundle for $\partial\bar\partial$-manifolds
Let $\pi:\mathcal X\to B$ be a holomorphic family of $\partial\bar\partial$-manifolds (compact complex manifolds satisfy $\partial\bar\partial$-lemma, e.g. Kähler manifolds, Fujiki class $\mathcal C$ …
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1
vote
0
answers
93
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Interpretation of $\mathcal H^k$
For a holomorphic family $\pi:\mathcal X\to B$ between complex manifolds, the map $\pi$ is a proper holomorphic submersion, $X_t:=\pi^{-1}(t)$, $t\in B$, $X=X_0$, we have the isomorphism $H^k(X,\mathb …
- 471
1
vote
0
answers
42
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Is the union of Fujiki cones open in $\mathcal H^{1,1}_{\mathbb R}$?
Let $\mathcal X\to B$ be a holomorphic family of compact Kähler manifolds, let $\mathcal K_t$ denote the Kähler cone of the fiber $X_t$, then the union $\cup_{t\in B}\mathcal K_t$ forms an open set in …
- 471
3
votes
2
answers
348
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Surjectivity of $H^2(X,\mathbb C)\to H^2(X,\mathcal O)$
Let $X$ be a complex manifold, there is a natural map $f:H^2(X,\mathbb C)\to H^2(X,\mathcal O)$ induced by the inclusion map $\mathbb C\hookrightarrow \mathcal O$ which coincides with the natural proj …
- 471
7
votes
1
answer
306
views
When $H^{p,q}_{\bar\partial}(X)$ can be seen as a subspace of $H^k(X,\mathbb C)$?
It is known that for a $\partial\bar\partial$-manifold $X$ (a compact complex manifold satisfies the $\partial\bar\partial$-lemma), the Bott-Chern cohomology $H_{BC}^{\bullet,\bullet}:=\frac{\ker \par …
- 471
5
votes
2
answers
380
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Does the Kähler form $\omega$ satisfy $d^*\omega=0$?
Let $X$ be a compact Kähler manifold with Kähler form $\omega$, then from Kodaira & Spencer's paper on deformations III, p.75, the authors state that the Kähler form satisfies the Laplace equation $\D …
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2
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1
answer
413
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Hermitian holomorphic line bundle and curvature Chern form in Demailly's book
In Demailly's book p.272, Theorem 13.9, there is:
Let $X$ be an arbitrary complex manifold.
(b) Let $\omega$ be a $\mathcal C^∞$ closed real (1, 1)-form such that ${ω}\in H^2_{dR}(X,\mathbb R)$ is th …
- 471
23
votes
2
answers
2k
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Is the complex structure of $\mathbb CP^n$ unique?
Let $\mathbb CP^n$ denotes the complex projective space of dimension $n$, we have a standard complex structure of $\mathbb CP^n$, and my question is: is this complex structure unique?
Or equivalently, …
- 471