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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.
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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 …
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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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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 …
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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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Obstruction map for holomorphic line bundle $\operatorname{Ob}_L:H^1(X,\mathcal O)\to H^2(X,...
$\DeclareMathOperator\Ob{Ob}$In Chan & Suen's paper A differential-geometric approach to deformation of pairs $(X, E)$ p.20, the authors define a Kuranishi map (or obstruction map) for the holomorphic …
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When the whole space $H^2(X,\mathbb Q)$ can be represented by $c(L_t)$ of $X_t$?
Let $X$ be a compact complex manifold, $\pi:\mathcal X\to B$ be a holomorphic family of $X$ with $X_t=\pi^{-1}(t),t\in B$, and $X_0=X$.
Let $L_t$ be a holomorphic line bundle over $X_t$, then its Cher …
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When Atiyah class and Chern class coincide?
Let $X$ be a compact complex manifold, $L$ be a holomorphic line bundle on $X$, then the exponential exact sequence $0\to \mathbb Z\hookrightarrow \mathcal O\to \mathcal O^*\to 0$ induces the map $c:H …
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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 …
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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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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 …
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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 …
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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} …
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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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Any manifold in Fujiki class $\mathcal C$ admits a Kähler deformation?
It is known that small deformations of a manifold $X$ in Fujiki class $\mathcal C$ might no longer be in $\mathcal C$, see This paper for example.
However, the following question is still open:
For a …
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Is Kähler current class representable by semipositive forms?
A compact complex manifold is called in Fujiki class $\mathcal C$ if it is bimeromorphic to a compact Kähler manifold, or equivalently, if there exists a proper holomorphic bimeromorphic map (i.e. a h …
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