Alberto Saracco, Giuseppe Tomassini
We generalize some of the results in [arXiv: math.CV/0503430], and prove a bump-lemma for closed sets in semi 1-coronae. From this we obtain some finite cohomology results and an extension theorem for analytic subsets in 1-coronae.
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We prove some extension theorems for analytic objects, in particular sections of a coherent sheaf, defined in semi q-coronae of a complex space. Semi q-coronae are domains whose boundary is the union of a Levi flat part, a q-pseudoconvex part and a q-pseudoconcave part. Such results are obtained mainly using cohomological techniques.
Let X be a (connected and reduced) complex space. A q-collar of X is a bounded domain whose boundary is a union of a strongly q-pseudoconvex, a strongly q-pseudoncave and two flat (i.e. locally zero sets of pluriharmonic functions) hypersurfaces. Finiteness and vanishing cohomology theorems obtained in math/0503490 and math/0701549 for semi q-coronae are generalized in this context and lead to results on extension problem and removable sets for sections of coherent sheaves and analytic subsets.
Among other things, we show that the ideal sheaf of a complex Hilbert submanifold of a pseudoconvex open subset of Hilbert space is acyclic over the ambient pseudoconvex open set. We also prove a vanishing theorem for a fairly general class of analytic sheaves over pseudoconvex open sets of a large class of Banach spaces. Some applications are also given.
Given a cohesive sheaf $\Cal S$ over a complex Banach manifold $M$, we endow the cohomology groups $H^q(M,\Cal S)$ of $M$ and $H^q(\frak U,\Cal S)$ of open covers $\frak U$ of $M$ with a locally convex topology. Under certain assumptions we prove that the canonical map $H^q(\frak U,\Cal S)\to H^q(M,\Cal S)$ is an isomorphism of topological vector spaces.
We introduce a cohomology, called extendable cohomology, for abstract complex singular varieties based on suitable differential forms. Beside a study of the general properties of such a cohomology, we show that, given a complex vector bundle, one can compute its topological Chern classes using the extendable Chern classes, defined via a Chern-Weil type theory. We also prove that the localizations of the extendable Chern classes represent the localizations of the respective topological Chern classes, thus obtaining an abstract residue theorem for compact singular complex analytic varieties. As …
Let X be a pure n-dimensional (n>1) complex analytic set in C^N with an isolated singularity at 0. In this paper we express the L2-(0,q)-d-bar-cohomology groups for all q with 0<q<n+1, of a sufficiently small deleted neighborhood of the singular point, in terms of resolution data. We also obtain identifications of the L2-(0,q)-d-bar-cohomology groups of the smooth points of X in terms of resolution data, when X is either compact or an open relatively compact complex analytic subset of a reduced complex space with finitely many isolated singularities.
Consider a holomorphic vector bundle $L\to X$ and an open cover ${\frak U}=\{U_a\colon a\in A\}$ of $X$, parametrized by a complex manifold $A$. We prove that the sheaf cohomology groups $H^q(X,L)$ can be computed from the complex $C^{\bullet}_{\text{hol}}$ $({\frak U},L)$ of cochains $(f_{a_0\ldots a_q})_{a_0,\ldots, a_q\in A}$ that depend holomorphically on the $a_j$, provided $S=\{(a,x)\in A\times X\colon x\in U_a\}$ is a Stein open subset of $A\times X$. The result is proved in the setting of Banach manifolds, and is applied to study representations on cohomology groups induced by a holomo…
Big and nef cohomology classes on compact Kähler spaces
本文证明了紧致弱Kähler空间上Demailly-Paun定理的版本，并证明紧致正规Kähler空间上big和nef Bott-Chern类的受限非Kähler轨迹等于通常的非Kähler轨迹。
Étale cohomology of Stein algebras
本文证明了有限维斯坦空间的有限系数奇异上同调与相应斯坦代数的层上同调同构，并探讨了上同调类的来源及消失性质。
We prove that the cohomology jump loci of rank one local systems on the complement in a small ball of a germ of a complex analytic set are finite unions of torsion translates of subtori. This is a generalization of the classical Monodromy Theorem stating that the eigenvalues of the monodromy on the cohomology of the Milnor fiber of a germ of a holomorphic function are roots of unity.
Étale cohomology of algebraic varieties over Stein compacta
本文证明Stein紧集上代数簇平展上同调与其解析化奇异上同调的比较定理，推导相关亚纯函数域上同调维数结论，并给出实解析空间紧子集上希尔伯特第17问题的定量版本。
In this paper, we investigate the Whitney--de Rham complex $Ω^\bullet_\text{W} (X)$ associated to a semi-analytic subset $X$ of an analytic manifold $M$. This complex is a commutative differential graded algebra, that is defined to be the quotient of the de Rham complex of smooth differential forms on $M$ by the differential graded ideal generated by all smooth functions which are flat on $X$. We use Hironaka's desingularization theorem to prove a Poincaré Lemma for $Ω^\bullet_\text{W} (X)$ holds true, which entails that its cohomology is isomorphic to the real cohomology of $X$. Furthermore, …
In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura, and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general $L^2$ extension theorem obtained by Demailly.
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