1 -convex manifolds are p- kähler

Alessandrini, Lucia; Bassanelli, Giovanni; Leoni, Marco

doi:10.1007/bf02941676

Cited by 3 publications

(19 citation statements)

References 16 publications

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“…Case (1) is very similar, it is enough to replace σ 2p−1 by σ 2q . This result was proved by Theorem 3.2(i) in [8].…”

Section: Duality On Non Compact Manifoldsmentioning

confidence: 73%

Forms and currents defining generalized p-Kähler structures

Alessandrini

2018

Abh. Math. Semin. Univ. Hambg.

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This paper is devoted, first of all, to give a complete unified proof of the Characterization Theorem for compact generalized p−Kähler manifolds (Theorem 3.2). The proof is based on the classical duality between "closed"positive forms and "exact"positive currents. In the last part of the paper we approach the general case of non compact complex manifolds, where "exact"positive forms seem to play a more significant role than "closed"forms. In this setting, we state the appropriate characterization theorems and give some interesting applications.

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“…Case (1) is very similar, it is enough to replace σ 2p−1 by σ 2q . This result was proved by Theorem 3.2(i) in [8].…”

Section: Duality On Non Compact Manifoldsmentioning

confidence: 73%

Forms and currents defining generalized p-Kähler structures

Alessandrini

2018

Abh. Math. Semin. Univ. Hambg.

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“…In case K, this is Corollary 4.9 in [7]; we consider here a more general context, namely, that of "p−Kähler"manifolds with p > dimY , non necessarily compact. In [9] we studied the case of 1-convex manifolds, where the cohomology groups H q (U, F ) are finite dimensional when q > 0. We proved there the following result: Adapting its proof, which is based on an accurate analysis of exact sequences of sheaves and cohomology groups, we get in our situation (where the cohomology groups vanish):…”

Section: Link Between "P−kähler"forms On M Andmmentioning

confidence: 99%

“…To give a hint of the first step (s = 0, p = 1) of the proof of this Claim, let us consider H, the sheaf of germs of real pluriharmonic functions, with the following well-known exact sequences of sheaves (see [9], p. 260):…”

Section: Link Between "P−kähler"forms On M Andmmentioning

confidence: 99%

“…In the non-compact case, we cannot use the classical characterization of Kähler manifolds by currents, which has been introduced by Sullivan [29] and by Harvey and Lawson [22]; hence we have no results about a generic modification. Nevertheless, in [9] we studied 1-convex manifolds (which are not compact but have a specific compact "soul") by positive currents. This tecnique can be used to get a partial result on proper modifications.…”

Section: Currents In the Non-compact Casementioning

confidence: 99%

“…Let us recall the notation and some results proved in [9], which we shall use in the proof. For every manifold X, P p c (X) denotes the closed convex cone of positive currents of bidimension (p, p) and compact support, while B p c (X) is the space of currents of bidimension (p, p) and compact support, which are (p, p)−components of a compactly supported boundary current, that is, the class of the current vanishes in…”

Section: Currents In the Non-compact Casementioning

confidence: 99%

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Proper Modifications of Generalized p-Kähler Manifolds

Alessandrini

2016

J Geom Anal

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In this paper, we consider a proper modification f :M → M between complex manifolds, and study when a generalized p−Kähler property goes back from M toM . When f is the blow-up at a point, every generalized p−Kähler property is conserved, while when f is the blow-up along a submanifold, the same is true for p = 1. For p = n−1, we prove that the class of compact generalized balanced manifolds is closed with respect to modifications, and we show that the fundamental forms can be chosen in the expected cohomology class. We get some partial results also in the non-compact case; finally, we end the paper with some examples of generalized p−Kähler manifolds.2010 Mathematics Subject Classification. Primary 53C55; Secondary 32J27, 32L05.

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Compact complex manifolds whose complement of an analytic subset is Kahler

Shimobe

2020

Preprint

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In this paper, we consider a class of balanced manifolds and provide a proof of a problem proposed by Silva -that compact complex manifolds that are Kähler outside an analytic subset are balanced. Next, as a special case of this theorem, we prove that compact complex manifolds which are Kähler outside a Kähler submanifold are class C. Finally, we prove that a blow up along a submanifold of Hironaka's examples are Kähler, and we establish that Hironaka's examples are examples of this theorem.

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1 -convex manifolds are p- kähler

Cited by 3 publications

References 16 publications

Forms and currents defining generalized p-Kähler structures

Forms and currents defining generalized p-Kähler structures

Proper Modifications of Generalized p-Kähler Manifolds

Compact complex manifolds whose complement of an analytic subset is Kahler

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