The Kaehler cone of a compact hyperkaehler manifold

Daniel Huybrechts

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The Kaehler cone of a compact hyperkaehler manifold

Daniel Huybrechts

Algebraic Geometry · Submitted 19 Sep 1999

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This is an attempt towards the understanding of the (birational) Kaehler cone of a compact hyperkaehler manifold in terms of the Beauville-Bogomolov form on its second cohomology.

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This is an attempt towards the understanding of the (birational) Kaehler cone of a compact hyperkaehler manifold in terms of the Beauville-Bogomolov form on its second cohomology. We discuss birational correspondences between hyperkaehler manifolds and their effects on the cohomology. Many of the results are conjectural in as much as they depend on a projectivity criterion for compact hyperkaehler manifolds contained in this paper's predecessor, but in which a serious mistake has oocured. An erratum is given in Sect. 6 and a way to rescue the approach is proposed in Sect. 7.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	This is an attempt towards the understanding of the (birational) Kaehler cone of a compact hyperkaehler manifold in terms of the Beauville-Bogomolov form on its second cohomology.	contribution 0.83	0.56	0.20	asserted 0.50
s1	We discuss birational correspondences between hyperkaehler manifolds and their effects on the cohomology.	contribution 0.93	0.78	0.06	asserted 0.91
s2	Many of the results are conjectural in as much as they depend on a projectivity criterion for compact hyperkaehler manifolds contained in this paper's predecessor, but in which a serious mistake has oocured.	limitation 1.00	0.38	0.06	speculative 0.68
s3	An erratum is given in Sect. 6 and a way to rescue the approach is proposed in Sect. 7.	contribution 0.79	0.86	0.05	asserted 0.98

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Original abstract

This is an attempt towards the understanding of the (birational) Kaehler cone of a compact hyperkaehler manifold in terms of the Beauville-Bogomolov form on its second cohomology. We discuss birational correspondences between hyperkaehler manifolds and their effects on the cohomology. Many of the results are conjectural in as much as they depend on a projectivity criterion for compact hyperkaehler manifolds contained in this paper's predecessor, but in which a serious mistake has oocured. An erratum is given in Sect. 6 and a way to rescue the approach is proposed in Sect. 7.

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14 arXiv papers cite this in 22 sentences

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Papers citing it · 15 sentences without a confident label

Cites it as support 3 papers

Proof: As follows from [ref] and [ref] (see [ref] for a precise statement), $(M,I)$ is a subset of positive cone given by a set of linear inequalities [equation] where $l_i$ is a countable set of rational $(1,1)$-classes.

Ergodic complex structures on hyperkahler manifolds · Density of non-Hausdorff orbits · 0.97
By [ref] , these two symplectic resolutions are deformation equivalent.

Birational Geometry of Singular Moduli Spaces of O'Grady Type · Main proof and consequence · 0.98
For any point $J ∈_η$ with the Picard group of $(M,J)$ has rank 1, the bundle $L$ is ample ( [ref] ).

Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratum · The monodromy group and the marked moduli space · 0.98

Cites it as background 3 papers

( [ref] , [ref] ).

Parabolic nef currents on hyperkaehler manifolds · Introduction · 0.89
( [ref] , see also [ref] ) Let $(M,ω)$ be a compact Kähler manifold, and ${(M_α, ϕ_α)}$ the set of all compact manifolds equipped with a birational morphism $ϕ_α: M_αM$.

Hyperkahler SYZ conjecture and semipositive line bundles · The birational Kähler cone · 0.95
In particular, there are many results about their birational geometry in the literature; see [ref] for instance.

Birational Geometry of Singular Moduli Spaces of O'Grady Type · Background · 1.00

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