On the Betti Numbers of Irreducible Compact Hyperkähler Manifolds of Complex Dimension Four

Guan, Daniel

doi:10.4310/mrl.2001.v8.n5.a8

Cited by 56 publications

(72 citation statements)

References 21 publications

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“…[2]). When dim C K i = 4, one has b 2 (K i ) 23 [21]. Therefore, any absolutely trianalytic subvariety in a 10-dimensional O'Grady manifold (if it exists) satisfies dim C Z 6, and has maximal holonomy.…”

Section: Remark 121 a Maximal Holonomy Hyperkähler Manifoldmentioning

confidence: 93%

k-symplectic structures and absolutely trianalytic subvarieties in hyperkähler manifolds

Soldatenkov

Verbitsky

2015

Journal of Geometry and Physics

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a b s t r a c t Let (M, I, J, K ) be a hyperkähler manifold, and Z ⊂ (M, I) a complex subvariety in (M, I).We say that Z is trianalytic if it is complex analytic with respect to J and K , and absolutely trianalytic if it is trianalytic with respect to any hyperkähler triple of complex structures (M, I, J ′ , K ′ ) containing I. For a generic complex structure I on M, all complex subvarieties of (M, I) are absolutely trianalytic. It is known that the normalization Z ′ of a trianalytic subvariety is smooth; we prove thatTo study absolutely trianalytic subvarieties further, we define a new geometric structure, called k-symplectic structure; this structure is a generalization of hypersymplectic structure. A k-symplectic structure on a 2d-dimensional manifold X is a k-dimensional space R of closed 2-forms on X which all have rank 2d or d. It is called non-degenerate if the set of all degenerate forms in R is a smooth, non-degenerate quadric hypersurface in R.We consider absolutely trianalytic tori in a hyperkähler manifold M of maximal holonomy. We prove that any such torus is equipped with a non-degenerate k-symplectic structure, where k = b 2 (M). We show that the tangent bundle TX of a k-symplectic manifold is a Clifford module over a Clifford algebra Cl(k − 1). Then an absolutely trianalytic torus in a hyperkähler manifold M with b 2 (M) 2r + 1 is at least 2 r−1 -dimensional.

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Section: Remark 121 a Maximal Holonomy Hyperkähler Manifoldmentioning

confidence: 93%

k-symplectic structures and absolutely trianalytic subvarieties in hyperkähler manifolds

Soldatenkov

Verbitsky

2015

Journal of Geometry and Physics

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“…By [6] this happens if and only if b 2 (X) = 23. This is the case for one of the two families of symplectic fourfolds known so far, namely the family of Hilbert schemes S [2] of a K3 surface S (and their deformations).…”

Section: Symplectic Four-foldsmentioning

confidence: 99%

Antisymplectic involutions of holomorphic symplectic manifolds

Beauville

2011

Journal of Topology

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“…We will use Guan's bounds [8] on the Betti numbers of X to restrict the possible values of d and deg∆. We can now state our final result.…”

Section: Fibrations On Four-foldsmentioning

confidence: 99%

“…In [8] Guan proved that the characteristic numbers of a holomorphic symplectic four-fold are bounded; so when X is a four-fold our formulae give bounds on the degree of ∆. We briefly indicate why such bounds might be useful.…”

Section: Introductionmentioning

confidence: 96%