Collections of Orbits of Hyperplane Type in Homogeneous Spaces, Homogeneous Dynamics, and Hyperkähler Geometry

Amerik, Ekaterina; Verbitsky, Misha

doi:10.1093/imrn/rnx319

Cited by 9 publications

(8 citation statements)

References 17 publications

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“…Since the connected component of the identity of the monodromy group is a subgroup of the generic Mumford-Tate group of the family and the latter commutes with the defect group, the local system P(X/S) becomes constant after a finite base change S → S. Remark 4. 4 Conjecturally, the group P(X ) is trivial for any hyper-Kähler variety X with b 2 > 3. In fact, the triviality of the defect group is equivalent to the conjecture which says that MT(H * X ) = G mot (H * X ) (i.e.…”

Section: Is a Smooth And Projective Morphism To A Non-singular Connected Variety Smentioning

confidence: 99%

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Galois representations on the cohomology of hyper-Kähler varieties

Floccari

2022

Math. Z.

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We show that the André motive of a hyper-Kähler variety X over a field $$K \subset {\mathbb {C}}$$ K ⊂ C with $$b_2(X)>6$$ b 2 ( X ) > 6 is governed by its component in degree 2. More precisely, we prove that if $$X_1$$ X 1 and $$X_2$$ X 2 are deformation equivalent hyper-Kähler varieties with $$b_2(X_i)>6$$ b 2 ( X i ) > 6 and if there exists a Hodge isometry $$f:H^2(X_1,{\mathbb {Q}})\rightarrow H^2(X_2,{\mathbb {Q}})$$ f : H 2 ( X 1 , Q ) → H 2 ( X 2 , Q ) , then the André motives of $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic after a finite extension of K, up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the étale cohomology of $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-Kähler varieties for which the Mumford–Tate conjecture is true.

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Section: Is a Smooth And Projective Morphism To A Non-singular Connected Variety Smentioning

confidence: 99%

“…In particular, if MBM(X ) = ∅ then Amp(X ) = NS + (X ). (ii) [4] Fix a deformation class of hyper-Kähler manifolds with b 2 ≥ 5. There exists a positive integer N , depending only on the deformation class, such that for any projective X of the given deformation type, every MBM class z on X satisfies −N < (z, z) < 0.…”

Section: Remark 52mentioning

confidence: 99%

Galois representations on the cohomology of hyper-Kähler varieties

Floccari

2022

Math. Z.

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“…The hyperkähler version of the so-called Morrison-Kawamata cone conjecture is proved recently by Amerik-Verbitsky ([AV17], [AV18]) based on their earlier work [AV15], using hyperbolic geometry and ergodic theory. Their result says that Aut(X) acts with finitely many orbits on the set of facets of the Kähler cone of X (see [AV16, Theorem 2.13 and the discussion after]).…”

Section: Cone Conjecturesmentioning

confidence: 99%

Finiteness of Klein actions and real structures on compact hyperkähler manifolds

Cattaneo

2019

Math. Ann.

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One central problem in real algebraic geometry is to classify the real structures of a given complex manifold. We address this problem for compact hyperkähler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence. We actually prove more generally that there are only finitely many, up to conjugacy, faithful finite group actions by holomorphic or anti-holomorphic automorphisms (the socalled Klein actions). In other words, the automorphism group and the Klein automorphism group of a compact hyperkähler manifold contain only finitely many conjugacy classes of finite subgroups. We furthermore answer a question of Oguiso by showing that the automorphism group of a compact hyperkähler manifold is finitely presented.

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“…This approach is generalized in the present paper. In [AV3] it is shown that for each hyperkähler manifold M there exists N > 0, depending only on the deformation class of M , such that for all MBM classes v one has −N < q(v, v) < 0. In the present paper, we prove that the lattice H 2 (M, Z) of a hyperkähler manifold M satisfying b 2 (M ) 5 (this is believed to hold always, but no proof exists today) contains a primitive sublattice Λ ⊂ H 2 (M, Z) which does not represent numbers smaller than N (that is, for any nonzero v ∈ Λ, one has |q(v, v)| N ).…”

mentioning

confidence: 99%

“…To construct the maximal holonomy hyperkähler (that is, ISH) manifolds with no MBM classes, we recall from [AV4] that the MBM classes have bounded square: 0 > q(z) > −N . Therefore to exhibit a lattice with a parabolic automorphism it is sufficient to find one representing zero and not representing numbers between zero and −N .…”

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confidence: 99%