Construction of automorphisms of hyperkähler manifolds

Abstract

Let M be an irreducible holomorphic symplectic (hyperkähler) manifold. If b 2 (M ) 5, we construct a deformation M ′ of M which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real (1, 1)-classes is hyperbolic. If b 2 (M ) 14, similarly, we construct a deformation which admits a parabolic automorphism.

Canonical Heights on Hyper-Kähler Varieties and the Kawaguchi–Silverman Conjecture

“…Remark 3.26. If λ 1 (f | V ) = 1, then λ dim V −1 (f | V ) = 1 by log concavity of dynamical degrees, so λ 1 (f −1 | V ) = 1 by Theorem 2.2 (1). As a result, B + (ν) contains all subvarieties of X for which the dynamical degree λ 1 (f | V ) drops to 1.…”

“…(2) If f : X X admits an invariant fibration π : X → Y as above, then λ 0 (f | π ) = 1 and λ 1 (f ) = max{λ 1…”

“…Applying condition (3) of [3, Proposition 3.1] to f • π yields a map f making the diagram Proof of Theorem 1.7. We first handle case (1). By [3, Lemma 7.1] we know that {D ∈ Nef(X) | c 2 (X) · D ≤ M} is compact for all M ≥ 0.…”

“…Proof of Theorem 1.9 (1). Let X be a threefold, f an automorphism of X, and π : X → S a Mori fiber space structure.…”

On the stable dynamical spectrum of complex surfaces

“…In Proposition 4.3, we give a sufficient condition for a chamber to be preserved. We expect that it will be useful to study the stable dynamical spectrum of supersingular K3 surfaces (as in [7]) and IHSM manifolds (as in [1]) as well.…”