Automorphisms of projective K3 surfaces with minimum entropy

Consider an arbitrary automorphism of an Enriques surface with its lift to the covering K3 surface. We prove a bound of the order of the lift acting on the anti‐invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially could be the minimal dynamical degree on Enriques surfaces and for which the existence of geometric automorphisms is unknown.

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“…If such two eigenvalues come up, they are real and reciprocal. Thus either

λ (f) = 1

or it is the largest real eigenvalue of the map

f^{*}

(for a precise discussion of the above notion and its properties see [, §.2.2.2], , and references therein).…”

Section: Dynamical Degreesmentioning

confidence: 99%

On automorphisms of Enriques surfaces and their entropy

Matsumoto

Ohashi

Rams

2018

Mathematische Nachrichten

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“…Instead of considering only the Salem degree of an automorphism, in this work we focus on the existence of automorphisms of (supersingular) K3 surfaces with a given entropy, and more precisely, logarithms of the minimal Salem numbers

λ_{d}

. For complex projective K3 surfaces, it is proved in that

λ_{d}

is the spectral radius of an automorphism for

d = 2, 4, 6, 8, 10

or 18, but not if

d = 14, 16

d ⩾ 20

, while the case

d = 12

is left open. As a byproduct of our work, we are able to realize also

λ_{12}

in the complex case (see Appendix), hence proving the following.…”

Section: Introductionmentioning

confidence: 99%

“…The values of

λ_{d}

for

d = 2, \dots, 20

can be found in [, Appendix A]. We have

λ_{22} \approx 1.235664580

with minimal polynomial

s_{22} = x^{22} - x^{20} - x^{19} + x^{15} + x^{14} - x^{12} - x^{11} - x^{10} + x^{8} + x^{7} - x^{3} - x^{2} + 1 .

…”

Section: Introductionmentioning

confidence: 99%

“…The crucial step is to check whether

f | N S (X) \otimes R

preserves a chamber representing the ample cone cut out by the nodal roots. In general it is hard to compute the (infinitely many) nodal roots, hence in an integer linear programming test is developed, which gives a sufficient but not necessary condition. To resolve this uncertainty we develop a (convex) quadratic integer program refining the linear one.…”

Section: Introductionmentioning

confidence: 99%

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Automorphisms of minimal entropy on supersingular K3 surfaces

Brandhorst

González-Alonso

2018

J. London Math. Soc.

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In this article we give a strategy to decide whether the logarithm of a given Salem number is realized as entropy of an automorphism of a supersingular K3 surface in positive characteristic. As test case it is proved that logλd, where λd is the minimal Salem number of degree d, is realized in characteristic 5 if and only if d⩽22 is even and d≠18. In the complex projective setting we settle the case of entropy logλ12, left open by McMullen, by giving the construction. A necessary and sufficient test is developed to decide whether a given isometry of a hyperbolic lattice, with spectral radius bigger than one, is positive, that is, preserves a chamber of the positive cone.

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“…While there are many examples of compact complex surfaces having automorphisms of positive entropies (works of Cantat [10], Bedford-Kim [5][6] [7], McMullen [27][28] [29][30], Oguiso [32][33], Cantat-Dolgachev [11], Zhang [46], Diller [17], Déserti-Grivaux [16], Reschke [38],...), there are few interesting examples of manifolds of higher dimensions having automorphisms of positive entropies (Oguiso [34][35], Oguiso-Perroni [31],...). In particular, for the class of smooth rational threefolds, there are currently only two known examples of manifolds with primitive automorphisms of positive entropy (see [36,13,12]).…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of blowups of threefolds being Fano or having Picard number 1

Truong¹

2016

Ergod. Th. Dynam. Sys.

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Abstract. Let X 0 be a smooth projective threefold which is Fano or which has Picard number 1. Let π : X → X 0 be a finite composition of blowups along smooth centers. We show that for "almost all" of such X, if f ∈ Aut(X) then its first and second dynamical degrees are the same. We also construct many examples of finite blowups X → X 0 , on which any automorphism is of zero entropy.The main idea is that because of the log-concavity of dynamical systems and the invariance of Chern classes under holomorphic automorphisms, there are some constraints on the nef cohomology classes.We will also discuss a possible application of these results to a threefold constructed by Kenji Ueno.

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Automorphisms of projective K3 surfaces with minimum entropy

Cited by 33 publications

References 30 publications

On automorphisms of Enriques surfaces and their entropy

On automorphisms of Enriques surfaces and their entropy

Automorphisms of minimal entropy on supersingular K3 surfaces

Automorphisms of blowups of threefolds being Fano or having Picard number 1

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