Dynamical degrees of automorphisms of non-projective hyperkähler manifolds
We determine the dynamical degrees of bimeromorphic automorphisms of hyperkähler manifolds of all known deformation types with irreducible characteristic polynomial.
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arXiv subjects
Matthias Schütt
Publications and source records attributed to Matthias Schütt.
Non-projective K3 surfaces with real or Salem multiplication
We determine the Hodge endomorphism algebras of non-projective complex K3 surfaces (and more generally, hyperkähler manifolds). We show that they are either totally real fields or number fields generated by Salem numbers. This is unlike the projective case, where the endomorphism fields are either totally real or CM. We also develop precise existence criteria and explore the relations to number theory and dynamics.
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K3 surfaces with real or complex multiplication
Let $E$ be a totally real number field of degree $d$ and let $m \geqslant 3$ be an integer. We show that if $md \leqslant 21$ then there exists an $(m-2)$-dimensional family of complex projective $K3$ surfaces with real multiplication by $E$. Analogous results are proved for CM number fields and also for all known higher-dimensional hyperkähler manifolds.
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Finite symplectic automorphism groups of supersingular K3 surfaces
We give a complete classification of finite groups acting symplectically on supersingular K3 surfaces of Artin invariant one. Using work of Dolgachev and Keum, this provides the full classification of tame finite symplectic automorphism groups on any K3 surface, and in particular of all finite symplectic automorphism groups on K3 surfaces in characteristic p>11.
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The 3-divisibility of divisors on K3 surfaces in characteristic 3
We describe the possible 3-divisible $A_2^n$ configurations of smooth rational curves on K3 surfaces in characteristic 3 and fully classify the resulting triple covers.
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On the numerically and cohomologically trivial automorphisms of elliptic surfaces II: $χ(S)>0$
In this second part we study first the group $Aut_{\mathbb Q}(S)$ of numerically trivial automorphisms of an algebraic properly elliptic surface $S$, that is, of a minimal algebraic surface with Kodaira dimension $κ(S)=1$, in the case $χ(S) \geq 1$. Our first surprising result is that, against what has been believed for over 40 years, there exist nontrivial such groups for $p_g(S) >0$. Indeed, we show even that $Aut_{\mathbb Q}(S)$ is always a 2-generated finite abelian group, but there is no absolute upper bound for its cardinality. At any rate, we give explicit and essentially optimal upper bounds for $|Aut_{\mathbb Q}(S)|$ in terms of the numerical invariants of $S$, as $χ(S)$, or the irregularity $q(S)$, or the bigenus $P_2(S)$. Moreover, we reach an almost complete description of the possible groups $Aut_{\mathbb Q}(S)$ and we give effective criteria for such surfaces to have trivial $Aut_{\mathbb Q}(S)$. Our second surprising results concern the quite elusive group $Aut_{\mathbb Z}(S)$ of cohomologically trivial automorphisms; we are able to give the explicit upper bounds for $|Aut_{\mathbb Z}(S)|$ in special cases: 9 when $p_g(S) =0$, and we achieve the sharp upper bound 3 when $S$ (i.e., the pluricanonical elliptic fibration) is isotrivial. Also in the non isotrivial case we produce subtle examples where $Aut_{\mathbb Z}(S)$ is a group of order 2 or 3.
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On the moduli space of stable surfaces with $p_g=1$ realizing the minimal volume
Let $M_1$ be the moduli space of the KSBA stable surfaces $X$ of geometric genus $p_g(X)=1$ realizing the minimal possible volume $K_X^2=\frac1{143}$. We show that its reduced part $M_{1,\rm red}$ is a $10$-dimensional projective variety isomorphic to the Baily--Borel compactification $\overline{F}_Λ^{\rm BB}$ of the moduli space of $Λ$-polarized K3 surfaces, where $Λ=II_{1,9}\simeq U\oplus E_8$ is a unimodular lattice of signature $(1,9)$. By a result of Brieskorn, $\overline{F}_Λ^{\rm BB}$ is a weighted projective space. We also verify the Viehweg hyperbolicity of the base of a Whitney equisingular family of stable surfaces in $M_1$. More generally, we prove that the same results hold for the moduli space $M_c$ of KSBA stable pairs $(X,B)$ with coefficients of $B$ belonging to a set $\mathcal C\subset [0,1]$ such that $\mathcal C\cup\{1\}$ attains a minimum, say $c$, and with $p_g(X)=1$, realizing the minimal possible volume $(K_X+B)^2=v(c)$. Indeed, we show that $M_{c,\rm red}$ is independent of $c$ and that for $c\le\frac7{13}$ $M_c$ is isomorphic to $\overline{F}_Λ^{\rm BB}$.
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Miyaoka's bound for conics on K3 surfaces and beyond
We show that Miyaoka's bound for the number of conics on a degree-2h K3 surface is attained for high h, and analogously for higher even degree (smooth) rational curves.
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On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$
In this first part we describe the group $Aut_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface $S$ with Kodaira dimension $κ(S)=1$), in the initial case $ χ(\mathcal{O}_S) =0$. In particular, in the case where $Aut_{\mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $Aut_{\mathbb{Z}}(S)$ is nontrivial, it is one of the following groups: $\mathbb{Z}/2, \mathbb{Z}/3, (\mathbb{Z}/2)^2$. We also show with easy examples that the groups $\mathbb{Z}/2, \mathbb{Z}/3$ do effectively occur. Respectively, in the case where $Aut_{\mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components.
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The 2-divisibility of divisors on K3 surfaces in characteristic 2
We show that K3 surfaces in characteristic 2 can admit sets of $n$ disjoint smooth rational curves whose sum is divisible by 2 in the Picard group, for each $n=8,12,16,20$. More precisely, all values occur on supersingular K3 surfaces, with exceptions only at Artin invariants 1 and 10, while on K3 surfaces of finite height, only $n=8$ is possible.
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Normal forms for quasi-elliptic Enriques surfaces and applications
We work out normal forms for quasi-elliptic Enriques surfaces and give several applications. These include torsors and numerically trivial automorphisms, but our main application is the completion of the classification of Enriques surfaces with finite automorphism groups started by Kondo, Nikulin, Martin and Katsura-Kondo-Martin.
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Low degree rational curves on quasi-polarized K3 surfaces
We prove that there are at most $(24-r_0)$ low-degree rational curves on high-degree models of K3 surfaces with at most Du Val singularities, where $r_0$ is the number of exceptional divisors on the minimal resolution. We also provide several existence results in the above setting (i.e. for rational curves on quasi-polarized K3 surfaces), which imply that for various values of $r_0$ our bound cannot be improved.
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On families of K3 surfaces with real multiplication
We exhibit large families of K3 surfaces with real multiplication, both abstractly using lattice theory, the Torelli theorem and the surjectivity of the period map, as well as explicitly using dihedral covers and isogenies.
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Singularities of normal quartic surfaces III (char=2, non-supersingular)
We show that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic 2 is at most 12, if the minimal resolution of $X$ is not a supersingular K3 surface. We also provide a family of explicit examples, valid in any characteristic.
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Singularities of normal quartic surfaces II (char=2)
We show, in this second part, that the maximal number of singular points of a quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic 2 is at most 14, and that, if we have 14 singularities, these are nodes and moreover the minimal resolution of $X$ is a supersingular K3 surface. We produce an irreducible component, of dimension 24, of the variety of quartics with 14 nodes. We also exhibit easy examples of quartics with 7 $A_3$-singularities.
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24 rational curves on K3 surfaces
Given d in IN, we prove that all smooth K3 surfaces (over any field of characteristic p other than 2,3) of degree greater than 84d^2 contain at most 24 rational curves of degree at most d. In the exceptional characteristics, the same bounds hold for non-unirational K3 surfaces, and we develop analogous results in the unirational case. For d at least 3, we also construct K3 surfaces of any degree greater than 4d(d+1) with 24 rational curves of degree exactly d, thus attaining the above bounds.
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12 rational curves on Enriques surfaces
Given d in IN, we prove that any polarized Enriques surface (over any field of characteristic different from 2 or with a smooth K3 cover) of degree greater than 12d^2 contains at most 12 rational curves of degree at most d. For d>2 we construct examples of Enriques surfaces of high degree that contain exactly 12 rational degree d curves.
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Q_l-cohomology projective planes and singular Enriques surfaces in characteristic two
We classify singular Enriques surfaces in characteristic two supporting a rank nine configuration of smooth rational curves. They come in one-dimensional families defined over the prime field, paralleling the situation in other characteristics, but featuring novel aspects. Contracting the given rational curves, one can derive algebraic surfaces with isolated ADE-singularities and trivial canonical bundle whose Q_l-cohomology equals that of a projective plane. Similar existence results are developed for classical Enriques surfaces. We also work out an application to integral models of Enriques surfaces (and K3 surfaces).
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