Narrow Results

Generalized Calabi–Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are both special cases of this general notion. The moduli space of generalized Calabi–Yau structures admits a canonical symplectic form with respect to which the moduli space of symplectic structures is Lagrangian. The standard theory of K3 surfaces implies surjectivity of the period map and a weak form of the Global Torelli theorem.

In this paper we construct an abelian fibration over P³ on the Hilbert cube of the primitive K3 surface of genus 9. After the abelian fibration constructed by Hassett and Tschinkel on the Hilbert square on the primitive K3 surface of genus 5, this is the second example where the abelian fibration is constructed directly on . The recent more general result of Sawon proves the existence of an abelian fibration on the Hilbert scheme of a primitive K3 surface S of degree 2g - 2 = m²(2n - 2). Our example provides an alternative proof in the case m = 2, n = 3. Furthermore we identify the general fiber with the Hilbert scheme of twisted cubic curves in a Fano 3-fold of genus 9, and interpret the addition law on this variety.

We study the Brill–Noether theory of the normalizations of singular, irreducible curves on a K3 surface. We introduce a singular Brill–Noether number ρ_sing and show that if Pic(K3) = ℤ[L], there are no 's on the normalizations of irreducible curves in |L|, provided that ρ_sing < 0. We give examples showing the sharpness of this result. We then focus on the case of hyperelliptic normalizations, and classify linear systems |L| containing irreducible nodal curves with hyperelliptic normalizations, for ρ_sing < 0, without any assumption on the Picard group.

Reflection walls of certain primitive vectors in the anti-invariant sublattice of the K3 lattice define Heegner divisors in the period space of Enriques surfaces. We show that depending on the norm of these primitive vectors, these Heegner divisors are either irreducible or have two irreducible components. The two components are obtained as the walls orthogonal to primitive vectors of the same norm but of different type as ordinary or characteristic.

Self-rational maps of generic algebraic K3 surfaces are conjectured to be trivial. We relate this conjecture to a conjecture concerning the irreducibility of the universal Severi varieties parameterizing nodal curves of given genus and degree lying on some K3 surface. We also establish a number of numerical constraints satisfied by such nontrivial rational maps, that is of topological degree > 1.

We prove that the gonality among the smooth curves in a complete linear system on a K3 surface is constant except for the Donagi–Morrison example. This was proved by Ciliberto and Pareschi under the additional condition that the linear system is ample. The constancy was originally conjectured by Harris and Mumford.

As a consequence we prove that exceptional curves on K3 surfaces satisfy the Eisenbud–Lange–Martens–Schreyer conjecture and explicitly describe such curves. They turn out to be natural extensions of the Eisenbud–Lange–Martens–Schreyer examples of exceptional curves on K3 surfaces.

Let S^[n] be the Hilbert scheme of length n subschemes of a K3 surface S. H²(S^[n],ℤ) is endowed with the Beauville–Bogomolov bilinear form. Denote by Mon the subgroup of GL[H*(S^[n],ℤ)] generated by monodromy operators, and let Mon² be its image in OH²(S^[n],ℤ). We prove that Mon² is the subgroup generated by reflections with respect to +2 and -2 classes (Theorem 1.2). Thus Mon² does not surject onto OH²(S^[n],ℤ)/(±1), when n - 1 is not a prime power.

As a consequence, we get counterexamples to a version of the weight 2 Torelli question for hyperKähler varieties X deformation equivalent to S^[n]. The weight 2 Hodge structure on H²(X,ℤ) does not determine the bimeromorphic class of X, whenever n - 1 is not a prime power (the first case being n = 7). There are at least 2^{ρ(n - 1) - 1} distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n - 1) is the Euler number of n - 1.

The second main result states, that if a monodromy operator acts as the identity on H²(S^[n],ℤ), then it acts as the identity on H^k(S^[n],ℤ), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism Mon → Mon², if n ≡ 0 or n ≡ 1 modulo 4 (Corollary 1.6).

Nikulin [On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by 2 reflections, J. Soviet. Math.22 (1983) 1401–1476; Surfaces of type $K3$ with a finite automorphism group and a Picard group of rank three, Proc. Steklov Institute of Math. (3) (1985) 131–155] and Vinberg [Classification of 2-reflective hyperbolic lattices of rank 4, Trans. Moscow Math. Soc. (2007) 39–66] proved that there are only a finite number of lattices of rank $\ge 3$ that are the Néron–Severi lattice of projective $K3$ surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric description of such $K3$ surfaces $X$, when these surfaces have moreover no elliptic fibrations. In that case, we show that such $K3$ surface is either a quartic with special hyperplane sections or a double cover of the plane branched over a smooth sextic curve which has special tangencies properties with some lines, conics or cuspidal cubic curves. We then study the converse, i.e. if the geometric description we obtained characterizes these surfaces. In four cases, the description is sufficient, in each of the four other cases, there is exactly another one possibility which we study. We obtain that at least five moduli spaces of $K3$ surfaces (among the eight we study) are unirational.

We discuss the isomorphism problem of projective schemes; given two projective schemes, can we algorithmically decide whether they are isomorphic? We give affirmative answers in the case of one-dimensional projective schemes, the case of smooth irreducible varieties with a big canonical sheaf or a big anti-canonical sheaf, and the case of K3 surfaces with a finite automorphism group. As related algorithmic problems, we also discuss decidability of positivity properties of invertible sheaves, and approximation of the nef cone and the pseudo-effective cone.

We investigate how the motive of hyper-Kähler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of Bülles to the O’Grady-10 situation. In the non-commutative setting, similar results are proved for the Chow motive of moduli spaces of (semi-)stable objects of the K3 category of a cubic fourfold. As a consequence, we provide abundant examples of hyper-Kähler varieties of O’Grady-10 deformation type satisfying the standard conjectures. In the second part, we study the André motive of projective hyper-Kähler varieties. We attach to any such variety its defect group, an algebraic group which acts on the cohomology and measures the difference between the full motive and its weight-2 part. When the second Betti number is not 3, we show that the defect group is a natural complement of the Mumford–Tate group inside the motivic Galois group, and that it is deformation invariant. We prove the triviality of this group for all known examples of projective hyper-Kähler varieties, so that in each case the full motive is controlled by its weight-2 part. As applications, we show that for any variety motivated by a product of known hyper-Kähler varieties, all Hodge and Tate classes are motivated, the motivated Mumford–Tate conjecture 7.3 holds, and the André motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results.

Let Ω be a non-zero holomorphic 2-form on a K3 surface S. Suppose that S is projective algebraic and is defined over . Let be the -vector space generated by the numbers given by all the periods ∫_γ Ω, γ ∈ H₂(S, ℤ). We show that, if , then S has complex multiplication, meaning that the Mumford–Tate group of the rational Hodge structure on H²(S, ℚ) is abelian. This result was announced in [P. Tretkoff, Transcendence and CM on Borcea–Voisin towers of Calabi–Yau manifolds, J. Number Theory 152 (2015) 118–155], without a detailed proof. The converse is already well known.

To pave the way for the journey from geometry to conformal field theory (CFT), these notes present the background for some basic CFT constructions from Calabi-Yau geometry. Topics include the complex and Kähler geometry of Calabi-Yau manifolds and their classification in low dimensions. I furthermore discuss CFT constructions for the simplest known examples that are based in Calabi-Yau geometry, namely for the toroidal superconformal field theories and their ℤ₂-orbifolds. En route from geometry to CFT, I offer a discussion of K3 surfaces as the simplest class of Calabi-Yau manifolds where non-linear sigma model constructions bear mysteries to the very day. The elliptic genus in CFT and in geometry is recalled as an instructional piece of evidence in favor of a deep connection between geometry and conformal field theory.