Projective K3 Surface Research Articles

K3 surface entropy and automorphism groups

We derive a characterization of the complex projective K3 surfaces which have automorphisms of positive entropy in terms of their Néron–Severi lattices. Along the way, we classify the projective K3 surfaces of zero entropy with infinite automorphism groups and we determine the projective K3 surfaces of Picard number at least five with almost abelian automorphism groups, which gives an answer to a long standing question of Nikulin.

The second integral cohomology of moduli spaces of sheaves on K3 and Abelian surfaces

In this paper we study the second integral cohomology of moduli spaces of semistable sheaves on projective K3 surfaces. If S is a projective K3 surface, v a Mukai vector and H a polarization on S that is general with respect to v, we show that H2(Mv,Z) is a free Z-module of rank 23 carrying a pure weight-two Hodge structure and a lattice structure, with respect to which H2(Mv,Z) is Hodge isometric to the Hodge sublattice v⊥ of the Mukai lattice of S. Similar results are proved for Abelian surfaces.

Gaps in the Support of Canonical Currents on Projective K3 Surfaces

Gaps in the Support of Canonical Currents on Projective K3 Surfaces

Counting elliptic fibrations on K3 surfaces

We solve the problem of counting jacobian elliptic fibrations on an arbitrary complex projective K3 surface up to automorphisms. We then illustrate our method with several explicit examples.

K3 surfaces with action of the group M20$M_{20}$

AbstractIt was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is 960 and if such a group has order 960, then it is isomorphic to the Mathieu group . In this paper, we are interested in projective K3 surfaces admitting a faithful symplectic action of the group . We show that there are infinitely many K3 surfaces with this action and we describe them and their projective models, giving some explicit examples.

Geometric description of 〈2〉-polarised Hilbert squares of generic K3 surfaces

A generic K3 surface of degree 2t is a general complex projective K3 surface S2t whose Picard group is generated by the class of an ample divisor H∈Div(S2t) such that H2=2t with respect to the intersection form. We show that if X is the Hilbert square of a generic K3 surface of degree 2t with t≠2 which admits an ample divisor D∈Div(X) with qX(D)=2, where qX is the Beauville–Bogomolov–Fujiki form, then X is a double EPW sextic.

Curves on K3 surfaces

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic 0, prove almost all cases in positive characteristic, and improve the proofs of the previously known cases. To achieve this, we introduce two new techniques in the deformation theory of curves on K3 surfaces. The first, regeneration, is a process opposite to specialization, which preserves the geometric genus and does not require the class of the curve to extend. The second, called the marked point trick, allows for a controlled degeneration of rational curves to integral ones in certain situations. Combining the two proves existence of integral curves of unbounded degree of any geometric genus g for any projective K3 surface in characteristic 0.

A gluing construction of projective K3 surfaces

We construct a non-Kummer projective K3 surface $X$ which admits compact Levi-flats by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective plane at nine general points.

Beauville–Voisin Filtrations on Zero-Cycles of Moduli Space of Stable Sheaves on K3 Surfaces

Abstract The Beauville–Voisin conjecture predicts the existence of a filtration on a projective hyper-Kähler manifold opposite to the conjectural Bloch–Beilinson filtration, called the Beauville–Voisin filtration. In [13], Voisin has introduced a filtration on zero-cycles of an arbitrary projective hyper-Kähler manifold. On the moduli space of stable objects on a projective K3 surface, there are other candidates constructed by Shen–Yin–Zhao and Barros–Flapan–Marian–Silversmith in [1, 10] and more recently by Vial in [11] from a different point of view. According to the work in [11], all of them are proved to be equivalent except Voisin’s filtration. In this paper, we show that Voisin’s filtration is the same as the other filtrations. As an application, we prove a conjecture in [1].

Lazarsfeld-Mukai Bundles on K3 Surfaces Associated with a Pencil Computing the Clifford Index

Let X be a smooth projective K3 surface over the complex numbers and let C be an ample curve on X. In this paper we will study the semistability of the Lazarsfeld-Mukai bundle EC,A associated to a line bundle A on C such that |A| is a pencil on C and computes the Clifford index of C. We give a necessary and sufficient condition for EC,A to be semistable.

Curves of maximal moduli on K3 surfaces

Abstract We prove that if X is a complex projective K3 surface and $g>0$ , then there exist infinitely many families of curves of geometric genus g on X with maximal, i.e., g-dimensional, variation in moduli. In particular, every K3 surface contains a curve of geometric genus 1 which moves in a nonisotrivial family. This implies a conjecture of Huybrechts on constant cycle curves and gives an algebro-geometric proof of a theorem of Kobayashi that a K3 surface has no global symmetric differential forms.

Stability of some vector bundles on Hilbert schemes of points on K3 surfaces

Let X be a projective K3 surfaces. In two examples where there exists a fine moduli space M of stable vector bundles on X, isomorphic to a Hilbert scheme of points, we prove that the universal family {mathcal {E}} on Xtimes M can be understood as a complete flat family of stable vector bundles on M parametrized by X, which identifies X with a smooth connected component of some moduli space of stable sheaves on M.

Extending Nirenberg–Spencer’s question on holomorphic embeddings to families of holomorphic embeddings

Nirenberg and Spencer posed the question whether the germ of a compact complex submanifold in a complex manifold is determined by its infinitesimal neighborhood of finite order when the normal bundle is sufficiently positive. To study the problem for a larger class of submanifolds, including free rational curves, we reformulate the question in the setting of families of submanifolds and their infinitesimal neighborhoods. When the submanifolds have no nonzero vector fields, we prove that it is sufficient to consider only first-order neighborhoods to have an affirmative answer to the reformulated question. When the submanifolds do have nonzero vector fields, we obtain an affirmative answer to the question under the additional assumption that submanifolds have certain nice deformation properties, which is applicable to free rational curves. As an application, we obtain a stronger version of the Cartan–Fubini-type extension theorem for Fano manifolds of Picard number 1. We also propose a potential application on hyperplane sections of projective K3 surfaces.

A motivic global Torelli theorem for isogenous K3 surfaces

We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic as Frobenius algebra objects; this can be regarded as a motivic Torelli-type theorem. We ask whether, more generally, twisted derived equivalent hyper-Kähler varieties have isomorphic Chow motives as (Frobenius) algebra objects and in particular isomorphic graded rational cohomology algebras. In the appendix, we justify introducing the notion of “Frobenius algebra object” by showing the existence of an infinite family of K3 surfaces whose Chow motives are pairwise non-isomorphic as Frobenius algebra objects but isomorphic as algebra objects. In particular, K3 surfaces in that family are pairwise non-isogenous but have isomorphic rational Hodge algebras.

Cox rings of K3 surfaces of Picard number three

Let X be a projective K3 surface over C. We prove that its Cox ring has a generating set whose degrees are either classes of smooth rational curves, sums of at most three elements of the Hilbert basis of the nef cone, or of the form 2(f+f′), where f,f′ are classes of smooth elliptic curves with f⋅f′=2. This result and techniques using Koszul's type exact sequences are then applied to determine a generating set for the Cox ring of all Mori dream K3 surfaces of Picard number three which is minimal in most cases. A presentation for the Cox ring is given in some special cases with few generators.

On the Homeomorphism Type of Smooth Projective Fourfolds

In this paper, we study smooth complex projective 4-folds which are topologically equivalent. First we show that Fano fourfolds are never oriented homeomorphic to Ricci-flat projective fourfolds and that Calabi-Yau manifolds and hyperkahler manifolds in dimension ≥ 4 are never oriented homeomorphic. Finally, we give a coarse classification of smooth projective fourfolds which are oriented homeomorphic to a hyperkahler fourfold which is deformation equivalent to the Hilbert scheme S[2] of two points of a projective K3 surface S.

Automorphisms of Hilbert schemes of points on a generic projective K3 surface

AbstractWe study automorphisms of the Hilbert scheme of n points on a generic projective K3 surface S, for any . We show that is either trivial or generated by a non‐symplectic involution and we determine numerical and divisorial conditions which allow us to distinguish between the two cases. As an application of these results we prove that, for any , there exist infinitely many admissible degrees for the polarization of the K3 surface S such that admits a non‐natural involution. This provides a generalization of the results of [7] for .

K3 surfaces with Picard number 2, Salem polynomials and Pell equation

If an automorphism of a projective K3 surface with Picard number 2 is of infinite order, then the automorphism corresponds to a solution of Pell equation. In this paper, by solving this equation, we determine all Salem polynomials of symplectic and anti-symplectic automorphisms of projective K3 surfaces with Picard number 2.

Formality conjecture for K3 surfaces

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra$\operatorname{RHom}^{\bullet }(F,F)$is formal for any sheaf$F$polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.

K3 surfaces with $\mathbb Z_2^2$ symplectic action

Let $G$ be a finite abelian group which acts symplectically on a K3 surface. The Néron-Severi lattice of the projective K3 surfaces admitting $G$ symplectic action and with minimal Picard number was computed by Garbagnati and Sarti. We consider a four-dimensional family of projective K3 surfaces with $\mathbb {Z}_2^2$ symplectic action which do not fall into the above cases. If $X$ is one of these K3 surfaces, then it arises as the minimal resolution of a specific $\mathbb {Z}_2^3$-cover of $\mathbb {P}^2$ branched along six general lines. We show that the Néron-Severi lattice of $X$ with minimal Picard number is generated by $24$ smooth rational curves and that $X$ specializes to the Kummer surface $\textrm{Km}(E_i\times E_i)$. We relate $X$ to the K3 surfaces given by the minimal resolution of the $\mathbb {Z}_2$-cover of $\mathbb {P}^2$, branched along six general lines, and the corresponding Hirzebruch-Kummer covering of exponent $2$ of $\mathbb {P}^2$.