Related papers: On the Shafarevich conjecture for Enriques surface…

Let $K$ be the field of fractions of a local Henselian DVR with perfect residue field. Assuming potential semi-stable reduction, we show that an unramified Galois-action on second $\ell$-adic cohomology of a K3 surface over $K$ implies that…

Let $f:X@>>>\Bbb P^1$ be a fibered surface with fibers of genus g>1. If f is semistable and non isotrivial we prove that X of non negative Kodaira dimension implies that the number s of singular fibers is at least 5. Information about the…

We compute the monodromy groups of real Enriques surfaces of hyperbolic type. The principal tools are the deformation classification of such surfaces and a modified version of Donaldson's trick, relating real Enriques surfaces and real…

We prove the KKV conjecture expressing Gromov-Witten invariants of K3 surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov-Witten/Pairs correspondence for K3-fibered…

In this paper, we prove the non-vanishing conjecture for cotangent bundles on isotrivial elliptic surfaces. Combined with the result by H\"{o}ring and Peternell, it completely solves the question for surfaces with Kodaira dimension at most…

We show that a 3-manifold containing an incompressible surface has topologically minimal surfaces of arbitrary high genus.

Kuratowski proved that a finite graph embeds in the plane if it does not contain a subdivision of either K_5 or K_{3,3}, called Kuratowski subgraphs. A conjectured generalization of this result to all nonorientable surfaces says that a…

In this paper we show the failure of the integral Hodge/Tate conjecture for the product of an Enriques surface with a smooth odd-dimensional projective hypersurface. To do this, we use a specialization argument of Colliot-Th\'el\`ene…

We prove that there exists a pencil of Enriques surfaces defined over $\mathbb{Q}$ with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with the trivial Chow group of zero-cycles on…

We determine the Gromov-Witten invariants of the local Enriques surfaces for all genera and curve classes and prove the Klemm-Mari\~{n}o formula. In particular, we show that the generating series of genus $1$ invariants of the Enriques…

In the present paper we describe the K3 surfaces admitting order 11 automorphisms and apply this to classify log Enriques surfaces of global index 11.

We prove the Noether-Lefschetz conjecture on the moduli space of quasi-polarized K3 surfaces. This is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal…

We construct a lot of K3 surface automorphisms of positive entropy having rotation domains of ranks 1 and 2. To carry out this construction, we first lay theoretical foundations concerning equivariant linearization of nonlinear maps under…

We prove the Shafarevich conjecture for very irregular varieties of dimension less than half the dimension of their Albanese variety, subject to some mild numerical conditions. Our proof relies on the Lawrence-Venkatesh method as in the…

We classify the minimal surfaces of general type with $K^2 \leq 4\chi-8$ whose canonical map is composed with a pencil, up to a finite number of families. More precisely we prove that there is exactly one irreducible family for each value…

Vojta's Conjectures are well known to imply a wide range of results, known and unknown, in arithmetic geometry. In this paper, we add to the list by proving that they imply that rational points tend to repel each other on algebraic…

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.

Using the theory of hyperkahler manifolds, we generalize the notion of Enriques surfaces to higher dimensions and construct several examples using group actions on Hilbert schemes of points or moduli spaces of stable sheaves.

Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a closed surface so that the complement of the surface is a connected essential surface for the knot complement. In this paper, we examine some variations on this…

Log Enriques surface is a generalization of K3 and Enriques surface. We will classify all the rational log Enriques surfaces of rank 18 by giving concrete models for the realizable types of these surfaces.