Related papers: Mukai duality on K3 surfaces from the differential…
Motivated by the question of rationality of cubic fourfolds, we show that a cubic X has an associated K3 surface in the sense of Hassett if and only if the variety F of lines on X is birational to a moduli space of sheaves on a K3 surface,…
We discuss the geometry of the genus one fibrations associated to an elliptic fibration on a K3 surface. We show that the two-torsion subgroup of the Brauer group of a general elliptic fibration is naturally isomorphic to the two-torsion of…
I have finalized my old (1979) results about enumeration of connected components of moduli of real polarized K3 surfaces. As an application, using recent results of math.AG/0312396, the complete classification of real polarized K3 surfaces…
The conjectural equivalence of curve counting on Calabi-Yau 3-folds via stable maps and stable pairs is discussed. By considering Calabi-Yau 3-folds with K3 fibrations, the correspondence naturally connects curve and sheaf counting on K3…
We show the birational boundedness of anti-canonical irreducible hypersurfaces which form 3-fold plt pairs. We also treat a collection of Du Val K3 surfaces which is birationally bounded but unbounded.
We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces. We show that integral transforms between derived categories of spaces induce,…
Let $X$ be an algebraic K3 surface, $v=(r,H,s)$ a primitive isotropic Mukai vector on $X$ and $M_X(v)$ the moduli of sheaves over $X$ with $v$. Let $N(X)$ be Picard lattice of $X$. In math.AG/0309348 and math.AG/0606289, all divisors in…
We study the field of moduli of singular abelian and K3 surfaces. We discuss both the field of moduli over the CM field and over $\Q$. We also discuss non-finiteness with respect to the degree of the field of moduli. Finally, we provide an…
We study Fourier-Mukai equivalences of (families of) abelian varieties and K3 surfaces in positive and mixed characteristics. We first prove in any characteristics that Fourier-Mukai partners of abelian varieties are again abelian…
In this technical note we describe a new (to the physics literature) construction of bundles on Calabi-Yaus. We primarily study this construction in the special case of K3 surfaces, for which interesting results can be obtained. For…
We investigate geometric invariants of cuspidal edges on focal surfaces of regular surface. In particular, we shall clarify the sign of the singular curvature at a cuspidal edge on a focal surface using singularities of parallel surface of…
A "reduced" differential geometry adapted to the presence of abelian isometries is constructed.Classical T-duality diagonalizes in this setting, allowing us to get conveniently the transformation of the relevant geometrical objects such as…
Let C be a general element in the locus of curves in M_g lying on some K3 surface, where g is congruent to 3 mod 4 and greater than or equal to 15. Following Mukai's ideas, we show how to reconstruct the K3 surface as a Fourier-Mukai…
We exhibit explicit examples of very general special cubic fourfolds with discriminant $d$ admitting an associated (twisted) K3 surface, which have non-isomorphic Fourier-Mukai partners. In particular, in the untwisted setting, we show that…
We show that the Fourier-Mukai transfortm on an abelian surface induces a birational map of the moduli space of stablke sheaves.
Tyurin degenerations of K3 surfaces are degenerations whose central fibre consists of a pair of rational surfaces glued along a smooth elliptic curve. We study the lattice theory of such Tyurin degenerations, establishing a notion of…
In part I of this paper we constructed certain fibered Calabi-Yaus by a quotient construction in the context of weighted hypersurfaces. In this paper look at the case of K3 fibrations more closely and study the singular fibers which occur.…
In this work we explore the physics associated to Calabi-Yau (CY) n-folds that can be described as a fibration in more than one way. Beginning with F-theory vacua in various dimensions, we consider limits/dualities with M-theory, type IIA,…
We construct several examples of genus-one fibered K3 surfaces without a global section with type $I_{n}$ fibers, by considering double covers of a special class of rational elliptic surfaces lacking a global section, known as Halphen…
We show that supersingular K3 surfaces in characteristic $p\geq5$ are related sequences of very special correspondences. This is not enough to conclude that they are unirational. As a byproduct, we exhibit a fibration structure on the…