4,733 research outputs found
Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds
We investigate a potential relationship between mirror symmetry for
Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and
Landau-Ginzburg models. More precisely, we show that if a Calabi-Yau admits a
so-called Tyurin degeneration to a union of two Fano varieties, then one should
be able to construct a mirror to that Calabi-Yau by gluing together the
Landau-Ginzburg models of those two Fano varieties. We provide evidence for
this correspondence in a number of different settings, including
Batyrev-Borisov mirror symmetry for K3 surfaces and Calabi-Yau threefolds,
Dolgachev-Nikulin mirror symmetry for K3 surfaces, and an explicit family of
threefolds that are not realized as complete intersections in toric varieties.Comment: v2: Section 5 has been completely rewritten to accommodate results
removed from Section 5 of arxiv:1501.04019. v3: Final version, to appear in
String-Math 2015, forthcoming volume in the Proceedings of Symposia in Pure
Mathematics serie
Families of lattice polarized K3 surfaces with monodromy
We extend the notion of lattice polarization for K3 surfaces to families over
a (not necessarily simply connected) base, in a way that gives control over the
action of monodromy on the algebraic cycles, and discuss the uses of this new
theory in the study of families of K3 surfaces admitting fibrewise symplectic
automorphisms. We then give an application of these ideas to the study of
Calabi-Yau threefolds admitting fibrations by lattice polarized K3 surfaces
Calabi-Yau Threefolds Fibred by Mirror Quartic K3 Surfaces
We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing
that any family of such K3 surfaces is completely determined by a map from the
base of the family to the moduli space of mirror quartic K3 surfaces. This is
then used to give a complete explicit description of all Calabi-Yau threefolds
fibred by mirror quartic K3 surfaces. We conclude by studying the properties of
such Calabi-Yau threefolds, including their Hodge numbers and deformation
theory.Comment: v2: Significant changes at the request of the referee. Section 3 has
been rearranged to accommodate a revised proof of Proposition 3.5 (formerly
3.2). Section 5 has been removed completely, it will instead appear as part
of Section 5 in arxiv:1601.0811
Normal Forms and Tyurin Degenerations of K3 Surfaces Polarised by a Rank 18 Lattice
We study projective Type II degenerations of K3 surfaces polarised by a
certain rank 18 lattice, where the central fibre consists of a pair of rational
surfaces glued along a smooth elliptic curve. Given such a degeneration, one
may construct other degenerations of the same kind by flopping curves on the
central fibre, but the degenerations obtained from this process are not usually
projective. We construct a series of examples which are all projective and
which are all related by flopping single curves from one component of the
central fibre to the other. The components of the central fibres obtained
include weak del Pezzo surfaces of all possible degrees. This shows that
projectivity need not impose any meaningful constraints on the surfaces that
can arise as components in Type II degenerations.Comment: 24 pages, comments welcome
Hodge Numbers from Picard-Fuchs Equations
Given a variation of Hodge structure over with Hodge numbers
, we show how to compute the degrees of the Deligne extension of
its Hodge bundles, following Eskin-Kontsevich-M\"oller-Zorich, by using the
local exponents of the corresponding Picard-Fuchs equation. This allows us to
compute the Hodge numbers of Zucker's Hodge structure on the corresponding
parabolic cohomology groups. We also apply this to families of elliptic curves,
K3 surfaces and Calabi-Yau threefolds
Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds
We investigate a potential relationship between mirror symmetry for Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and Landau-Ginzburg models. More precisely, we show that if a Calabi-Yau admits a so-called Tyurin degeneration to a union of two Fano varieties, then one should be able to construct a mirror to that Calabi-Yau by gluing together the Landau-Ginzburg models of those two Fano varieties. We provide evidence for this correspondence in a number of different settings, including Batyrev-Borisov mirror symmetry for K3 surfaces and Calabi-Yau threefolds, Dolgachev-Nikulin mirror symmetry for K3 surfaces, and an explicit family of threefolds that are not realized as complete intersections in toric varieties
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