Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2405.12009

Degenerations and Fibrations of K3 Surfaces: Lattice Polarisations and Mirror Symmetry

Luca Giovenzana·Alan Thompson
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 lattice polarisation that is compatible with existing definitions for the general fibre and the rational surfaces comprising the central fibre. We separately consider elliptically fibred K3 surfaces, where the base of the fibration admits a splitting into a pair of discs with specified monodromy around the boundary. In this setting we establish a notion of lattice polarisation for the induced elliptic fibrations over discs, which is compatible with the existing definition for K3 surfaces. Finally, we discuss the mirror symmetric correspondence between these two settings.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Luca Giovenzana, Alan Thompson. 2024-07-22. Degenerations and Fibrations of K3 Surfaces: Lattice Polarisations and Mirror Symmetry. https://arxiv.org/abs/2405.12009

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

G3-Criteria and Applications

The G3-property of a subvariety was introduced by Hironaka-Matsumura, and plays an important role for deducing connectedness and extension results. Unfortunately, it's a rather elusive notion, which is not always easy to establish. Most of the existing work is concentrated on subvarieties of homogeneous varieties. The first goal of this article is to show that mobility assumptions on the subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a certain partial positivity property, slightly stronger than G3, previously introduced by the author. Second, we apply the result to prove that, in numerous situations, the splitting of the normal bundle of a smooth two-codimensional subvariety implies that it is a complete intersection.

math.AG↗
arXiv2026

Nodal degeneration of chiral algebras I: Global structure and gluing formula

We define a natural extension of a universal factorization algebra $\mathcal{A}$ to families of stable punctured curves, by integrating over all semistable modifications. We prove that the resulting sheaf of factorization homology satisfies a natural gluing formula, by tensoring over a certain derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, generalizing the Verlinde formula for gluing of conformal blocks.

math.AG↗