Counting special Lagrangian classes and semistable Mukai vectors for K3 surfaces

TL;DR

This paper investigates the asymptotic behavior of special Lagrangian classes and semistable Mukai vectors on K3 surfaces, providing bounds and exact formulas related to geometric and stability properties.

Contribution

It establishes an upper bound on the growth of irreducible special Lagrangian classes and derives the exact leading term for the count of semistable Mukai vectors under Bridgeland stability.

Findings

Upper bound on irreducible special Lagrangian classes

Exact leading term for Mukai vectors count

Results relate geometric classes to stability conditions

Abstract

Motivated by the study of the growth rate of the number of geodesics in flat surfaces with bounded lengths, we study generalizations of such problems for K3 surfaces. In one generalization, we give a result regarding the upper bound on the asymptotics of the number of classes of irreducible special Lagrangians in K3 surfaces with bounded period integrals. In another generalization, we give the exact leading term in the asymptotics of the number of Mukai vectors of semistable coherent sheaves on algebraic K3 surfaces with bounded central charges, with respect to generic Bridgeland stability conditions.