Some examples of tilt-stable objects on threefolds

TL;DR

This paper explores properties and examples of tilt-stable objects on smooth complex threefolds, providing structure theorems, conditions for stability, and analyzing behavior as polarization varies.

Contribution

It offers new structure results for slope semistable sheaves, describes conditions for Bridgeland stability, and studies tilt stability under large polarization.

Findings

Structure theorem for slope semistable sheaves with vanishing discriminant

Conditions for tilt-stability of certain sheaves as polarization grows large

Identification of specific Chern classes ensuring tilt stability

Abstract

We investigate properties and describe examples of tilt-stable objects on a smooth complex projective threefold. We give a structure theorem on slope semistable sheaves of vanishing discriminant, and describe certain Chern classes for which every slope semistable sheaf yields a Bridgeland semistable object of maximal phase. Then, we study tilt stability as the polarisation $\omega$ gets large, and give sufficient conditions for tilt-stability of sheaves of the following two forms: 1) twists of ideal sheaves or 2) torsion-free sheaves whose first Chern class is twice a minimum possible value.