Some Examples of Tilt-Stable Objects on Threefolds

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 polarization ω 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.

Key Words:

• Algebraic geometry
• Bridgeland stability

2010 Mathematics Subject Classification:

• Primary: 14D20
• Secondary: 14F05
• 14J10
• 14J30

ACKNOWLEDGMENTS

The authors would like to thank Ziyu Zhang for helpful discussions, Emanuele Macrì for kindly answering our questions, and an anonymous referee for suggesting a more efficient proof of Theorem 3.14.