Presented By: Algebraic Geometry Seminar - Department of Mathematics
Algebraic Geometry Seminar: Bogomolov-Gieseker inequality and Bridgeland stability conditions
Shengxuan Liu (University of Michigan)
Bridgeland stability conditions are powerful tools to study the derived categories of algebraic varieties. On smooth projective surfaces, the classical Bogomolov inequality gives a bound for the second Chern character of stable sheaves. This thus implies the existence of the Bridgeland stability conditions on surfaces by the tilting technique. However, to construct a Bridgeland stability condition on higher dimensional varieties, a Bogomolov-Gieseker type bound for higher Chern characters is expected, whose proof usually requires a stronger Bogomolov-Gieseker inequality. Besides Fano varieties and K3 surfaces, it is difficult to prove a stronger Bogomolov-Gieseker inequality. In this talk, I will start with definitions of Bridgeland stability conditions, and then show how to use the Bogomolov inequality to construct the Bridgeland stability conditions on surfaces. Then I will show one way to prove a stronger Bogomolov-Gieseker inequality on some Calabi-Yau threefolds, and as an application, the existence of the Bridgeland stability conditions on such Calabi-Yau threefolds. If time permits, I will mention another application of this stronger inequality by Feyzbakhsh-Thomas on enumerative invariants.
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