Mirror Symmetry, Autoequivalences, and Bridgeland Stability Conditions

Abstract

The present thesis studies various aspects of Calabi-Yau manifolds, including mirror symmetry, systolic geometry, and dynamical systems. We construct the mirror operation of Atiyah flop in symplectic geometry. We construct the mirror metric of the Weil-Petersson metric on the complex moduli space of Calabi-Yau manifolds, in terms of derived categories and Bridgeland stability conditions. We propose a new generalization of Loewner’s torus systolic inequality from the perspective of Calabi-Yau geometry, and prove a generalized systolic inequality for generic K3 surfaces. We study the dynamical systems formed by autoequivalences on the derived categories of Calabi-Yau manifolds, and find the first counterexamples of Kikuta-Takahashi’s conjecture on categorical entropy.