Executive Summary : | The main open question in algebraic geometry is to complete the classification of algebraic varieties up to isomorphism over algebraically closed base fields of arbitrary characteristics. Classification of projective curves and surfaces is complete over algebraically closed fields of any characteristic, but the classification problem is wide open in higher dimensions, especially over fields of positive characteristic. This project proposes understanding the classification problem for irreducible symplectic varieties over fields of positive characteristic. Irreducible symplectic varieties over complex numbers or their analytic avatar compact hyperkähler manifolds have been studied extensively over the last half-century using techniques from algebraic geometry, differential geometry, and even physics. However, they have been only studied sporadically as test cases for Tate conjecture. The project aims to study their geometry via their derived categories, focusing on developing an understanding of the consequences of mirror symmetry on such varieties. To overcome the lack of differential geometry tools, the project will use deformation theory as a main tool and develop the theory of canonical lifts for ordinary irreducible symplectic varieties. These characteristic zero lifts are expected to be unique and serve as a bridge to apply Hodge theoretic techniques to conclude results on varieties of positive characteristic. The project's main motivation is to exemplify what a positive characteristic analog of kähler metric and the Fukaya category should look like. |