About Homological Mirror Symmetry

TL;DR

This paper explores the B-side of Homological Mirror Symmetry, focusing on derived categories, sheaf theory, and algebraic structures, with applications to elliptic curves and K3 surfaces, providing foundational insights into the conjecture.

Contribution

It offers a comprehensive, self-contained study of derived categories and sheaf theory related to the B-side of homological mirror symmetry, including specific cases like elliptic curves and K3 surfaces.

Findings

Detailed analysis of derived categories and homological algebra.

Application to elliptic curves and K3 surfaces.

Foundational framework for the B-side of homological mirror symmetry.

Abstract

The B-side of Kontsevich's Homological Mirror Symmetry Conjecture is discussed. We give first a self-contained study of derived categories and their homological algebra, and later restrict to the bounded derived category of schemes and ultimately Calabi--Yau manifolds, with particular emphasis on the basics of the underlying sheaf theory, and the algebraic features therein. Finally, we loosely discuss the lowest dimensional manifestations of homological mirror symmetry, namely for elliptic curves and $K3$ surfaces. The present work is a sequel to the author's survey "Towards Homological Mirror Symmetry" on the A-side of homological mirror symmetry.