Homological Projective Duality

TL;DR

This paper introduces Homological Projective Duality, extending classical duality into a homological framework, and demonstrates its implications for semiorthogonal decompositions and categories of singularities in algebraic geometry.

Contribution

It defines Homological Projective Duality for smooth varieties and proves key properties relating dual varieties and their linear sections.

Findings

Orthogonal linear sections have equivalent semiorthogonal decompositions.

Categories of singularities of these sections are equivalent.

Homological Duality applies to projectivizations of vector bundles.

Abstract

We introduce a notion of Homological Projective Duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties $X$ and $Y$ in dual projective spaces are Homologically Projectively Dual, then we prove that the orthogonal linear sections of $X$ and $Y$ admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate Homological Projective Duality for projectivizations of vector bundles.