Minifolds and Phantoms

TL;DR

This paper classifies minifolds up to dimension 4, explores their derived categories, and constructs new phantom categories with vanishing Hochschild homology and Grothendieck groups, advancing understanding of semi-orthogonal decompositions.

Contribution

It classifies low-dimensional minifolds, proposes a conjecture for fake projective spaces, and constructs new phantom categories with vanishing invariants.

Findings

Classification of minifolds in dimensions up to 4

Proof of semi-orthogonal decomposition conjecture for certain fake projective planes

Construction of new phantom categories with vanishing Hochschild homology and Grothendieck group

Abstract

A minifold is a smooth projective $n$-dimensional variety such that its bounded derived category of coherent sheaves $\D^b(X)$ admits a semi-orthogonal decomposition into an exceptional collection of $n+1$ exceptional objects. In this paper we classify minifolds of dimension $n \leq 4$. We conjecture that the derived category of fake projective spaces have a similar semi-orthogonal decomposition into a collection of $n+1$ exceptional objects and a category with vanishing Hochschild homology. We prove this for fake projective planes with non-abelian automorphism group. We construct new examples of phantom categories with both Hochschild homology and Grothendieck group vanishing.