Tensor triangulated category structures in the derived category of a variety with big (anti)canonical bundle

Angel Toledo

Vol. 327 (2023), No. 2, 359–377

DOI: 10.2140/pjm.2023.327.359

Abstract

Let $X$ be a smooth projective variety over $ℂ$ with big (anti)canonical bundle. It is known that for such $X$ the Balmer spectrum of the tensor triangulated category of perfect complexes $\mathrm{Perf}<!--FUNCTION\; APPLICATION-->(X)$, equipped with the derived tensor product ${\otimes }_X^{\mathbb{𝕃}}$, recovers the space $X$. We study the possible tensor triangulated category structures one can put on $\mathrm{Perf}<!--FUNCTION\; APPLICATION-->(X)$. As an application, we prove a monoidal version of the well-known Bondal–Orlov reconstruction theorem.

Keywords

tensor triangulated category, Balmer spectrum, Bondal–Orlov reconstruction, derived categories

Mathematical Subject Classification

Primary: 14F08, 18G80

Milestones

Received: 23 September 2023

Revised: 28 January 2024

Accepted: 31 January 2024

Published: 12 March 2024