Relative Monoidal Bondal-Orlov

TL;DR

This paper extends the Bondal-Orlov reconstruction theorem to a relative monoidal setting, establishing uniqueness of tensor structures on derived categories of certain smooth projective varieties over a base scheme.

Contribution

It introduces a relative monoidal version of the Bondal-Orlov theorem and constructs a stack of dg-bifunctors to analyze tensor structures on derived categories.

Findings

Uniqueness of tensor triangulated structures on derived categories of fibers.

Construction of a stack of dg-bifunctors parametrizing local homotopical behavior.

Applicability to varieties with ample (anti-)canonical bundles over a base scheme.

Abstract

In this article we study a relative monoidal version of the Bondal-Orlov reconstruction theorem. We establish an uniqueness result for tensor triangulated category structures $(\boxtimes,\mathbb{1})$ on the derived category $D^{b}(X)$ of a variety $X$ which is smooth projective and faithfully flat over a quasi-compact quasi-separated base scheme $S$ in the case where the fibers $X_{s}$ over any point $s\in S$ all have ample (anti-)canonical bundles. To do so we construct a stack $\Gamma$ of dg-bifunctors which parametrize the local homotopical behaviour of $\boxtimes$, and we study some of its properties around the derived categories of the fibers $X_{s}$.