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

Angel Israel Toledo Castro

Abstract

Let $X$ be a smooth projective variety over $\mathbb{C}$ with big (anti-)canonical bundle. It is known that in this situation the Balmer spectrum of the tensor triangulated category of perfect complexes $Perf(X)$ of $X$ equipped with the derived tensor product $\otimes_{X}^{\mathbb{L}}$ recovers the space $X$. In this work we study the possible tensor triangulated category structures one can put on $Perf(X)$. As an application we prove a monoidal version of the well-known Bondal-Orlov reconstruction theorem.

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

Abstract

Let be a smooth projective variety over with big (anti-)canonical bundle. It is known that in this situation the Balmer spectrum of the tensor triangulated category of perfect complexes of equipped with the derived tensor product recovers the space . In this work we study the possible tensor triangulated category structures one can put on . As an application we prove a monoidal version of the well-known Bondal-Orlov reconstruction theorem.
Paper Structure (4 sections, 30 theorems, 19 equations)

This paper contains 4 sections, 30 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Derived categories and the Balmer reconstruction
  3. Tensor triangulated geometry
  4. TTC's and Picard groups

Key Result

Theorem 1.1

bondal2001reconstruction Let X be an irreducible smooth projective variety with ample (anti-)canonical bundle. If $D^{b}(X)\simeq D^{b}(Y)$ for some other smooth algebraic variety Y, then $X \cong Y$.

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Corollary 1.4
  • Definition 2.1
  • Example 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 55 more