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Semiorthogonal decompositions of projective varieties with isolated rational singularities

Yuto Arai

Abstract

We develop the method of inducing semiorthogonal decompositions of projective varieties with isolated rational singularities from those of small resolutions of singularities, which generalizes semiorthogonal decompositions for singular surfaces by Karmazyu-Kuznetsov-Shinder. \par We first explain the classical generator of the null category, which is a kind of triangulated full subcategory, and prove that the orthogonal decompositions of the null category is induced through its generator. Next, we prove that the candidate of induced semiorthogonal decomposition behaves well with respect to the inverse image of resolution morphism, and as a corollary, we obtain the required semiorthogonal decomposition.

Semiorthogonal decompositions of projective varieties with isolated rational singularities

Abstract

We develop the method of inducing semiorthogonal decompositions of projective varieties with isolated rational singularities from those of small resolutions of singularities, which generalizes semiorthogonal decompositions for singular surfaces by Karmazyu-Kuznetsov-Shinder. \par We first explain the classical generator of the null category, which is a kind of triangulated full subcategory, and prove that the orthogonal decompositions of the null category is induced through its generator. Next, we prove that the candidate of induced semiorthogonal decomposition behaves well with respect to the inverse image of resolution morphism, and as a corollary, we obtain the required semiorthogonal decomposition.
Paper Structure (5 sections, 11 theorems, 53 equations)

This paper contains 5 sections, 11 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Birational geometry
  3. Null categories
  4. Semiorthogonal decompositions
  5. The proof of the main theorem

Key Result

Theorem 1.1

Let $X$ be a surface with rational singularities, and $\pi \colon Y \to X$ be a resolution. Let be a semiorthogonal decomposition of $\mathop{\mathrm{D}}\nolimits ^{\mathrm{b}} (Y)$ which is compatible with $\pi$, in the sense of Definition 4.1. Then, $\mathcal{A}_i := \mathop{\mathrm{R\pi}}\nolimits _* \widetilde{\mathcal{A}}_i$ for $i \in \{ 1,\ldots,r \}$ form a semiorthogonal decomposition

Theorems & Definitions (27)

  • Theorem 1.1: MR4382477
  • Definition 2.1: rational singularity
  • Corollary 2.2
  • proof
  • Lemma 2.3: MR4382477, MR4276320
  • proof
  • Remark 2.4
  • Corollary 2.5: MR4382477, MR4276320
  • proof
  • Proposition 2.6: MR2057015
  • ...and 17 more