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Fully faithful functors, skyscraper sheaves, and birational equivalence

Chunyi Li, Xun Lin, Xiaolei Zhao

Abstract

Let $X$ and $Y$ be two smooth projective varieties such that there is a fully faithful exact functor from $D^b(\mathrm{Coh}(X))$ to $D^b(\mathrm{Coh}(Y))$. We show that $X$ and $Y$ are birational equivalent if the functor maps one skyscraper sheaf to a skyscraper sheaf. Further assuming that $X$ and $Y$ are of the same dimension, we show that if $X$ has ample canonical bundle and $H^0(X ,K_X)\neq 0$, or if $X$ is a K3 surface with Picard number one, then $Y$ is birational to a Fourier--Mukai partner of $X$.

Fully faithful functors, skyscraper sheaves, and birational equivalence

Abstract

Let and be two smooth projective varieties such that there is a fully faithful exact functor from to . We show that and are birational equivalent if the functor maps one skyscraper sheaf to a skyscraper sheaf. Further assuming that and are of the same dimension, we show that if has ample canonical bundle and , or if is a K3 surface with Picard number one, then is birational to a Fourier--Mukai partner of .
Paper Structure (13 sections, 30 theorems, 72 equations)

This paper contains 13 sections, 30 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Support of Semiorthogonal Factor
  3. Adjoint functors
  4. Semiorthogonal factor
  5. Definition of support
  6. Restriction of a functor
  7. Composition of restricted functors
  8. Fully faithful functors and birational equivalence
  9. Main result
  10. Isomorphic of Zariski local neighborhood
  11. Isomorphism between two open subsets
  12. K3 surface case
  13. An Alternative approach to Corollary \ref{['cor:pgneq0bir']}

Key Result

Theorem 1.1

Let $X$ and $Y$ be two irreducible smooth projective varieties with a fully faithful exact functor $\mathsf{F}\colon \mathrm{D}^{b}(X)\rightarrow \mathrm{D}^{b}(Y)$. Assume that there exists a closed point $x\in X$ such that $\mathsf{F}(\mathcal{O}_x)=\mathcal{O}_y$ for some closed point $y\in Y$. T

Theorems & Definitions (56)

  • Theorem 1.1: Theorem \ref{['thm:pointtoglobal']}
  • Theorem 1.2: Theorem \ref{['thm:K3pic1']}
  • Corollary 1.3: Corollary \ref{['cor:pgneq0bir']}
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4: Bridgeland-Maciocia:K3Fibrations
  • proof
  • Lemma 2.5
  • ...and 46 more