Flops and spherical functors

Compositio mathematica

article

Agnieszka Bodzenta¹  Alexey Bondal² 

[1] Faculty of Mathematics, Informatics and Mechanics University of Warsaw;Steklov Mathematical Institute of Russian Academy of Sciences

关键词: flops;    derived categories;    coherent sheaves;    spherical functors;    14E05;    14F08;    14E30;    18N10;   

DOI  :  10.1112/S0010437X22007497

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

We study derived categories of Gorenstein varieties $X$ and $X^+$ connected by a flop. We assume that the flopping contractions $f\colon X\to Y$ , $f^+ \colon X^+ \to Y$ have fibers of dimension bounded by one and $Y$ has canonical hypersurface singularities of multiplicity two. We consider the fiber product $W=X\times _YX^+$ with projections $p\colon W\to X$ , $p^+\colon W\to X^+$ and prove that the flop functors $F = Rp^+_*Lp^* \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X^+)$ , $F^+= Rp_*L{p^+}^* \colon {\mathcal {D}}^b(X^+) \to {\mathcal {D}}^b(X)$ are equivalences, inverse to those constructed by Van den Bergh. The composite $F^+ \circ F \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X)$ is a non-trivial auto-equivalence. When variety $Y$ is affine, we present $F^+ \circ F$ as the spherical cotwist of a spherical couple $(\Psi ^*,\Psi )$ which involves a spherical functor $\Psi$ constructed by deriving the inclusion of the null category $\mathscr {A}_f$ of sheaves ${\mathcal {F}} \in \mathop {{\rm Coh}}\nolimits (X)$ with $Rf_*({\mathcal {F}} )=0$ into $\mathop {{\rm Coh}}\nolimits (X)$ . We construct a spherical pair ( ${\mathcal {D}}^b(X)$ , ${\mathcal {D}}^b(X^+)$ ) in the quotient ${\mathcal {D}}^b(W) /{\mathcal {K}}^b$ , where ${\mathcal {K}}^b$ is the common kernel of the derived push-forwards for the projections to $X$ and $X^+$ , thus implementing in geometric terms a schober for the flop. A technical innovation of the paper is the $L^1f^*f_*$ vanishing for Van den Bergh's projective generator. We construct a projective generator in the null category and prove that its endomorphism algebra is the contraction algebra.

【 授权许可】

CC BY