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- W1917161576 abstract "We study derived categories of Gorenstein varieties $X$ and $X^+$ connected by a flop. We assume that the flopping contractions $fcolon Xto Y$ , $f^+ colon X^+ to Y$ have fibers of dimension bounded by one and $Y$ has canonical hypersurface singularities of multiplicity two. We consider the fiber product $W=Xtimes _YX^+$ with projections $pcolon Wto X$ , $p^+colon Wto X^+$ 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." @default.
- W1917161576 created "2016-06-24" @default.
- W1917161576 creator A5043068443 @default.
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- W1917161576 date "2022-05-01" @default.
- W1917161576 modified "2023-10-05" @default.
- W1917161576 title "Flops and spherical functors" @default.
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- W1917161576 doi "https://doi.org/10.1112/s0010437x22007497" @default.
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