Abstract
Given a contraction of a variety X to a base Y, we enhance the locus in Y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings D, under mild assumptions which hold in the case of (1) crepant partial resolutions admitting a tilting bundle with trivial summand, and (2) all contractions with fibre dimension at most one. In all cases, this produces a global invariant. In the crepant setting, we then apply this to study derived autoequivalences of X. We work generally, dropping many of the usual restrictions, and so both extend and unify existing approaches. In full generality we construct a new endofunctor of the derived category of X by twisting over D, and then, under appropriate restrictions on singularities, give conditions for when it is an autoequivalence. We show that these conditions hold automatically when the non-isomorphism locus in Y has codimension 3 or more, which covers determinantal flops, and we also control the conditions when the non-isomorphism locus has codimension 2, which covers 3-fold divisor-to-curve contractions.
Highlights
Given a contraction of a variety X to a base Y, we enhance the locus in Y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings D, under mild assumptions which hold in the case of (1) crepant partial resolutions admitting a tilting bundle with trivial summand, and (2) all contractions with fibre dimension at most one
In full generality we construct a new endofunctor of the derived category of X by twisting over D, and under appropriate restrictions on singularities, give conditions for when it is an autoequivalence. We show that these conditions hold automatically when the non-isomorphism locus in Y has codimension 3 or more, which covers determinantal flops, and we control the conditions when the non-isomorphism locus has codimension 2, which covers 3-fold divisor-to-curve contractions
Further we show in 2.5(2) that our framework applies to all contractions with fibres of dimension at most one
Summary
We turn our attention to crepant contractions and autoequivalences. One benefit of the construction of E on X is that, using the natural exact sequence. (2) D is relatively spherical (in the sense of 5.6) for all closed points z ∈ Z If these conditions hold, and they are automatic provided that dim Z ≤ dim Y − 3, the functor TwistX is an autoequivalence of Db(coh X). They are automatic provided that dim Z ≤ dim Y − 3, the functor TwistX is an autoequivalence of Db(coh X) We remark that both parts of 1.5(1) can fail in general, and if Z is not equidimensional, D is not relatively spherical for some z ∈ Z. We establish results for one-dimensional fibre contractions which are not an isomorphism in codimension two, where the conditions in 1.5 are not automatic Amongst others, this includes partial resolutions of Kleinian singularities, and 3-fold crepant divisor-to-curve examples.
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