Weight truncation for wall-crossings in birational cobordisms

Abstract

We develop the technique of weight truncation in the context of wall-crossings in birational cobordisms, parallel to that in [7,4]. More precisely, for each such wall-crossing, we embed the bounded above derived category of coherent sheaves of the semistable part as a semi-orthogonal summand of that of the stack in question. Our construction does not require any smoothness assumptions, and exhibits a strong symmetry across the two sides of the wall-crossing. As an application, we show that for wallcrossings satisfying suitable regularity conditions, a certain duality of local cohomology complexes implies the existence of a fully faithful functor/equivalence between the derived categories under wall-crossings.

Survey on homological flips and homological flops

“…where the middle component is equivalent to D − coh (X + ). A proof may be found in [14] (a revised version is under preparation where the arguments will be simplified and put in a broader scope), and is based on the tensor form (4.3) below. We will skip the proof in this article.…”

“…The following result says that this condition is also sufficient. A proof may be found in [14] (a revised version is under preparation, for which the arguments will be simplified). In this case, the functor (3.13) restricts to (4.2)…”

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We give a survey for the results in [13,14,15], which attempts to relate the derived categories under general classes of flips and flops. We indicate how the approach fails because of what appears to be a formal problem.

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“…In this article, we give a survey for an approach developed in [13,14,15] to the problem of relating derived categories under flips and flops. The notions of flips and flops are sometimes a source of confusion because in the literature (especially those on derived categories) the terminology is sometimes used to refer to the closely related notions of K-dominant and K-equivalent birational maps.…”

“…14. A Z-graded ring A is said to satisfy canonical vanishing at a ∈ Z if we have RΓ I + (A) ∈ D <a (Gr(A)) and RΓ I − (A) ∈ D >a (Gr(A)).…”