Wonderful resolutions and categorical crepant resolutions of singularities

Abstract

International audienc

Categorical crepant resolutions for quotient singularities

“…Note however that if the categorical resolution is of "geometric origin", then all these notions coincide with the classical definition of crepancy. Namely, we have the proposition ( The existence of weakly crepant resolutions of singularities has been proved in a quite general context (see [2,3]). For instance, it is proved in [2] that all Gorenstein determinantal varieties (general, symmetric, skew-symmetric) admit weakly crepant resolution of singularities.…”

“…Namely, we have the proposition ( The existence of weakly crepant resolutions of singularities has been proved in a quite general context (see [2,3]). For instance, it is proved in [2] that all Gorenstein determinantal varieties (general, symmetric, skew-symmetric) admit weakly crepant resolution of singularities. The existence of strongly crepant resolution seems to be a much more delicate issue.…”

“…The existence of weakly crepant resolutions of singularities has been proved in a quite general context (see [Abu13c,Abu13a]). For instance, it is proved in [Abu13c] that all Gorenstein determinantal varieties (general, symmetric, skew-symmetric) admit weakly crepant resolution of singularities. The existence of strongly crepant resolution seems to be a much more delicate issue.…”

Semiorthogonal decompositions in algebraic geometry

“…Another interesting question is to find new methods of construction of minimal categorical resolutions. An interesting development in this direction is the work [Ab12] in which a notion of a wonderful resolution of singularities (an analogue of wonderful compactifications) is introduced and it is shown that a wonderful resolution gives rise to a weakly crepant categorical resolution. This can be viewed as an advance on the first part of Proposition 3.7.…”

Comparing the commutative and non-commutative resolutions for determinantal varieties of skew symmetric and symmetric matrices

“…The construction of Λ also works when n is even but then Λ is not an NCCR, albeit very close to one. In particular one may show that D(Λ) is a "weakly crepant categorical resolution" of Perf(Y ), again in the sense of [12] (see [1] for an entirely different construction of such resolutions).…”