Mukai pairs and simple $K$-equivalence

Abstract

A K-equivalent map between two smooth projective varieties is called simple if the map is resolved in both sides by single smooth blow-ups. In this paper, we will provide a structure theorem of simple K-equivalent maps, which reduces the study of such maps to that of special Fano manifolds. As applications of the structure theorem, we provide examples of simple Kequivalent maps, and classify such maps in several cases, including the case of dimension at most 8.

Manifolds with two projective bundle structures

“…Within Birational Geometry, they appear as the exceptional divisors of simple K-equivalent maps (cf. [7]), a class containing Mukai flops. Classification results for manifolds with two projective bundle structures could be useful in the study of this type of maps.…”

The generalized roof F(1,2,n): Hodge structures and derived categories

“…The existence of two different projective bundle structures on a Fano variety has recently raised attention: in [ORS20] a link with the Campana-Peternell conjecture has been highlighted, while in [Kan18] the study of a broad class of K-equivalences has been reduced to the classification of roofs of projective bundles, i.e. special Fano varieties of Picard rank two which admit two different projective bundle structures of the same rank.…”

“…Generalized homogeneous roofs. In [Kan18], the notion of roof has been introduced. Instead of the original definition, we will recall the following characterization which is the most suitable for the purpose of this paper: Proposition 2.1.…”

Calabi-Yau fibrations, simple K-equivalence and mutations

“…In fact, while some constructions like the Pfaffian-Grassmannian above and the intersection of two translates of G(2, 5) [OR17,BCP17] can be now explained by the homological projective duality and categorical joins programs [Kuz07,KP19], there exists a class of conjecturally derived equivalent pairs of Calabi-Yau varieties [KR20, Conjecture 2.6] for which a general argument is missing. In the context of Kequivalence, the notion of roof of projective bundles has been introduced by Kanemitsu to define special Fano manifolds which admit two projective bundle structures [Kan18]. It has been shown that from the data of a hyperplane section of a roof of projective bundles one can define two equidimensional Calabi-Yau varieties [KR20]: several instances of this problem had been previously investigated [Muk98, IMOU1606, Kuz16, KR17] but a working general approach to prove derived equivalence has yet to be found.…”

“…The main motivation for this construction arises in light of the DK-conjecture [BO02], [Kaw02]. In fact, Kanemitsu showed that for a simple K-equivalent map, which is a birational morphism µ : X 1 X 2 between smooth projective varieties resolved by a single blowup X 0 such that the pullbacks of the canonical bundles of X 1 and X 2 to X 0 are isomorphic, the exceptional loci are both isomorphic to a family of roofs [Kan18]. If we assume the exceptional locus to be a roof bundle, we construct fully faithful embeddings of D b coh(X 1 ) and D b coh(X 2 ) in the derived category of X 0 and we prove derived equivalence of X 1 and X 2 for some classes of such birational pairs.…”