The double point relation defines a natural theory of algebraic cobordism for bundles on varieties. We construct a simple basis (over the rationals) of the corresponding cobordism groups over Spec(C) for all dimensions of varieties and ranks of bundles. The basis consists of split bundles over products of projective spaces. Moreover, we prove the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism.
Taiwang DengYau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing, 100084, ChinaBin XuYau Mathematical Sciences Center and Department of Mathematics, Tsinghua University, Beijing, China
In this article we study a conjecture of Geiss-Leclerc-Schröer, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type quiver. In the general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic.
We realize the moduli spaces of cubic fourfolds with specified group actions as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. We prove the geometric (GIT) compactifications are naturally isomorphic to the Hodge theoretic (Looijenga, in many cases Baily–Borel) compactifications. The key ingredients of the proof are the global Torelli theorem by Voisin, the characterization of the image of the period map given by Looijenga and Laza independently, and the functoriality of Looijenga compactifications proved in the Appendix.
We study the non-klt locus of singularities of pairs. We show that
given a pair (X, B) and a projective morphism X → Z with connected fibres such
that −(KX +B) is nef over Z, the non-klt locus of (X, B) has at most two connected
components near each fibre of X → Z. This was conjectured by Hacon and Han.
In a different direction we answer a question of Mark Gross on connectedness
of the non-klt loci of certain pairs. This is motivated by constructions in Mirror
Nathan PriddisInstitut für Algebraische Geometrie, Universität Hannover, D-30060 HannoverYuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, UT 84112Mark ShoemakerDepartment of Mathematics, Colorado State University Fort Collins, CO 80523
We establish a new relationship (the MLK correspondence) between twisted FJRW theory and local Gromov-Witten theory in all genera. As a consequence, we show that the Landau-Ginzburg/Calabi-Yau correspondence is implied by the crepant transformation conjecture for Fermat type in genus zero. We use this to then prove the Landau-Ginzburg/Calabi-Yau correspondence for Fermat type, generalizing the results of A. Chiodo and Y. Ruan.