Twisted jets, motivic measures and orbifold cohomology

By Takehiko Yasuda. Compositio Mathematica, 2001

97 citations recorded by the source corpus.

We introduce the notion of twisted jets. For a Deligne–Mumford stack $\mathscr{X}$ of finite type over $\mathbb{C}$, a twisted $\infty$-jet on $\mathscr{X}$ is a representable morphism $\mathscr{D} \to \mathscr{X}$ such that $\mathscr{D}$ is a smooth Deligne–Mumford stack with the coarse moduli space Spec$\mathbb{C}[[t]]$. We study a motivic measure on the space of the twisted $\infty$-jets on a smooth Deligne–Mumford stack. As an application, we prove that two birational minimal models with Gorenstein quotient singularities have the same orbifold cohomology as a Hodge structure.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2ccd59aef1f47468645060322a1ed214758ffc86

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. T. Graber
2. L. Borisov
3. Sam Payne
4. Gregory G. Smith
5. Linda Chen
6. Jim Bryan
7. M. Mustaţă
8. Chi Li

In Geometry and Topology

1. Sam Payne
2. M. Mustaţă
3. L. Borisov
4. H. Iritani
5. L. Fu
6. T. Graber
7. Gregory G. Smith
8. Charles Vial

Citations in the corpus

• Motivic hyper-Kähler resolution conjecture, I : Generalized Kummer varieties (2016, background): “the orbifold Euler number and the orbifold Hodge numbers, should be related to the corresponding invariants of a crepant resolution ([4], [5], [63], [42]).”
• Motivic integration over Deligne–Mumford stacks (2003, unknown): “In [ Yas1 ], we defined a barely faithful morphism to be a morphism f : Y → X of DM stacks such that for every object ξ of Y, the map Aut(ξ) → Aut(f(ξ)) is bijective.”
• Ehrhart polynomials and stringy Betti numbers (2005, methodology): “t) td(1−t−1)d = δQ(t). 6 MUSTATAANDPAYNE 3. δ-vectors via orbifold cohomology The orbifold cohomology of a Gorenstein variety Y with quotient singularities was defined by Chen and Ruan [CR] and Yasuda [Ya], as follows. There is a canonically associated orbifold (smooth Deligne-Mumford stack) Y whose ”
• The Crepant Resolution Conjecture (2006, background): “If X is a Gorenstein orbifold whose coarse moduli scheme X admits a crepant resolution Y → X , Yasuda has proven that H orb (X ,C) and H(Y,C) are isomorphic as graded vector spaces [29].”
• The orbifold Chow ring of toric Deligne-Mumford stacks (2003, result): “. The product structure is induced by the degree zero part of the quantum product; in particular, it involves Gromov-Witten invariants. Inspired by string theory and results in Batyrev [3] and Yasuda [27], one expects that, in nice situations, the orbifold Chow ring coincides with the Chow ring of a”