Gromov-Witten theory of Deligne-Mumford stacks

Abstract

Introduction 1 2. Chow rings, cohomology and homology of stacks 5 3. The cyclotomic inertia stack and its rigidification 10 4. Twisted curves and their maps 18 5. The boundary of moduli 22 6. Gromov-Witten classes 27 7. The age grading 35 8. A few useful facts 41 9. An example: the weighted projective line 44 Appendix A. Gluing of algebraic stacks along closed substacks 48 Appendix B. Taking roots of line bundles 52 Appendix C. Rigidification 54 References 57 Research of D.A.

Ice cream and orbifold Riemann - Roch

“…However, at present, it is not well understood how to use these abstract results in practice to compute the dimension of RR spaces. Toën's result was applied to weighted projective spaces by F. Nironi [10], to quasismooth varieties in weighted projective spaces by S. Zhou [2] and to twisted curves by D. Abramovich and A. Vistoli [11]. Edidin's recent treatment [12] clarifies orbifold RR considerably; our results and Zhou's thesis [2] provide many practical exercises.…”

Stokes matrices for the quantum cohomologies of a class of orbifold projective lines

“…The Gromov-Witten theory for orbifolds developed by Abramovich-Graber-Vistoli [1] and Chen-Ruan [5] gives us the following. For simplicity, we shall denote by M P 1 A the complex manifold M together with the Frobenius structure on M obtained in Proposition 3.3 and call it the Frobenius manifold constructed from the Gromov-Witten theory for P 1 A .…”

“…This idea comes from physics and attracted the interests of mathematicians because of spectacular predictions for classical enumeration problems in algebraic geometry through the mirror symmetry. Now the theory of quantum cohomology and the Gromov-Witten invariants are extended to cases when X is an orbifold [1,5].…”

Quantum Cohomology and Periods

“…For example, Batyrev's mirror may not admit a full crepant resolution for dimension bigger than 3. By the development of orbifold GromovWitten theory [14,13,1], we can now work with partial resolutions with orbifold singularities. In this paper, we encounter a phenomenon of multigeneration (1) of orbifold quantum D-modules.…”

“…By the development of orbifold GromovWitten theory [14,13,1], we can now work with partial resolutions with orbifold singularities. In this paper, we encounter a phenomenon of multigeneration (1) of orbifold quantum D-modules. This phenomenon was first observed by Guest-Sakai [29] (in a different language) for a degree 3 Fano hypersurface in P(1, 1, 1, 2).…”