The cohomology ring of crepant resolutions of orbifolds.

*(English)*Zbl 1105.14078
Jarvis, Tyler J. (ed.) et al., Gromov-Witten theory of spin curves and orbifolds. AMS special session, San Francisco, CA, USA, May 3–4, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3534-3/pbk). Contemporary Mathematics 403, 117-126 (2006).

The orbifold cohomology of complex orbifolds was introduced by W. Chen and Y. Ruan [Commun. Math. Phys. 248, No. 1, 1–31 (2004; Zbl 1063.53091)]. The article under review proposes a set of conjectures relating orbifold cohomology and the cohomology of certain resolutions. Supporting evidence for the conjectures is then offered in a number of interesting examples.

In short, for those complex orbifolds admitting hyperkähler resolutions, Ruan conjectures that the cohomology of the resolution is isomorphic to the orbifold cohomology. This is termed “the cohomological hyperkähler resolution conjecture”. It applies, for instance, to the case of the Hilbert scheme of points on a torus or a \(K3\) surface, when viewed as a hyperkähler resolution of the symmetric power of the surface. This particular case was in fact recently confirmed by M. Lehn and C. Sorger [Invent. Math. 152, No. 2, 305–329 (2003; Zbl 1035.14001)], B. Fantechi and L. Göttsche [Duke Math. J. 117, No. 2, 197–227 (2003; Zbl 1086.14046)] and B. Uribe [Commun. Anal. Geom. 13, No. 1, 113–128 (2005; Zbl 1087.32012)].

A similar conjecture is formulated for general crepant resolutions of Gorenstein orbifolds. In this case one needs to include quantum corrections in the cohomology of the resolution. These quantum corrections encode certain Gromov-Witten invariants of exceptional fibers of the resolution, and vanish in the hyperkähler case. Finally, for \(K\)-equivalent manifolds, a cohomological miminmal model conjecture is formulated, also after including the quantum corrections in cohomology.

For the entire collection see [Zbl 1091.14002].

In short, for those complex orbifolds admitting hyperkähler resolutions, Ruan conjectures that the cohomology of the resolution is isomorphic to the orbifold cohomology. This is termed “the cohomological hyperkähler resolution conjecture”. It applies, for instance, to the case of the Hilbert scheme of points on a torus or a \(K3\) surface, when viewed as a hyperkähler resolution of the symmetric power of the surface. This particular case was in fact recently confirmed by M. Lehn and C. Sorger [Invent. Math. 152, No. 2, 305–329 (2003; Zbl 1035.14001)], B. Fantechi and L. Göttsche [Duke Math. J. 117, No. 2, 197–227 (2003; Zbl 1086.14046)] and B. Uribe [Commun. Anal. Geom. 13, No. 1, 113–128 (2005; Zbl 1087.32012)].

A similar conjecture is formulated for general crepant resolutions of Gorenstein orbifolds. In this case one needs to include quantum corrections in the cohomology of the resolution. These quantum corrections encode certain Gromov-Witten invariants of exceptional fibers of the resolution, and vanish in the hyperkähler case. Finally, for \(K\)-equivalent manifolds, a cohomological miminmal model conjecture is formulated, also after including the quantum corrections in cohomology.

For the entire collection see [Zbl 1091.14002].

Reviewer: Dragos Oprea (Palo Alto)