Stringy Chern classes

Work of Dixon, Harvey, Vafa and Witten in the 80’s ([DHVW85]) introduced a notion of Euler characteristic (for quotients of a torus by a finite group) which became known as the physicist’s orbifold Euler number. In the 90’s V. Batyrev introduced a notion of stringy Euler number ([Bat99b]) for ‘arbitrary Kawamata log-terminal pairs’, proving that this number agrees with the physicist’s orbifold Euler number for algebraic varieties with a regular action of a finite group, and thereby proving a strong form of the McKay correspondence as conjectured by Miles Reid. The stringy Euler number is one of a series of stringy invariants, defined in different contexts and at different levels of generality. Among many others, the work of Lev Borisov and Anatoly Libgober ([BL03], [BL]) stands out as probably the most advanced. A natural question is whether invariants such as the stringy Euler number are numerical manifestations of more refined invariants. For example, recall that the conventional Euler characteristic of a compact nonsingular complex variety is the degree of the total Chern class of its tangent bundle (PoincaréHopf). Is there a ‘stringy’ Poincaré-Hopf theorem? That is, is the stringy Euler number the degree of a stringy Chern class naturally defined for singular varieties?