Tautological Sheaves On Hilbert Schemes

Many invariants of the Hilbert schemes of points on a nonsingular projective surface, including Betti numbers[1], Hodge numbers[2], cobordism classes[3], and elliptic genus[4], can be determined explicitly by the corresponding invariants of the surfaces.We use the following strategymy[5] to compute some generating series of (equivari-ant) integrals of some vector bundles constructed from the tautological bundles on Hilbert schemes of points on surfaces:1. Reduce to the cases of P2and P1x P1, using a result due to Ellingsrud-Gottsche-Lehn[3] on the universal formula for integrals of multiplicative classes on Hilbert schemes.2. P2and P1x P1are toric surfaces, so one can use their natural torus actions and the induced actions on their Hilbert schemes to further reduce to the equivariant case of C2.3. Use a localization calculation to establish the equivariant case.We establish the explicit formula onå'Œand also propose some conjectural generating series of In particular, we propose the equivariant version of Lehn’s conjecture. And we also make some approach to higher rank case.When calculating, we relate the results after local-ization with Macdonald polynomials, and it is a new relationship between Hilbert schemes and symmetric functions.