Macdonald formula for curves with planar singularities
-
Davesh Maulik
and
Zhiwei Yun
Abstract
We generalize Macdonald's formula for the cohomology of Hilbert schemes of points on a curve from smooth curves to curves with planar singularities: we relate the cohomology of the Hilbert schemes to the cohomology of the compactified Jacobian of the curve. The new formula is a consequence of a stronger identity between certain perverse sheaves defined by a family of curves satisfying mild conditions. The proof makes essential use of Ngô's support theorem for compactified Jacobians and generalizes this theorem to the relative Hilbert scheme of such families. As a consequence, we give a cohomological interpretation of the numerator of the Hilbert-zeta function of curves with planar singularities.
The authors would like to thank A. Oblomkov for bringing this problem to their attention and V. Shende for helpful discussions. They are also grateful to an anonymous referee for helpful suggestions. D.M. is supported by the Clay Research Fellowship. Z.Y. is partially supported by the NSF grant DMS-0969470.
© 2014 by De Gruyter
Articles in the same Issue
- Frontmatter
- The Oseen–Navier–Stokes flow in the exterior of a rotating obstacle: The non-autonomous case
- Macdonald formula for curves with planar singularities
- Geometry of orbit spaces of proper Lie groupoids
- Lefschetz trace formula for open adic spaces
- Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K-theory
- Almost harmonic Maass forms and Kac–Wakimoto characters
- Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space
- Unique Cartan decomposition for II1 factors arising from arbitrary actions of hyperbolic groups
- Addendum: Hybrid bounds for twisted L-functions
Articles in the same Issue
- Frontmatter
- The Oseen–Navier–Stokes flow in the exterior of a rotating obstacle: The non-autonomous case
- Macdonald formula for curves with planar singularities
- Geometry of orbit spaces of proper Lie groupoids
- Lefschetz trace formula for open adic spaces
- Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K-theory
- Almost harmonic Maass forms and Kac–Wakimoto characters
- Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space
- Unique Cartan decomposition for II1 factors arising from arbitrary actions of hyperbolic groups
- Addendum: Hybrid bounds for twisted L-functions
