The geometry of the parabolic Hilbert schemes

Let X be a smooth projective surface and D be a smooth divisor over an algebraically closed field k. In this paper, we discuss the moduli schemes of the ideals of points of X with parabolic structures at D. They are called parabolic Hilbert schemes. The first result is that the parabolic Hilbert schemes are smooth. And then some of the studies of Ellingsrud-Strømme, Göttsche, Cheah, Nakajima and Grojnowski on the Hilbert schemes can be naturally generalized in the parabolic case. We determine the class of the parabolic Hilbert schemes in the Grothendieck ring of k-varieties: The class is described in terms of products of the symmetric powers of X and D and the affine spaces. Thus we obtain a formula for the generating functions of the E-polynomials or the Poincaré polynomials of the parabolic Hilbert schemes of a smooth projective surface X and a divisor D over the complex number field C. Moreover we obtain the extension of the Nakajima-Grojnowski theory for the parabolic Hilbert schemes: i.e., we introduce the incidence varieties which induce the operations on the cohomology groups of the parabolic Hilbert schemes. We see that the Heisenberg relations holds. Thus we obtain a representation of Heisenberg algebra.