Exact formulas for invariants of Hilbert schemes

A theorem of Göttsche establishes a connection between cohomological invariants of a complex projective surface S and corresponding invariants of the Hilbert scheme of n points on S . This relationship is encoded in certain infinite product q -series which are essentially modular forms. Here we make use of the circle method to arrive at exact formulas for certain specializations of these q -series, yielding convergent series for the signature and Euler characteristic of these Hilbert schemes. We also analyze the asymptotic and distributional properties of the q -series’ coefficients.