Poincaré invariants

Abstract

We construct an obstruction theory for relative Hilbert schemes in the sense of [K. Behrend, B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1) (1997) 45–88] and compute it explicitly for relative Hilbert schemes of divisors on smooth projective varieties. In the special case of curves on a surface $V$, our obstruction theory determines a virtual fundamental class $\left[\left[{\mathrm{Hilb}}_V^m\right]\right]\in A_{\frac{m(m-k)}{2}}\left({\mathrm{Hilb}}_V^m\right)$, which we use to define Poincaré invariants $(P_V^{+},P_V^{-}):H^2(V,Z)⟶{\Lambda }^{\ast }{H}^1(V,Z)\times {\Lambda }^{\ast }{H}^1(V,Z)\text{.}$ These maps are invariant under deformations, satisfy a blow-up formula, and a wall crossing formula for surfaces with $p_g\left(V\right)=0$. For the case $q\left(V\right)\ge 1$, we calculate the wall crossing formula explicitly in terms of fundamental classes of certain Brill–Noether loci for curves. We determine the invariants completely for ruled surfaces, and rederive from this classical results of Nagata and Lange. The invariant $(P_V^{+},P_V^{-})$ of an elliptic fibration is computed in terms of its multiple fibers.

When the fibered product ${\mathrm{Hilb}}_V^m{\times }_{{\mathrm{Pic}}_V^m}{\mathrm{Hilb}}_V^{k-m}$ is empty, there exists a more geometric obstruction theory, which gives rise to a second virtual fundamental class $\left\{{\mathrm{Hilb}}_V^m\right\}\in A_{\frac{m(m-k)}{2}+p_g\left(V\right)}\left({\mathrm{Hilb}}_V^m\right)$. We show that $\left\{{\mathrm{Hilb}}_V^m\right\}=\left[\left[{\mathrm{Hilb}}_V^m\right]\right]$ when $p_g\left(V\right)=0$, and use the second obstruction theory to prove that $\left[\left[{\mathrm{Hilb}}_V^m\right]\right]=0$ when $p_g\left(V\right)>0$ and ${\mathrm{Hilb}}_V^m{\times }_{{\mathrm{Pic}}_V^m}{\mathrm{Hilb}}_V^{k-m}=0̸$.

We conjecture that our Poincaré invariants coincide with the full Seiberg–Witten invariants of [Ch. Okonek, A. Teleman, Seiberg–Witten invariants for manifolds with $b_{+}=1$, and the universal wall crossing formula, Internat. J. Math. 7 (6) (1996) 811–832] computed with respect to the canonical orientation data. The main evidence for this conjecture is based on the existence of an Kobayashi–Hitchin isomorphism which identifies the moduli spaces of monopoles with the corresponding Hilbert schemes. We expect this isomorphism to identify also the corresponding virtual fundamental classes. This more conceptual conjecture is true in the smooth case. Using the blow-up formula, the wall crossing formula, and a case by case analysis for surfaces of Kodaira dimension less than 2, we are able to reduce our conjecture to the following assertion: $deg\left[\left[{\mathrm{Hilb}}_V^m\right]\right]={(-1)}^{\chi \left(O_V\right)}$ for minimal surfaces $V$ of general type with $p_g\left(V\right)>0$ and $q\left(V\right)>0$.