Some new invariants of vector bundles on smooth projective surfaces

Abstract

Let $\mathcal{E}$ be a rank-2 bundle over a smooth complex projective surface X . Whenever the subscheme of zeros of a global section of $\mathcal{E}$ is zero-dimensional, it gives a geometric realization of the second Chern class $c_2(\mathcal{E})$ . Taking the incidence correspondence \[\xymatrix{{\mathcal{Z} =\{(x,[e]) \in X\times \bf{P} (H^0(\mathcal{E}))\mid e(x)= 0\}} \ar[r]^>>>>>{p_2}& {\bf{P} (H^0(\mathcal{E}))}}\] and considering the Zariski open subset $U_f \subset \bf{P} (H^0(\mathcal{E}))$ over which the morphism p 2 is finite, we have \[\mathcal{Z}_f = p^{-1}_2(U_f)\longrightarrow U_f,\] a family of zero-dimensional subschemes of X of length $\deg (c_2(\mathcal{E}))$ . This can be viewed as a distinguished geometric representative of $c_2(\mathcal{E})$ . We define a new invariant of $\mathcal{E}$ which can be viewed as a ‘lifting’ of $c_2(\mathcal{E})$ to U f . This invariant is a sequence of sections of some coherent sheaves on U f . These sheaves are built from the following cohomology cup-products: \begin{align}\gamma_1 &: H^1 (\mathcal{E}^{\ast}) \otimes H^0 (\mathcal{E}) \longrightarrow H^1 (\mathcal{O}_X), \gamma_2&: S^2 H^1 (\mathcal{E}^{\ast})\longrightarrow H^2 (\mathcal{O}_X (\det(\mathcal{E}^{\ast})) ).\end{align} The main property of our invariant is as follows: either it determines the family $\mathcal{Z}_f \stackrel{p_2}{\longrightarrow} U_f$ , or the vector bundle $\mathcal{E}$ is ‘special’. The speciality is expressed in terms of the special geometry of zero-loci of global sections of $\mathcal{E}$ and the special geometry of X . The sequence of sections entering the definition of our invariant is obtained by starting with the one defined by the cup-product $\gamma_2$ and deriving others inductively by using a geometric interpretation of a part of the cup-product $\gamma_1$ together with the Grothendieck residue map. So one can view our invariant as $\gamma_2$ together with some kind of higher-order cohomology cup-products. The emergence of these higher-order cohomology cup-products is explained conceptually as higher-order derivatives of a natural deformation of $\gamma_2$ associated to a certain ‘natural’ deformation of the complex structure on $\mathcal{E}$ . The variety of these natural deformations of $\mathcal{E}$ has all the features of the classical Jacobian of curves: it carries a distinguished divisor which either determines the family $\mathcal{Z}_f \stackrel{p_2}{\longrightarrow} U_f$ or ‘sees’ that $\mathcal{E}$ is special in the aforementioned sense. An essentially new feature of this Jacobian of $\mathcal{E}$ is that it also carries a variation of Hodge-like structures which arises naturally from our invariant.