[For the entire collection see Zbl 0683.00009.]
The purpose of this paper is to outline a construction of a non-abelian candidate for Abel-Jacobi theory in higher dimensions. Let L be a nef and big divisor on a surface X. To each integer \(d\geq 1\) can be associated the following objects:
(1) a stratification \(X^{(d)} (=0\)-dimensional subschemes of length \(d)\supset \Gamma^ 0_ d\supset \Gamma^ 1_ d\supset..\). induced by the incidence cycle \(Z_ d\subset X^{(d)}\times X;\)
(2) a sequence \(Y^ r_ d=P(Ext^ 1) (= \)torsion free sheaves whose Chern invariants are \({\mathcal O}_ X(L)\) and \(d\)), \(\pi: Y^ r_ d\to \Gamma^ r_ d,\) \(r=0,1,...;\)
(3) a natural map \(\alpha: Y^ r_ d\to F\), where F is an appropriate flag variety in \(H^ 0(K_ X+L).\)
Let \(\pi: Y=Y^ r_ d\to \Gamma =\Gamma^ r_ d\), and let \(Z\subset \Gamma \times X\) be the incidence cycle. There is a torsion free sheaf \(\tilde T_ Y=T_ Y\oplus {\mathcal O}_ Y\), and a divisor \(D_{Y/\Gamma}\) on \(\tilde Y=P(\tilde T_ Y)\), which define a polarization \((Y,D_{Y/\Gamma})\) for \((\alpha,Y,F)\), such that:
(5) the image \(\tilde Z'\) of the natural canonical map \(k: Z=Z\times_{\Gamma}Y\to P(\tilde T^*_ Y)\) is determined by the polarization \((\tilde Y,D_{Y/\Gamma})\) (theorem 1);
(6) if k is generically “1 to 1” then \((\tilde Y,D_{_ Y/\Gamma})\) determines the incidence cycle Z on \(\Gamma\times Y.\)
The analogy is extended to the infinitesimal variation of Hodge structure (IVHS). The triple \((\alpha,Y,F)\) defines: (i) a filtration \(M_ Y=(M^ 0\supset M^ 1\supset...)\) of sheaves; (ii) natural maps \(p^ 0\) and p; in particular, let U be a Zariski open subset of Y, then \(p: \Theta_{Y/\Gamma}\otimes {\mathcal O}_ U\to \oplus Hom(M^{\ell}/M^{\ell +1},M^{\ell -1}/M^{\ell}).\) The data \((T_ Y,M_ Y,p^ 0,p)\) is the infinitesimal polarization of \((\alpha,Y,F)\); moreover, the morphism p coincides with the differential of the period map \(\alpha\) (see remark 1.10).
The introduction of the infinitesimal polarization is in accordance with the Griffiths philosophy: an IVHS type data should contain the geometry of the underlying geometric object. An illustration of this principle is theorem 2, which can be viewed as the infinitesimal Torelli for 0-cycles on the surface X.