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A relation between the parabolic Chern characters of the de Rham bundles. (English) Zbl 1132.14006
Let \(\pi:X_U \to U\) be a smooth projective morphism of relative dimension \(n\) between nonsingular varieties. The \(i^{th}\) hypercohomology bundle of the complex \(\Omega^{\bullet}(X_U/U)\), denoted by \({\mathcal H}^i:= R^i\pi_*\Omega^{\bullet}(X_U/U)\), is equipped with the flat Gauss-Manin connection \(\nabla\). The pair \(({\mathcal H}^i, \nabla)\) is called the de Rham bundle or Gauss-Manin bundle of weight \(i\). Let \(S\) denote a nonsingular compactification of \(U\) such that \(D=S-U\) is a normal crossing divisor. The authors consider parabolic bundles \(\overline{\mathcal H}^i(X_U/U))\) on \(S\) associated to the logarithmic connection instead of the canonical extension on \(S\) studied by H. Esnault, S. Bloch and others [Am. J. Math. 119, No. 4, 903–952 (1997; Zbl 0928.14009); Ann. Math. (2) 151, No. 3, 1025–1070 (2000; Zbl 0985.14005)]. While the monodromy of the local system ker \(\nabla\) in the canonical case is unipotent (with nilpotent residues), in the parabolic case the monodromy is quasi-unipotent and the residues have rational eigenvalues.
Theorem 1. The Chern character (in the rational Chow group) of the alternating sum of parabolic bundles \(\sum^{2n}_{i=0} (-1)^i \mathrm{ch}(\overline{\mathcal H}^i(X_U/U))\) lies in \(\mathrm{CH}^0(S)_{\mathbb Q}\) (or equivalently, the pieces in all of the positive-codimension Chow groups with rational coefficients vanish).
In fact the theorem is proved in greater generality in case \(U\) is the open set over which \(\pi\) is topologically a fibration. As a corollary, it is shown that if \(\pi\) is a family of surfaces and \(S\) a good compactification (i.e. \(D\) is smooth) then \(\mathrm{ch}(\overline{\mathcal H}^i(X_U/U))\in \mathrm{CH}^0(S)\) for each \(i\geq 0\). In the case of canonical extensions, an analogue of Theorem 1 was conjectured by Esnault and is known to be true in many cases.

MSC:
14C25 Algebraic cycles
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14D20 Algebraic moduli problems, moduli of vector bundles
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