All regulators of flat bundles are torsion. (English) Zbl 0865.14003

In a previous paper [J. Differ. Geom. 43, No. 3, 674-692 (1996)], we proved the Bloch conjecture on rationality of secondary characteristic classes of flat rank-two vector bundles over a compact Kähler manifold, and announced the full higher-dimensional generalization, which we now state.
Theorem 1.1. Let \(X\) be a smooth complex projective variety. Let \(\rho:\pi_1(X)\to\text{SL}_n(\mathbb{C})\), \(n\geq 2\), be a representation of the fundamental group and let \(E_\rho\) be the corresponding holomorphic flat vector bundle over \(X\). For \(i\geq 2\) let \(c_i\in H_{\mathcal D}^{2i} (X,\mathbb{Z}(i))\) be the Chern class in the Deligne cohomology group of \(X\). Then \(c_i\) is a torsion class.
Theorem 1.1 follows from:
Main theorem. Let \(X\) be a compact Kähler manifold and let \(\rho\): \(\pi_1(X)\to \text{SL}_n ({\mathcal O}_S)\). Then the image under \(\widehat\rho_*\) of \(H_{2i-1} (X,\mathbb{Z})\) in \(H_{2i-1} (\text{BLS} ({\mathcal O}_S),\mathbb{Z})\) is torsion.
The strategy chosen in the present paper will be to use the refined version of J. H. Sampson’s theorem in Contemp. Math. 49, 125-134 (1986; Zbl 0605.58019) and the deformation theory of flat bundles over projective bases, Goldman-Millson and Simpson, for proving the following key result. Two different proofs will be given.
Theorem. Let \(X\) be a compact Kähler manifold and let \(\rho\) be as in 1.1. Then for any \(j\geq 2\) the volume regulator \(\widehat\rho^* (\text{Vol}_{2j-1}) \in H^{2j-1} (X,\mathbb{R})\) is zero.
For \(j=2\), this is proved in parts 4.3-4.5 of the author’s paper cited above. In the course of the proof we will rely on the geometrical description of the regulators, given in 3.1-3.3 of the same paper.


14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)


Zbl 0605.58019
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