zbMATH — the first resource for mathematics

Regulators of Chow cycles on Calabi-Yau varieties. (English) Zbl 1057.14017
Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3355-3/hbk). Fields Inst. Commun. 38, 87-117 (2003).
For a projective manifold \(X\), one has various maps \(\text{cl}_{k,m}: \text{CH}^k(X, m)\to H^{2k- m}_{\mathcal D}(X,\mathbb{A}(k)),\) called regulators, from the Bloch higher Chow group to Deligne cohomology, where \(\mathbb{A}\) is a subring of \(\mathbb{R}\), \(\mathbb{A}(k)= \mathbb{A}(2\pi\sqrt{-1})^k\). In this paper the author gives explicit formulas for the regulators. In particular, he obtains a formula for the regulator \(\text{cl}_{k,2}: \text{CH}^k(X, 2)\to H^{2k- 2}_{\mathcal D}(X, \mathbb{Z}(k))\) in terms of Milnor \(K\)-theory. He also gives some evidence in support of the Hodge-\({\mathcal D}\)-conjecture for general \(K3\) surfaces.
For the entire collection see [Zbl 1022.00014].

14C25 Algebraic cycles
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
19E20 Relations of \(K\)-theory with cohomology theories