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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].

##### MSC:
 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
##### Keywords:
Chow group; Deligne cohomology