On the dilogarithmic cycle class map (Q702909)

Let \(X\) be a smooth projective surface over \(\mathbb{C}\). It is known by a famous result of Mumford that, if \(p_g(X)= \dim H^0(X,\Omega^2_X)\neq 0\), then the group CH\(^2(X)_0\) of zero cycles of degree \(0\) contains a ``big'' Albanese kernel \(T(X)\). In particular \(T(X)\) contains uncountable many cycles (which are non torsion due to a theorem of Roitmann). One possible way to detect such elements in \(T(X)\) is the one suggested by Bloch: he defines an analytic complex of sheaves \({\mathcal B}_X(2)\) such that the corresponding hypercomology \(B_2(X)= \mathbb{H}^4(X_{\text{an}},{\mathcal B}_X(2))\) is equipped with a cycle map CH\(^2(X)@>{\text{cl}_{\text{di}}}>>B_2(X)\), called the dilogarithmic cycle class map, wich commutes with the cycle map CH\(^2(X)@>{\text{cl}_{\mathcal D}}>> H^4_{\mathcal D}(X,\mathbb{Z}(2))\) to Deligne cohomology. In this paper the author proves the following result. Theorem. Assume that there exists an abelian variety \(A\) and a correspondence \(\Gamma\in \text{CH}^2(X\times A)\), such that the induced map \(\Gamma_*: H^0(X, \Omega^2_X)\to H^0(A, \Omega^2_A)\) is not zero. Then \(\text{Ker\,cl}_{\text{di}}\) contains uncountably many non torsion cycles. In the case \(X= E\times E\) where \(E\) is an elliptic curve then \(\text{Ker\,cl}_{\text{di}}= T(X)\).