A regulator formula for Milnor \(K\)-groups
Let \(X\) be a projective variety, \(\dim X=d\). In the paper under review the author constructs some new formulas for the regulator maps from Milnor \(K\)-groups \(K_n^M( \mathbb{C}(X))\) to Deligne cohomology \(H^n_\mathcal{D}(\eta_X, \mathbb{Z}(n))\) which allows to make these maps more accessible to computations in many interesting cases. The first part of the paper is devoted to geometric motivation and construction of cycle-class and Abel-Jacobi maps. Let to an element \({f}= \sum m_j f_{1j}\otimes \cdots \otimes f_{nj} \in \otimes^n\mathbb{Z}\{\mathbb{C}(X)\}\) correspond the subvariety \(V_{{f}}:=\bigcup_{ij}| (f_{ij})| \) and the ``graph cycle'' \[ \gamma_{ {f}}=\big[\sum m_j ( \text{id}_X; f_{1j}, \ldots, f_{nj})_*(X\setminus V_{ {f}}))\big]\cap ((X\setminus V_{ {f}}) \times \square^n), \] where \(\square^n:=( \mathbb{P}^1_\mathbb{C}\setminus \{1\})^n\), thus one gets an isomorphism \(K_n^M( \mathbb{C}(X)) \overset{\cong}{\to} CH^n(\eta_X,n)\). On the other hand, the map \( {f}\to \partial {f}\) induces the map \( \text{Tame}: K_n^M( \mathbb{C}(X)) \to \coprod_{x\in X^1}K_n^M( \mathbb{C}(x))\). Further, the author defines the Abel-Jacobi map \({AJ}\) on graphs which takes graphs into functionals on forms \(\omega\in \Gamma_{X^\infty,c}^{2d-n+1}\), and shows that \({AJ}(\gamma_{f})= \int_X R'_{f}\wedge \omega\) for some \(n-1\)-current \(R'_{f}\). It turns out (Proposition 4.4) that if \(\text{Ker}\text{Tame}=0\) or \(n>d\), then the rule \({f} \mapsto R'_{f}\) induces a well defined map \(R':\;K_n^M( \mathbb{C}(X)) \to H^{n-1}(\eta_X, \mathbb{C}/\mathbb{Q}(n))\). The second part of the paper is devoted to employ a local-global spectral sequence to describing the residues of the Milnor currents. Here the author constructs the higher residue maps of the Milnor currents and the maps \(\text{Tame}^i\), generalizing the map \(\text{Tame}\) from Part I, investigates the relationships between these maps and defines the group \(K_n^M(X)=\bigcap_i \text{ker}(\text{Tame}^i)\). The restriction \[ R: K_n^M(X) \to \text{im}(H^{n-1}(X, \mathbb{C}/\mathbb{Q}(n)) \to H^{n-1}(\eta_X, \mathbb{C}/\mathbb{Q}(n))) \] of the Milnor regulator gives the holomorphic Milnor regulator \(R\) which is related to the arithmetic on \(X\), and is a correct generalization to higher \(n (>d)\) of Bloch's construction for \(n=2, d=1\). Part 3 of the paper is devoted to the study of regulator maps in families. Here the author proves a rigidity theorem (Theorem 7.1) which asserts that for a smooth \((n-1)\)-dimensional family \(X_s\) and for \( \{{f}_s\}\in \text{ker}( \text{Tame}) \subseteq K_n^M( \mathbb{C}(X))\), \( \nabla [R_{ {f}_s}]=0\) unless \(X_s\) are elliptic curves. Another important result is the following vanishing theorem (Theorem 8.3): Let \(X_0 \subset \mathbb{P}^{n+r}\) be a very general smooth complete intersection of multidegree \((D_0,\ldots, D_r)\), where if \(n=2\) then \(\sum D_j\neq n+r+1\). Then the image of the holomorphic Milnor regulator \( R: K_n^M(X) \to \underline{H^{n-1}}(\eta_X, \mathbb{C}/\mathbb{Q}(n))\) is zero. Finally, in Part 4 a demonstration of a Milnor regulator computation for a degenerate elliptic curve is given. The paper contains also many remarks, motivations, questions, geometric interpretations and examples of the considered concepts.
scientific article; zbMATH DE number 2037060