Beilinson regulator

Definition

Let $X$ be a smooth quasi-projective variety over $\C$, let $K_n(X)$ be Quillen's algebraic K-group, and let $\psi^k$ be the Adams operations on $K_n(X) \otimes \Q$. For integers $n \ge 0$ and $i$, let $$K_n(X)^{(i)}_\Q = \{ x \in K_n(X) \otimes \Q : \psi^k(x) = k^i x \text{ for all } k \ge 1 \}$$ be the weight $i$ eigenspace. Let $H^{j}_{\mathcal{D}}(X, \R(i))$ be the real Deligne–Beilinson cohomology of $X$, defined with differential forms of logarithmic growth along a smooth compactification. The Beilinson regulator is the family of homomorphisms $$r_{\mathcal{D}} \colon K_n(X)^{(i)}_\Q \to H^{2i-n}_{\mathcal{D}}(X, \R(i))$$ induced by Beilinson's Chern character from algebraic K-theory to Deligne–Beilinson cohomology. For a variety over $\R$ or over $\Q$, one first passes to $\C$. Then one takes invariants under complex conjugation, acting on both $X(\C)$ and the coefficients.