Comparing \(K\)-theories for complex varieties (Q2762636)

Let \(X\) be smooth quasiprojective variety over \(\mathbb{C}\) and let \(KU^{-*}(X)= K^{-*}_{\text{top}} (X^{\text{an}})\) denote the topological \(K\)-theory of complex vector bundles on the underlying analytic space \(X(\mathbb{C})\), \(KU^* (X;\mathbb{Z}/m)\) the corresponding theory with \(\mathbb{Z}/m\) coefficients and \(K_* (X;\mathbb{Z}/m)\) the algebraic \(K\)-theory of \(X\) with \(\mathbb{Z}/m\) coefficients. Then the so-called Quillen-Lichtenbaum conjecture for \(X\) asserts that the canonical homomomorphism NEWLINE\[NEWLINE\rho_n(X): K_n(X;\mathbb{Z}/m) \to KU^{-n} (X;\mathbb{Z}/m)NEWLINE\]NEWLINE is an isomorphism for all \(n\geq\dim X-1\) and an injection for \(n=\dim X-2\). The above conjecture is now known to be true for any \(m\), thanks to recent results by \textit{V. Voevodsky} on the norm residue homomorphism. The case \(m=2\) may be found in \textit{C. Pedrini} and \textit{C. Weibel} [\(K\)-Theory 21, 356--385 (2000; Zbl 0999.19002)].NEWLINENEWLINENEWLINEThe paper essentially deals with the case of any complex quasiprojective variety \(X\): after recalling the definitions of a theory of ``semi-topological \(K\)-theory'' \(K_*^{\text{semi}}(X)\) previously defined by the same authors -- related to Friedlander-Lawson motivic cohomology \(L^*H^* (X)\) -- which has the property that the natural map \(K_*(X)\to K^{-*}_{\text{top}} (X^{\text{an}})\) factors through \(K_*^{\text{semi}}(X)\), they introduce a new theory \(K_*(\Delta^\bullet_{\text{top}}\times X)\) which is more accessible to computations than the previous one and coincides with \(K_*^{\text{semi}}(X)\), for \(X\) projective. This paper contains the following results:NEWLINENEWLINENEWLINEFor any \(m>0\) there exists a natural isomorphism for all quasiprojective varieties \(X\): NEWLINE\[NEWLINEK_* (X;\mathbb{Z}/m) \simeq K_*(\Delta^\bullet_{\text{top}} \times X;\mathbb{Z}/m).NEWLINE\]NEWLINE Moreover also the map: NEWLINE\[NEWLINEK_*^{\text{semi}} (X)[1/\beta]\to K^{-*}_{\text{top}} (X^{\text{an}})NEWLINE\]NEWLINE is an isomorphism, where \(\beta\in K_2^{\text{semi}}(\text{spec} \mathbb{C})\) is the Bott element. The above results fit together yielding an isomorphism: NEWLINE\[NEWLINEK_*(X;\mathbb{Z}/m)\to K_* ^{\text{semi}} (X;\mathbb{Z}/m).NEWLINE\]NEWLINE An interesting consequence of the above isomorphism is a new proof of the Quillen-Lichtenbaum conjecture for smooth complete curves.