Varieties over a finite field with trivial Chow group of 0-cycles have a rational point. (English) Zbl 1092.14010

Let \(X\) be a \(d\)-dimensional smooth projective variety over a field \(k\). The author investigates more deeply a result of S. Bloch: if the Chow group of 0-cycles \(\text{CH}_0(X\times_k\overline{k(X)})\) is \(\mathbb Z\), then there exists \(N\in{\mathbb N}\setminus\{0\}\), a 0-dimensional subscheme \(\xi\) of \(X\), a divisor \(D\subset X\), and a \(d\)-dimensional cycle \(\Gamma\subset X\times D\) such that \(N\cdot\Delta\equiv \xi\times X+\Gamma\), where \(\Delta\in\text{CH}^d(X\times_k X)\otimes_{\mathbb Z}\mathbb Q\) is the diagonal. This result has several consequences. In particular it follows that \(H^i(X,\mathcal{O}_X)=0\) for \(i\geq 1\). The author’s main purpose is to show that Bloch’s method still works fine when applied to the rigid cohomology of P. Berthelot.


14C25 Algebraic cycles
14F30 \(p\)-adic cohomology, crystalline cohomology
14G05 Rational points
14G15 Finite ground fields in algebraic geometry
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