## 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.

### MSC:

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

### Keywords:

Chow group; rational point; finite fields
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