If we replace projective variety with algebraic variety in the statement of the Weil conjectures what happens? To me it seems the statement still makes sense. But is it still true?



Weil Conjectures for nonprojective algebraic varieties
Correctly restated, the conjectures hold for any variety $V$ (not necessarily complete or nonsingular) over a finite field $k$.


Dwork proved that the zeta function $Z(V,t)$ of $V$ is a rational function of $t$.


Grothendieck (et al.) expressed $Z(V,t)$ as the alternating product of the characteristic
polynomials of the Frobenius map $F$ acting on the etale cohomology groups of $V$ with compact support.


Deligne showed (Weil II) that for each positive integer $i$ and each eigenvalue $a$ of $F$ acting on the $i$th etale cohomology group of $V$ with compact support, there exists an integer $j\leq i$ such that all the complex conjugates of $a$ have absolute value $q^{j/2}$ where $q=|k|$.


When $V$ is nonsingular and complete, these statements together with Poincare duality, give Weil's original conjectures.


