Serre's Analogue of the Weil Conjectures for Non-Compact Kahler Manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T13:17:37Z http://mathoverflow.net/feeds/question/77611 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77611/serres-analogue-of-the-weil-conjectures-for-non-compact-kahler-manifolds Serre's Analogue of the Weil Conjectures for Non-Compact Kahler Manifolds Abtan Massini 2011-10-09T17:03:32Z 2012-12-20T01:01:23Z <p>The classical Riemann Hypothesis concerns the locations of zeroes of the Riemann zeta-function, or more generally the Dedekind zeta-functions of number fields. Its analogue for varieties defined over finite fields is part of the Weil conjectures, which concerns the absolute values of the zeroes and poles of the zeta-functions of these varieties.</p> <p>The zeta-functions for varieties have analogues for complex manifolds. When a complex manifold is compact Kahler, the ablosute values of the zeroes and poles of its zeta-functions then satisfy similar properties as the zeta-functions for varieties. This theorem is due to Serre, and his proof is based on Hodge theory on compact Kahler manifolds. </p> <p>Does anyone know if this can be extended to non-compact Kahler manifolds?</p> http://mathoverflow.net/questions/77611/serres-analogue-of-the-weil-conjectures-for-non-compact-kahler-manifolds/116832#116832 Answer by Daniel Litt for Serre's Analogue of the Weil Conjectures for Non-Compact Kahler Manifolds Daniel Litt 2012-12-20T01:01:23Z 2012-12-20T01:01:23Z <p>Since the link in the comments is broken, here is a link to Serre's paper: <a href="http://www.jstor.org/stable/1970088?seq=2" rel="nofollow">Analogues Kahleriennes de Certaines Conjectures de Weil</a>. There seems to be some misunderstanding about what Serre actually proves implicit in the question, so I'll clarify it a bit, explain an easy analogue for some non-compact varieties, and indicate what should be true in general.</p> <p>First of all, Serre's result is <em>not</em> for arbitrary compact Kahler manifolds. Rather, he starts with a smooth complex projective variety $X$, and an endomorphism $f: X\to X$, with an ample divisor $E$ so that $f^{-1}(E)$ is algebraically equivalent to $qE$ for some integer <code>$q&gt;0$</code>. Then he shows that the eigenvalues of $f^*$ acting on $H^r(X, \mathbb{C})$ have absolute value $q^{r/2}$. Contrary to the statement of the question, the algebraicity of $X$ is built into the very result, since it requires the existence of an ample divisor.</p> <p>Serre doesn't explicitly define the zeta function of $(X, f)$, but by analogy to the Weil conjectures, one may define <code>$$Z(X, f, t)=\prod_i \det(1-f^*t\mid H^i(X, \mathbb{C}))^{(-1)^{i+1}}.$$</code> Then this zeta function satisfies the desired "Riemann hypothesis," by the result above.</p> <p>When looking for a non-compact analogue, the motto one should have in mind is that zeta functions should behave well under "cutting-and-pasting." For example, if $Z(X, t)$ is the zeta function from the Weil conjectures, where <code>$X/\mathbb{F}_q$</code> is an arbitrary variety, and <code>$Y\subset X$</code> is a closed subscheme, then <code>$$Z(X, t)=Z(X\setminus Y, t) \cdot Z(Y, t).$$</code></p> <p>So here's an analogue of this fact in the setting of smooth quasi-projective varieties. Suppose $X$ is a smooth projective variety over <code>$\mathbb{C}$</code>, and $f: X\to X$ is a self-map satisfying the conditions of Serre's result. Suppose <code>$Y\subset X$</code> is a smooth closed subvariety, so that <code>$f(Y)\subset Y$</code>. Then <code>$Y, f|_Y$</code> also satisfy the conditions of the theorem, and so <code>$Z(X, f, t)$</code> and <code>$Z(Y, f|_Y, t)$</code> satisfy the "Riemann Hypothesis." Now suppose <code>$f(X\setminus Y)\subset X\setminus Y$</code> as well, and <code>$f|_{X\setminus Y}$</code> is proper. Then we may define <code>$$Z(X\setminus Y, f|_{X\setminus Y}, t)=\prod_i \det(1-f|_{X\setminus Y}^*t\mid H^i_c(X\setminus Y, \mathbb{C}))^{(-1)^{i+1}}.$$</code></p> <p>Here $H^i_c(X, \mathbb{C})$ denotes the cohomology of $X$ with compact support. Does this zeta function satisfy some kind of Riemann hypothesis? Well, from the long exact sequence relating the compactly supported cohomology of $X\setminus Y$ to that of $X$ and $Y$, we have that $$Z(X\setminus Y, f|_{X\setminus Y}, t)=Z(X, f, t)/Z(Y, f|_Y, t).$$</p> <p>So certainly all of the poles and zeroes of $Z(X\setminus Y, f|_{X\setminus Y}, t)$ have absolute value $q^{-r/2}$ for some integer $r$. This gives a sort of "Riemann hypothesis" for quasi-projective varieties admitting a particularly nice compactification. What you'll notice immediately though is that the $r$ does not match up with the cohomological degree, as it does in the compact case---there is some "slippage" coming from the boundary map in the long exact sequence. Rather, the $r$ is related to the "weight filtration" on the cohomology of $X\setminus Y$.</p> <p>Now suppose $U$ is an arbitrary quasiprojective variety, and $f: U\to U$ is a proper map. $U$ might not have the ridiculously nice compactification we need to run the above argument, but the zeta function defined using cohomology with compact support still makes sense. I doubt that it will in general satisfy a "Riemann hypothesis" of the type you're looking for, since the condition on ample divisors may not makes sense. What is true, though, is that if $U$ admits a smooth compactification $X$ so that $f$ extends to a map $g: X\to X$ satisfying the conditions of Serre's result, so that $g(X\setminus U)\subset X\setminus U$, (where $X\setminus U$ is not necessarily smooth!) this zeta function will satisfy a "Riemann hypothesis." Namely, the zeroes and poles of $Z(U, f, t)$ will be of the form $q^{-r/2}$, with the $r$ coming from the weight filtration on the compactly supported cohomology of $U$.</p> <p>To see this, one may imitate Serre's argument using the mixed Hodge structure on the cohomology of $X\setminus Z$. I don't immediately see how to give an analogue of Serre's condition on the existence of the ample divisor $E$ for $U$ without reference to a compactification, though there should be a way to do so; then one should be able to imitate Serre's argument, just working with the mixed Hodge structure on the compactly supported cohomology of $U$.</p> <p><code></code></p>