Mixed structures on Hom spaces induced by mixed sheaves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T08:28:24Z http://mathoverflow.net/feeds/question/108984 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108984/mixed-structures-on-hom-spaces-induced-by-mixed-sheaves Mixed structures on Hom spaces induced by mixed sheaves Reladenine Vakalwe 2012-10-06T06:38:35Z 2012-10-08T16:06:52Z <p>Let $D^b_m(X)$ (resp $D^b(X)$) denote the derived category of mixed Hodge modules (resp. constructible sheaves) on a complex variety $X$. Let </p> <p>$rat\colon D^b_m(X)\to D^b(X)$</p> <p>be the forgetful' functor. This is t-exact for the perverse t-structure on the right. Write $MHM(X)$ for the abelian category of mixed Hodge modules on $X$. Then $MHM(pt)$ is the category of graded polarizable mixed Hodge structures, and $rat\colon MHM(pt) \to VectorSpaces$ is the evident forgetful functor. </p> <p>Now let $M,N\in D^b_m(X)$. Set </p> <p>$\mathcal{H}om(M,N) = \Delta^!(\mathbb{D}M \boxtimes N)$, </p> <p>where $\Delta\colon X\to X\times X$ is the diagonal map, and $\mathbb{D}$ is Verdier duality. </p> <p>Let $a\colon X \to pt$ be the evident map. Then</p> <p><code>$rat ( H^0(a_*\mathcal{H}om(M,N))) = H^0(rat(a_*\mathcal{H}om(M,N))) = Hom(rat(M), rat(N))$</code></p> <p>and in this way we get a Hodge structure on $Hom(rat(M),rat(N))$. All functors are derived.</p> <p><b>My question: </b> If $M,N$ are <s>pure</s> <b>pointwise pure (see Geordie Williamson's comment below)</b>, then is the induced structure on $Hom(rat(M), rat(N))$ pure?</p> <p>My gut answer is no (even if $X$ is complete, the $\Delta^!$ should be messing weights up), but it would make me happier if the answer is yes!</p> <p>If the answer is no, under what additional conditions (other than requiring $X$ to be smooth and complete plus $M,N$ being the constant' sheaf) can the answer be converted to yes? </p> <p>I guess one could also ask the same sort of question for mixed $\ell$-adic sheaves. But I am even less familiar with that setting.</p>