On polarized (pure) Hodge structures - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:48:25Z http://mathoverflow.net/feeds/question/42006 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42006/on-polarized-pure-hodge-structures On polarized (pure) Hodge structures Mikhail Bondarko 2010-10-13T10:16:31Z 2010-10-13T12:04:32Z <p>Some simple questions, for which I know no precise reference (and would be deeply grateful for a nice one!):</p> <ol> <li><p>Is it true that the category of (pure) polarized Hodge structures is abelian semi-simple, whereas the whole category of pure Hodge structures is not? </p></li> <li><p>Should one only consider those morphisms of polarized Hodge structures that respect polarizations in order to obtain an abelian category?</p></li> <li><p>Is it true that all pure Hodge structures 'that come from geometry' (for example, the graded pieces of the weight filtration of the singular cohomology of varieties and motives) are polarized?</p></li> </ol> http://mathoverflow.net/questions/42006/on-polarized-pure-hodge-structures/42008#42008 Answer by Donu Arapura for On polarized (pure) Hodge structures Donu Arapura 2010-10-13T11:29:41Z 2010-10-13T12:04:32Z <p>Fortunately, these questions are easy to answer. First of all, it helps to distinguish between polarizable Hodge structures and polarized structures. For polarizable, we merely require that a polarization exists, but it is not fixed. Let Hodge structure mean pure rational Hodge structure below. Then </p> <ul> <li><p>The category of polarizable pure Hodge structures is abelian and semisimple (morphisms are not required to respect polarizations). This is essentially proved in Theorie de Hodge II.</p></li> <li><p>The category of arbitrary Hodge structures is abelian but not semisimple. To see the nonsemisimplicity, we can use a theorem of Oort-Zarhin [Endmorphism algebras of complex tori, Math Ann 1995], that any finite dimensional \$\mathbb{Q}\$-algebra is the endomorphism algebra of some complex torus, and therefore of some Hodge structure.</p></li> <li><p>All pure Hodge structures of geometric origin are polarizable. So this is a very reasonable condition to impose. For \$Gr_WH^*(X)\$, this is explained for example in Beilinson's <em>Notes on absolute Hodge cohomology</em> (although this was already implicit in Deligne's construction).</p></li> </ul>