Timeline for A Hodge substructure with ''nice' weight factors that does not correspond to a mixed submotif?
Current License: CC BY-SA 2.5
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| Nov 17, 2010 at 23:38 | comment | added | AFK | So you also have a surjection $Gr^W H_1(\mathfrak{u}_{MHS}) \to Gr^W H_1(\mathfrak{u}_{MM(F)})$ between abelianizations of their pro-unipotent radicals. And this induces the dual to the map between $Ext^1$'s as $\mathfrak{u}_{MM(F)} = \bigoplus_M Ext^1(\mathbb{Q},M)^\vee \otimes M$ where $M$ ranges over a class of representative of simple objects. | |
| Nov 17, 2010 at 23:38 | comment | added | AFK | If you assume that the Hodge realization is fully faithful and its image is stable by subojects, then the injectivity of $Ext^1_{MM(F)}(\mathbb{Q},M) \to Ext^1_{MHS}(\mathbb{Q}^H,M^H)$ for $M$ simple follows by the tannakian formalism. Indeed the assumption is equivalent to saying that you have a surjective map between the tannakian fundamental groups $MHS$ and $MM(F)$. | |
| Nov 17, 2010 at 23:02 | comment | added | Emerton | Sorry, it may be that what I suggested above about being an injection in general is nonsense; if so, ignore it. Hopefully I can get my thoughts straight and try to say something with more confidence (and more chance of being correct). | |
| Nov 17, 2010 at 22:55 | comment | added | Emerton | ... the abelian variety, and this is an injection. Might the more general case be similar?) | |
| Nov 17, 2010 at 22:55 | comment | added | Emerton | Dear Mikhail, Could you explain in more detail what you mean by having "a subobject for each weight factor"? Based on the question in your second sentence, I guess you don't mean submotives. By the way, are you sure the $Ext^1$s don't inject? (The Hodge conjecture says something like the realization map induces an isomorphism on Hom's, and then perhaps general nonsense implies that you have an injection on $Ext^1$s? At least in the case of $Ext^1$ of the trivial motive by the $H^1$ of an abelian variety, the realization map sends the Mordell--Weil group into the $\mathbb C$ points of ... | |
| Nov 17, 2010 at 22:07 | history | edited | Mikhail Bondarko | CC BY-SA 2.5 |
edited tags; edited title
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| Nov 17, 2010 at 21:57 | history | asked | Mikhail Bondarko | CC BY-SA 2.5 |