Timeline for Hodge structures generated by cohomology groups of varities with dimension less than $n$
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 29, 2015 at 8:35 | comment | added | Mostafa - Free Palestine | @MikhailBondarko thanks! You have completely resolved my problem. | |
| Jan 29, 2015 at 8:13 | comment | added | Mikhail Bondarko | So, $H^{n-1}(X,Q)$ is a substructure in the corresponding cohomology of a smooth hyperplane section of $X$. Now you should just recall that the category of polarizable (pure) Hodge structures is Abelian semi-simple. | |
| Jan 29, 2015 at 8:10 | comment | added | Mostafa - Free Palestine | @abx I mean it is a subquotient of a direct sum of Hodge structures of cohomology groups of curves. I think lefschetz hyperplane implies the above claim for $i< n-1$ but I am curious about the case $i=n-1$ | |
| Jan 29, 2015 at 7:57 | comment | added | abx | I am not sure I understand the question. Let us take $n=2$, $i=1$, so $X$ is a surface; what do you mean by "the Hodge structure on $H^1(X,\mathbb{Q})$ is generated by Hodge structures of curves"? | |
| Jan 29, 2015 at 6:05 | comment | added | Venkataramana | This will follow from Lefschetz hyperplane section theorem | |
| Jan 28, 2015 at 21:28 | comment | added | dhy | Not for $n=2$ and $i=1$; take $X$ the variety of lines on a smooth cubic threefold, then this is basically the main result of the Clemens-Griffiths proof of irrationality of smooth cubic threefolds. My guess is that for $i$ at least $2$ this is a hard question. | |
| Jan 28, 2015 at 21:11 | history | asked | Mostafa - Free Palestine | CC BY-SA 3.0 |