Timeline for cohomology of configuration space of punctured variety
Current License: CC BY-SA 3.0
12 events
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| Sep 28, 2016 at 17:20 | comment | added | Cepu | @DanPetersen Hi, do you think that is possible that the dimension of $H^{2}(F(X,n))$ is $5\binom n 2$? I do my calculation using Totaro's paper but I'm not sure. | |
| Jul 15, 2016 at 20:11 | comment | added | Alex Suciu | Indeed, the cdga that Totaro describes computes the (rational) cohomology of $F(X,n)$. But why is it a model for the configuration space? That requires a separate (and non-trivial) argument. | |
| Jul 15, 2016 at 18:45 | vote | accept | Cepu | ||
| Jul 15, 2016 at 16:31 | comment | added | Dan Petersen | A useful reference for you could be my paper "The structure of the tautological ring in genus one", specifically the final section, which does some very explicit computations about the cohomology of $F(X,n)$ where $X$ is a punctured elliptic curve. | |
| Jul 15, 2016 at 16:29 | comment | added | Dan Petersen | BTW - note that $H^2(X^n)$ is isomorphic to $\binom n 2$ copies of $V \otimes V$, and it's easy to see that the class of the symplectic form in each of these copies of $V \otimes V$ goes to zero in $H^2(F(X,n))$. Indeed, the "$(i,j)$th" copy of the symplectic form is exactly the class of the diagonal $\Delta_{ij}$ in $X^n$, so of course it goes to zero in the cohomology of $F(X,n)$ where we have removed all the diagonals. | |
| Jul 15, 2016 at 16:27 | comment | added | Dan Petersen | Not exactly. If $V$ is generated by $a,b$ then $V \otimes V$ has the subspace $V_{0,2}$ (the symmetric square) gen'd by $a \otimes a$, $b \otimes b$, and $a \otimes b + b \otimes a$. The exterior square is spanned by $a \otimes b - b \otimes a$, which is also the class of the symplectic form, so $V_{1,1}=0$. | |
| Jul 15, 2016 at 13:02 | comment | added | Cepu | ups $a\wedge b$ is symplectic, hence $V_{1,1}=0$ in this case. | |
| Jul 15, 2016 at 12:14 | comment | added | Cepu | Thanks, I have some difficult to visualize $V_{1,1}$. Here an example. assume that $X$ is a punctured elliptic curve (i.e. genus 1). Then $V$ is generated by $a,b$; $V_{0,2}$ is generated by $a^2, b^2, ab$ and $V_{1,1}=a \wedge b$. Is that correct? | |
| Jul 15, 2016 at 9:37 | history | edited | Dan Petersen | CC BY-SA 3.0 |
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| Jul 15, 2016 at 6:19 | history | edited | Dan Petersen | CC BY-SA 3.0 |
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| Jul 15, 2016 at 6:04 | history | edited | Dan Petersen | CC BY-SA 3.0 |
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| Jul 15, 2016 at 5:58 | history | answered | Dan Petersen | CC BY-SA 3.0 |