0
$\begingroup$

Let $S$ be a compact algebraic surface with du Val singularities and $T$ be the minimal resolution of it. Is it true that $$ H_2(T,\mathbb{Z})/(\oplus_{i}\mathbb{Z}E_i)\cong H_2(S,\mathbb{Z}), $$ where $E_i$s are the exceptional curves of the resolution? Or under what condition does it hold?

The claim is true for the local case. That is, let $G \subset SL(2,\mathbb{Z})$ be a finite group and $X$ be the minimal resolution of $\mathbb{C}^2/G$. Then we have $$ H_2(X,\mathbb{Z})=\oplus_{i}\mathbb{Z}E_i $$ where $E_i$s are again the exceptional curves of the resolution. In other words, $E_i$s form a $\mathbb{Z}$-basis of $H_2(X,\mathbb{Z})$.

I think my problem is well-known and I would appreciate any help. I would also appreciate it if someone could teach me how other (co)homology groups change under minimal resolutions.

Improve this question
$\endgroup$
4
  • $\begingroup$ You mean $H_2(T,\mathbb{Z})=H_2(S,\mathbb{Z})\oplus(\oplus_{i}\mathbb{Z}E_i) $. But how do you see $H_2(S,\mathbb{Z})$ into $H_2(T,\mathbb{Z})$? Don't you want to look at cohomology instead? $\endgroup$
    – abx
    Commented Dec 6, 2013 at 9:05
  • $\begingroup$ Thank you for the comment. I fixed the problem. I would rather study homology of $S$. I think the Poince duality fails on $S$, so I use the quotient as above. $\endgroup$
    – Nail Haber
    Commented Dec 6, 2013 at 9:21
  • $\begingroup$ Did you try to write down Mayer Vietoris sequences for both $S$ and $T$ (choosing small tubular neighbourhoods of the exceptional fibers of the minimal resolution) ? It seems to me that it can work. I can add that the $H_1$ in unchanged ; in fact the $\pi_1$ itself is preserved for this kind of singularities (you can have a look at this post mathoverflow.net/questions/50580/…) $\endgroup$
    – Benoit
    Commented Dec 6, 2013 at 10:52
  • $\begingroup$ @Benoit I miscalculated the Mayer Vietoris sequences. You are right and it in particular works for $H_2$. Thanks! $\endgroup$
    – Nail Haber
    Commented Dec 6, 2013 at 16:28

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .