Timeline for Are there symplectic 4-folds with $b_+>1$, $b_-=0$?
Current License: CC BY-SA 2.5
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| Jan 2, 2011 at 19:13 | history | edited | Tim Perutz | CC BY-SA 2.5 |
fixed statement of symplectic BMY
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| Jan 2, 2011 at 19:11 | comment | added | Tim Perutz | Ah yes, I was in a muddle about what happens in the ruled case. As you say, it should really question about what happens when $K\cdot \omega \geq 0$. But I'll continue by email. | |
| Jan 2, 2011 at 18:29 | comment | added | Dmitri Panov | Tim, happy New Year for you too! I am really curious to know this plausibility argument. I guess this conjecture should be formulated only for symplectic 4-folds of non-negative Kodaira dimension, because indeed for $S=\mathbb CP^1\times \Sigma_g$ we have $c_1^2=-4(2g-2)$, while $c_2=-2(2g-2)$, so $c_1^2>4c_2$ if $g>1$... | |
| Jan 1, 2011 at 17:14 | comment | added | Tim Perutz | Happy New Year, Dima! It's not necessarily an open question - someone might be able to give a clear "no" as the answer. About symplectic BMY: first, the conjecture that $c_1^2\leq 4c_2$ is a variant of a conjecture of Fintushel-Stern, I think (if you hope for 3 as constant, you have to take care of ruled surfaces). I know one plausibility argument, but the world of symplectic 4-manifolds has frequently been found to be larger than expected... | |
| Jan 1, 2011 at 11:31 | comment | added | Dmitri Panov | Tim, thanks for the answer!! So this question is open modulo this folkloric conjecture that for symplectic 4-folds $c_1^2\le 3c_2$. By the way, do you believe this conjecture? :) | |
| Dec 31, 2010 at 23:45 | history | answered | Tim Perutz | CC BY-SA 2.5 |