Timeline for Examples of Calabi-Yau that are birational to each other?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 20, 2012 at 7:49 | comment | added | temp | What if I want to find projective examples that are birational but not isomorphic? | |
| Sep 10, 2012 at 10:21 | comment | added | Sasha | @Rhys Davies: this argument only shows that the birational isomorphism is not biregular, or that $X$ and $X'$ are not isomorphic OVER their common contraction. Sometimes, manifolds $X$ and $X'$ related by a flop are ABSTRACTLY isomorphic, for example two small resolutions of a quadratic cone are. | |
| Sep 10, 2012 at 7:26 | comment | added | Rhys Davies | @temp: Suppose $X$ and $X'$ are connected by flopping a single rational curve $C$ to $C'$. Since they are birational, the divisor class groups of $X$ and $X'$ are naturally isomorphic, and you can use this to show that they're not biholomorphic. For example, if $D$ is an effective divisor on $X$, satisfying $D\cdot C = 1$, then its proper transform on $X'$ (which I'll also call $D$), satisfies $D\cdot C' = -1$. The second Chern class also changes: $$ c_2(X')\cdot D = c_2(X)\cdot D + 2 $$ | |
| Sep 10, 2012 at 2:52 | comment | added | Sasha | Usually some of them are projective and the other are not. It depends on existence of Weil divisors passing through all singular points. | |
| Sep 9, 2012 at 23:02 | comment | added | temp | Why are they not isomorphic to each other? | |
| Sep 9, 2012 at 7:29 | history | answered | Sasha | CC BY-SA 3.0 |