Timeline for The canonical line bundle of a normal variety
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 5, 2011 at 15:07 | vote | accept | Jesus Martinez Garcia | ||
| Nov 19, 2010 at 18:23 | answer | added | Sándor Kovács | timeline score: 56 | |
| Sep 27, 2010 at 22:26 | answer | added | JRG | timeline score: 12 | |
| Aug 17, 2010 at 12:46 | comment | added | Jesus Martinez Garcia | So, the formula is not true in general if $X$ is not toric? I have checked what J.C. said and it is true, it also appears in Cox-Little-Schenck (I have no Fulton with me to check). Mmmh, that is an answer, but I cannot take a comment as a an answer (I may write it myself tbut that is not fair either). If J.C. wants to write it I will take that. | |
| Aug 16, 2010 at 14:06 | comment | added | user5117 | "casw" --> "case" | |
| Aug 16, 2010 at 14:02 | comment | added | user5117 | Dear Jesus, Just to clarify, the point of my comment was that no projective (or more generally as Chris says, proper) toric variety is Calabi--Yau. Also, as John Christian says in the first comment, the formula for toric varieties is K_X = O(-\sum D_i) where the sum is over the torus invariant divisors --- not representatives of all divisors in the class group. Indeed, the class group will almost always be infinite, in which casw such a sum will not be defined. | |
| Aug 16, 2010 at 13:33 | comment | added | Jesus Martinez Garcia | Hey, I rephrased the whole question but I am asking the same thing. I never said that all toric varieties are Calabi-Yau. I said only if they lie on a hyperplane, for instance the one whose cone is generated by Cone($e_1,e2,−e1+2e2$), but that is not the point. I want to know if the formula is true, and not just for toric cases | |
| Aug 16, 2010 at 13:29 | history | edited | Jesus Martinez Garcia | CC BY-SA 2.5 |
Rephrase the whole content for clarity and fixed a couple of typos
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| Aug 16, 2010 at 12:47 | comment | added | Chris Brav | Hi Artie. I just wanted to clarify for the question-asker. Also, yes, you are right: since toric varieties are rational, proper ones do not admit any non-trivial forms of top degree, so they can't be Calabi-Yau, even in the weak sense. | |
| Aug 16, 2010 at 12:11 | comment | added | user5117 | Dear Chris, Of course you're right. I was assuming say projective (maybe proper is enough). In that case I think my argument above suffices to prove your assertion. | |
| Aug 16, 2010 at 12:01 | comment | added | Chris Brav | To clarify the comments above about toric varieties not being Calabi-Yau: by Calabi-Yau, sometimes people colloquially mean that the canonical bundle is trivial and sometimes they mean more specifically that the variety should be projective, have trivial canonical, and that the intermediate cohomology groups of the structure sheaf be trivial (so that the total cohomology looks like that of a sphere). There are non-proper toric varieties with trivial canonical bundle (for instance a torus), but I don't think it is possible to have a proper toric variety with trivial canonical bundle. | |
| Aug 16, 2010 at 10:09 | comment | added | user5117 | Talking off the top of my head: a toric variety has a dense open subset isomorphic to a torus, so (over an algebraically closed field say) is rational, and in particular never Calabi--Yau. Also a minor point "since the smooth locus has codimension 1" is a confusing way to say what you mean. "since the singular locus has codimension 2" would be better. | |
| Aug 16, 2010 at 9:06 | history | asked | Jesus Martinez Garcia | CC BY-SA 2.5 |