Timeline for The Relationship between Complex and Algebraic Geomety
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 11, 2009 at 4:02 | comment | added | David E Speyer | No to the second. Kaehler plus Moishezon = projective, and not all smooth toric varieties are projective. I think no to the first, but I need to think about it. (Or wait for some else to answer.) | |
| Nov 10, 2009 at 23:37 | comment | added | Kevin H. Lin | Is it true that (compact if needed) smooth toric varieties are always symplectic? Are all (compact if needed) smooth toric varieties Kaehler? | |
| Nov 10, 2009 at 22:56 | comment | added | David E Speyer | I should have said "compact n-dimensional smooth toric variety" in the above answer. | |
| Nov 10, 2009 at 22:53 | comment | added | David E Speyer | The cohomology of a n-dimensional smooth toric variety is given by the same recipe whether or not it is projective: one generator x(i), in degree 2, for each ray rho(i) of the fan. The relations are that $x(i\_1) ... x(i\_k) =0$ whenever the rays rho(i_1) ... rho(i_k) do not lie in a common cone, combined with n linear relations that are a little too complicated to write here. (See Fulton's book.) The first set of relations is in degree larger than 2, so H^2 has dimension (number of rays)-n, which is always greater than 0. | |
| Nov 10, 2009 at 22:49 | comment | added | David E Speyer | A compact complex manifold without $H^2$ is not algebraic. See a sketch of a proof here: sbseminar.wordpress.com/2008/02/14/… | |
| Nov 10, 2009 at 22:27 | comment | added | Greg Kuperberg | A projective structure on a toric variety is given by a rational convex polytope structure on its fan. When its fan is complete but non-regular, the variety is proper but not projective. But I don't know how to compute H^2 of this, although I'm sure lots of people do. However, an even-dimensional simple Lie group such as SU(3) has left-invariant complex structures, and these immediately have no H^2. | |
| Nov 10, 2009 at 20:52 | history | answered | Kevin H. Lin | CC BY-SA 2.5 |