7
$\begingroup$

In complex dimension $2$, if a surface $S$ is a blowup of a surface $S'$, one has the following relation between their Chern numbers :

$c_1^2(S) + 1 = c_1^2(S')$

$c_2(S) - 1 = c_2(S')$

By using this fact one gets : for the Chern numbers $c_1^2$ and $c_2$ of the tangent bundle of a compact complex surface $S$ to be zero implies for $S$ to be minimal, and to belong to only some specific families among those listed in the Enriques-Kodaira classification according to this table for example : https://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification#/media/File:Geography_of_surfaces.jpg

And if the Chern numbers are both negative, then the surface has to be either ruled or a blowup of a ruled surface.

My question is : are there similar conclusions one could draw in bigger dimension ?

$\endgroup$
0

2 Answers 2

10
$\begingroup$

When a compact Kahler manifold satisfies $c_1=0$, it admits a Ricci-flat Kahler metric by Calabi-Yau, hence its tangent bundle is polystable (direct sum of stable bundles of the same slope). Then its discriminant $\int_M 2nc_2\wedge \omega^{n-2}$ is non-negative, by Bogomolov inequality, and positive when the curvature of $TM$ is non-zero. Therefore, a compact Kahler manifold with $c_1, c_2=0$ admits a flat Kahler metric, hence by Bieberbach's solution of Hilbert XVIII it is a quotient of a compact torus by a finite group which acts freely.

When $M$ is non-Kahler, this cannot be applied, and there are many examples of complex manifolds with vanishing Chern classes, such as Hopf manifolds, Calabi-Eckmann manifolds, Inoue surfaces, complex nilmanifolds, holomorphic parallelizable manifolds, $SL(2,{\Bbb C})/\Gamma$ which Ben McKay mentioned, and so on, and so forth.

$\endgroup$
6
$\begingroup$

On the complex torus, all Chern numbers vanish, but the same is true on the compact complex manifold $G/\Gamma$, given by quotienting a complex Lie group by a cocompact lattice. Such lattices exist in all complex semisimple Lie groups, I believe by results of Mostow. See E. Ghys, Deformation des structures complexes sur les espaces homogènes de $SL_2\mathbb{C}$, J. Riene Angew. Math., 468 (1995), p. 113-138. These complex manifolds admit a holomorphic connection on the tangent bundle. They are not Kähler except when $G$ is a complex torus and $\Gamma=\{1\}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.