How restrictive is having zero Chern numbers for a compact complex manifold ? Same for negative Chern number?

Question

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 ?

Answer 1

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.

Answer 2

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 $\Gamma=\{1\}$.