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

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 ?

## 2 Answers

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.

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 $$\Gamma=\{1\}$$.