Return to Question

9 edited tags; [made Community Wiki]
8 added 18 characters in body

Let $X$ be an $n$-dimensional compact Kahler manifold with negative first Chern class. Are its Chern numbers $\prod_{i=1}^{n-1} c_i^{k_i}$, over $k _i \geq 0$ with $\sum ik_i = n$, bounded in terms of $(-1)^n e(X) = (-1)^nc_n = c_n(\Omega_X^1)$?

Are there, moreover, only finitely many deformation types for $|e(X)| = (-1)^{n}e(X)$ bounded?

For $n = 2$, we have of course the famous Bogomolov-Miyaoka-Yau bound $c_1^2 \leq 3c_2$.

Remark. One may also include the convention $c_0 := 1$, for which the answer is positive since $(-1)^nc_n > 0$, in order to cover the prototypical observation that the Euler number of a hyperbolic surface is negative. In view of this example, I would also like to extend the original question to the realm of log manifolds and hyperbolicity.

7 added 96 characters in body; added 41 characters in body

Let $X$ be an $n$-dimensional compact Kahler manifold with positive negative first Chern class. Are its Chern numbers $\prod_{i=1}^{n-1} c_i^{k_i}$, over $k _i \geq 0$ with $\sum ik_i = n$, bounded in terms of $c_n (-1)^n e(X) = (-1)^n e(X)$-1)^nc_n = c_n(\Omega_X^1)$? Are there only finitely many deformation types for$|e(X)| = (-1)^{n}e(X)$bounded? For$c_n$?n = 2$, we have of course the Bogomolov-Miyaoka-Yau bound $c_1^2 \leq 3c_2$.

Remark. One may also include the convention $c_0 := 1$, for which the answer is positive since $c_n (-1)^nc_n > 0$, in order to cover the prototypical observation that the Euler number of a hyperbolic surface is negative. In view of this example, I would also like to extend the original question to the realm of log manifolds and hyperbolicity.

6 added 65 characters in body
5 added 8 characters in body; deleted 8 characters in body
4 added 16 characters in body
3 added 24 characters in body; added 23 characters in body; added 18 characters in body
2 added 325 characters in body; edited title; deleted 4 characters in body
1