Miyaoka-Yau inequality

In complex geometry , the Miyaoka-Yau inequality (also Bogomolov-Miyaoka inequality ) is used to characterize ball quotients .

Miyaoka-Yau inequality for complex surfaces

Let be a compact complex surface of general type . Then for the Chern classes and the inequality ${\displaystyle S}$ ${\displaystyle c_{1}}$ ${\displaystyle c_{2}}$

${\displaystyle c_{1}^{2}\leq 3c_{2}}$

and equality only applies if a ball quotient, i.e. a complex hyperbolic surface . ${\displaystyle S}$

Generalizations

Let be a -dimensional complex projective variety whose canonical divisor is traffic light . Then the inequality holds ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle K_{X}}$

${\displaystyle \left(c_{2}-{\frac {n}{2(n+1)}}c_{1}^{2}\right)\left[K_{X}\right]^{n-2}\geq 0}$

and equality is only valid if it is a ball quotient, i.e. a complex-hyperbolic manifold . ${\displaystyle X}$

Individual evidence

1. ↑ Y. Miyaoka : On the Chern numbers of surfaces of general type , Inventiones Mathematicae 42, 225-237, 1977.
2. ^ ST Yau : Calabi's conjecture and some new results in algebraic geometry , Proceedings of the National Academy of Sciences of the United States of America 74, 1798-1799, 1977.
3. ^ D. Greb , S. Kebekus , T. Peternell , B. Taji : The Miyaoka-Yau inequality and uniformization of canonical models , Annales scientifiques de l'École normal supérieure, 2019.