Bogomolov–Miyaoka–Yau inequality

Remarks

Miyaoka and Yau proved the inequality independently, Miyaoka by algebraic methods and Yau by constructing Kähler–Einstein metrics; Bogomolov had earlier proved the weaker bound $c_1^2 \le 4 c_2$. Yau's proof shows that equality holds exactly when $X$ is a quotient of the unit ball in $\C^2$ by a discrete group acting freely. Examples are the fake projective planes, with $c_1^2 = 9$ and $c_2 = 3$. Example: a smooth quintic surface in $\mathbb{P}^3$ has $c_1^2 = 5$ and $c_2 = 55$, and $5 \le 165$. The hypothesis on the Kodaira dimension is needed: for $X = C \times \mathbb{P}^1$ with $C$ of genus $2$, one has $c_1^2 = 8(1 - 2) = -8$ and $c_2 = 4(1 - 2) = -4$, so $c_1^2 > 3 c_2 = -12$. The inequality fails in positive characteristic in general.

Going deeper

For a student

Together with Noether's inequality $c_1^2 \ge 2 \chi(\mathcal{O}_X) - 6$ for minimal surfaces, the inequality confines the Chern numbers of surfaces of general type to a wedge, the domain of the geography of surfaces. It is the surface case of the Chern number inequalities for varieties with Kähler–Einstein metrics. Yau used the equality case to prove that every compact complex surface homotopy equivalent to $\mathbb{P}^2$ is isomorphic to $\mathbb{P}^2$.