
## 2. The symplectic BMY conjecture

The Bogomolov–Miyaoka–Yau (BMY) inequality states that the Chern numbers of a compact complex surface $S$ of general type satisfy $c_1^2(S) \leq 3 c_2(S)$. Moreover, if equality holds then $S$ is the quotient of the unit ball in $\mathbb{C}^2$ by an infinite discrete group. Alternatively, the result can be expressed as an inequality $3\sigma(S) \leq \chi(S)$, where $\chi$ denotes the Euler characteristic and $\sigma$ the signature of the intersection form.
1. ### Symplectic BMY Inequality

#### Conjecture 2.1.

If $(X,\omega)$ satisfies $c_1^2 \geq 0$, then $c_1^2 \leq 3 c_2$.
• #### Problem 2.2.

[Casals] Does Miyaoka’s inequality follow by showing that some homology class in the projectivization $\mathbb{P}(T^*X)$ does not admit a symplectic representative?

Cite this as: AimPL: Symplectic four-manifolds through branched coverings, available at http://aimpl.org/symplecticfour.