Bogomolov–Miyaoka–Yau inequality

For a smooth minimal complex projective surface of general type, the self-intersection of its first Chern class is at most three times its second Chern number: c₁² ≤ 3c₂.

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Chern numbers: Topological integers obtained by integrating characteristic classes of a complex manifold. The inequality compares the surface’s first Chern number squared with its second.
Algebraic surface: A two-dimensional algebraic variety. The inequality is a numerical constraint on a particular class of algebraic surfaces.
Fedor Bogomolov: A Russian mathematician known for work in algebraic geometry and mathematical physics. His 1970s work established an important predecessor to the surface inequality.
Fake projective plane: A smooth complex surface with the same Betti numbers as the projective plane but not isomorphic to it. These ball quotients attain equality despite having the Betti numbers of the projective plane.
Geography of algebraic surfaces: The study of which numerical invariants can occur for algebraic surfaces. The inequality supplies an upper boundary for canonical self-intersection relative to Euler characteristic.
Kähler–Einstein metric: A Kähler metric whose Ricci curvature is proportional to the metric. Differential-geometric proofs translate curvature restrictions into bounds on Chern numbers.
Minimal model (algebraic geometry): A birational model with no exceptional curves that can be contracted while remaining smooth. Minimality removes blow-up contributions that otherwise alter the Chern numbers.
Shigeru Miyaoka: A Japanese mathematician whose work includes major results on algebraic surfaces and threefolds. He proved the inequality for surfaces of general type using algebraic-geometric methods.
Compact complex ball quotient: A compact complex manifold obtained by quotienting a complex ball by a suitable discrete group of automorphisms. For surfaces, these are the geometric models of equality in the inequality.
Uniformization theorem: A theorem describing simply connected covering spaces of Riemann surfaces by standard geometries. The surface equality case is a higher-dimensional analogue of geometric uniformization.