Generalisation of the Bogomolov-Miyaoka-Yau Inequality to Singular Surfaces

The paper considers pairs (X, B) where X is a normal projective surface over C, and B is a Q-divisor whose coefficients are 1 or 1−1/m for some natural number m. A log canonical singularity on such a pair is a quotient by a finite or infinite group, so if (X, B) has log canonical singularities, the orbifold Euler number eorb(X, B) can be defined. The main result is a Bogomolov-Miyaoka-Yau-type inequality which implies that if (X, B) has log canonical singularities and κ(X, KX + B) ⩾ 0 then (KX+B)2 ⩽ 3eorb(X, B). The actual inequality proved is somewhat stronger and it also implies all the previously published versions of the Bogomolov-Miyaoka-Yau inequality. The proof involves the Log Minimal Model Program, Q-sheaves when KX+B is nef, and a study of the changes in the two sides of the inequality under a contraction. The paper also contains a further generalisation where the coefficients of B can be arbitrary rational numbers in 0, 1, a different condition is imposed on the singularities and KX+B is required to be nef. Some applications of the inequalities are also given, for example, estimating the number of singularities or certain kinds of configurations of curves on surfaces. 1991 Mathematics Subject Classification: 14J17, 14J60, 14C17.