In this chapter we study two topics that have important applications to flips and to moduli questions.

In Section 4.1 we studied the log canonical centers of an lc pair (X, Δ); these are centers of divisors of discrepancy −1.

Here we study a larger class of interesting subvarieties called log centers, which are centers of divisors of negative discrepancy. As a general principle, the closer the discrepancy is to −1, the more a log center behaves like a log canonical center. Thus log canonical centers are the most special among the log centers.

The case when X is normal is treated in Section 7.1 and the general semi-log canonical version is derived from it in Section 7.2.

The depth of the structure sheaf and of the dualizing sheaf of a semi-log canonical pair is studied in Section 7.3.

Assumptions In this chapter we work with schemes (or algebraic spaces) that are of finite type over a base scheme S that is essentially of finite type over a field of characteristic 0.

Log centers

Definition 7.1 Let f: (X, Δ) → Z be a weak crepant log structure and W ⊂ Z an irreducible subvariety. The minimal log discrepancy of (or over) W is defined as the infimum of the numbers 1 + a(E, X, Δ) where E runs through all divisors over X such that f (centerX(E)) = W. It is denoted by

if the choice of f: (X, Δ) → Z is clear.