By János Kollár

This e-book offers a entire therapy of the singularities that seem within the minimum version software and within the moduli challenge for types. The research of those singularities and the advance of Mori's software were deeply intertwined. Early paintings on minimum versions depended on precise learn of terminal and canonical singularities yet many later effects on log terminal singularities have been acquired as outcomes of the minimum version application. fresh paintings at the abundance conjecture and on moduli of sorts of common variety depends on refined houses of log canonical singularities and conversely, the sharpest theorems approximately those singularities use newly built specified circumstances of the abundance challenge. This ebook untangles those interwoven threads, providing a self-contained and entire thought of those singularities, together with many formerly unpublished effects.

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**Example text**

Then (X, ) has a dlt, Q-factorial, minimal model over S. Warning The proofs of the next two cases use some of the results that we develop in Chapters 4–5. Thus we will avoid using them until Chapter 7. 15) of (X, ) intersects f −1 (S 0 ) and (f −1 (S 0 ), |f −1 (S 0 ) ) → S 0 has a canonical model. Then (X, ) has a dlt, Q-factorial, minimal model over S and also a canonical model over S. 8 (n-folds) (Birkar, 2011; Hacon and Xu, 2011a) Assume that f : X → S such that (X, + ) is dlt is projective and there is an effective divisor and KX + + ∼Q,f 0.

Thus F is effective. Every exceptional divisor appears on some resolution, hence a(E, X) ≥ 0 for every divisor E. 2 Assume that KX is Q-Cartier. If a(E, X) > −1 for every exceptional divisor E then f∗ ωY = ωX . The converse usually does not hold. Proof If mKX is Cartier, we have an equality OY (mKY ) = f ∗ OX (mKX )(mF ) where F = i a(Ei , X)Ei . Set G := i Ei . Since ma(Ei , X) > −m is an integer, we know that (m − 1)G + mF is effective. Thus the previous argument yields that f∗ OY (mKY + mG)(−G) = OX (mKX ).

2) A canonical model of (X , ) is also a canonical model of (X, ). (3) If a(E, X , ) < a(E, X, ) for every π-exceptional divisor E then every weak canonical (resp. minimal) model of (X , ) is also a weak canonical (resp. minimal) model of (X, ). Proof Let (X w , w ) be a weak canonical model of (X, ). If E is any divisor on X , then a(E, X , ) ≤ a(E, X, ) (and equality holds if E is not πexceptional). 1), a(E, X , ) ≤ a(E, X, ) ≤ a(E, Xw , w ). 19) hold automatically, hence (X w , w ) is also a weak canonical model of (X , ).