Singularities of pairs.

*(English)*Zbl 0905.14002
Kollár, János (ed.) et al., Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9–29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62(pt.1), 221-287 (1997).

Development of higher dimensional algebraic geometry in the last ten years frequently led to consider pairs \((X,D)\) where \(X\) is an algebraic variety, \(D\) is a \(\mathbb{Q}\)-linear combination of divisors and both \(X\) and \(D\) may be singular. The class of all such pairs is currently called the log category. The paper under review is a nicely written survey on the major results concerning the log category and their applications, including new results as well as many new simpler proofs of old results. It can be read by algebraic geometers that are non-familiar with techniques of higher dimensional geometry.

After presenting generalizations of the Kodaira vanishing theorem (section 2) the basic definitions concerning the log category appear in section 3, including discrepancy, a measure of how singular a pair \((X,D)\) is. Sections 4 and 5 are devoted, respectively, to Bertini-type theorems about singularities of generic members of linear systems and to studying linear systems \(K_X+L\), \(K_X\) the canonical system and \(L\) ample. Section 6 deals with the construction of divisors in a fixed numerical equivalence class, which are rather singular at a point \(x\) but not too singular near \(x\). In section 7 singularities of a pair \((X,D)\) are compared to those of \((H,D\mid H)\), \(H \subset X\) a hypersurface (“inversion of adjunction”). Section 8 to 10 introduce the log canonical threshold, a new measure of the singularities of \((X,D)\), suitable for the case in which no information is given by discrepancy, and relate it to previously know invariants. Finally, section 11 presents a new proof of the rationality of canonical singularities avoiding the use of Grothendieck’s general duality.

For the entire collection see [Zbl 0882.00032].

After presenting generalizations of the Kodaira vanishing theorem (section 2) the basic definitions concerning the log category appear in section 3, including discrepancy, a measure of how singular a pair \((X,D)\) is. Sections 4 and 5 are devoted, respectively, to Bertini-type theorems about singularities of generic members of linear systems and to studying linear systems \(K_X+L\), \(K_X\) the canonical system and \(L\) ample. Section 6 deals with the construction of divisors in a fixed numerical equivalence class, which are rather singular at a point \(x\) but not too singular near \(x\). In section 7 singularities of a pair \((X,D)\) are compared to those of \((H,D\mid H)\), \(H \subset X\) a hypersurface (“inversion of adjunction”). Section 8 to 10 introduce the log canonical threshold, a new measure of the singularities of \((X,D)\), suitable for the case in which no information is given by discrepancy, and relate it to previously know invariants. Finally, section 11 presents a new proof of the rationality of canonical singularities avoiding the use of Grothendieck’s general duality.

For the entire collection see [Zbl 0882.00032].

Reviewer: E.Casas-Alvero (Barcelona)