Log canonical singularities are Du Bois. (English) Zbl 1202.14003
Log canonical singularities are a fundamental class of singularities of the minimal model program. In fact, they are the most severe singularities where it is hoped that the techniques of the minimal model program might hold. In this very important and fundamental paper, the authors prove that log canonical singularities satisfy a useful condition, explicitly they show that log canonical singularities are Du Bois singularities.
Having Du Bois singularities, a term coined by Steenbrink based on the work of Deligne and Du Bois, is the natural local condition on a variety \(X / \mathbb{C}\) which implies that the natural map \(H^i(X^{\text{an}}, \mathbb{C}) \to H^i(X, \mathcal{O}_X)\) surjects for all \(i\) [P. Du Bois, Bull. Soc. Math. Fr. 109, 41–81 (1981; Zbl 0465.14009); J. H. M. Steenbrink, Compos. Math. 42, 315–320 (1981; Zbl 0428.32017)] or this paper for a more precise definition and additional discussion). This surjectivity condition is naturally desirable when studying families of varieties with singular fibers or when trying to prove vanishing theorems on singular varieties.
It had been an open question for more than 20 years whether or not log canonical singularities are Du Bois, see for example Conjecture 1.13 in [J. Kollár (ed.), Flips and abundance for algebraic threefolds. A summer seminar at the University of Utah, Salt Lake City, 1991. Astérisque. 211. Paris: Société Mathématique de France (1991; Zbl 0782.00075)].
There have been a number of partial results in this direction including work of Ishii, the second author of this paper, Karen Smith, and also the reviewer of this paper; see [S. Ishii, Math. Ann. 270, 541–554 (1985; Zbl 0541.14002)], [S. J. Kovács, Compos. Math. 118, No. 2, 123–133 (1999; Zbl 0962.14011)], [K. Schwede, Compos. Math. 143, No. 4, 813–828 (2007; Zbl 1125.14002)], and [S. J. Kovács, K. Schwede and K. E. Smith, Adv. Math. 224, No. 4, 1618–1640 (2010; Zbl 1198.14003)], respectively.
The proof of this result relies on subtle vanishing theorems (and generalizations thereof) as well as on the recently completed minimal model program, see [C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)]. In fact, the authors show a more general result, that on a log canonical variety, any union of log canonical centers has Du Bois singularities. This generalizes results of Ambro (that such a union is seminormal, see [http://front.math.ucdavis.edu/9806.5067]) and also of the reviewer [loc. cit.].
In the last section of this paper, the authors apply their results to the study of families whose fibers have log canonical singularities. They show that, under mild hypotheses, the condition that fibers are Cohen-Macaulay is a closed condition. This application will help describe the geometry of the moduli spaces of higher dimensional algebraic varieties.
Having Du Bois singularities, a term coined by Steenbrink based on the work of Deligne and Du Bois, is the natural local condition on a variety \(X / \mathbb{C}\) which implies that the natural map \(H^i(X^{\text{an}}, \mathbb{C}) \to H^i(X, \mathcal{O}_X)\) surjects for all \(i\) [P. Du Bois, Bull. Soc. Math. Fr. 109, 41–81 (1981; Zbl 0465.14009); J. H. M. Steenbrink, Compos. Math. 42, 315–320 (1981; Zbl 0428.32017)] or this paper for a more precise definition and additional discussion). This surjectivity condition is naturally desirable when studying families of varieties with singular fibers or when trying to prove vanishing theorems on singular varieties.
It had been an open question for more than 20 years whether or not log canonical singularities are Du Bois, see for example Conjecture 1.13 in [J. Kollár (ed.), Flips and abundance for algebraic threefolds. A summer seminar at the University of Utah, Salt Lake City, 1991. Astérisque. 211. Paris: Société Mathématique de France (1991; Zbl 0782.00075)].
There have been a number of partial results in this direction including work of Ishii, the second author of this paper, Karen Smith, and also the reviewer of this paper; see [S. Ishii, Math. Ann. 270, 541–554 (1985; Zbl 0541.14002)], [S. J. Kovács, Compos. Math. 118, No. 2, 123–133 (1999; Zbl 0962.14011)], [K. Schwede, Compos. Math. 143, No. 4, 813–828 (2007; Zbl 1125.14002)], and [S. J. Kovács, K. Schwede and K. E. Smith, Adv. Math. 224, No. 4, 1618–1640 (2010; Zbl 1198.14003)], respectively.
The proof of this result relies on subtle vanishing theorems (and generalizations thereof) as well as on the recently completed minimal model program, see [C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)]. In fact, the authors show a more general result, that on a log canonical variety, any union of log canonical centers has Du Bois singularities. This generalizes results of Ambro (that such a union is seminormal, see [http://front.math.ucdavis.edu/9806.5067]) and also of the reviewer [loc. cit.].
In the last section of this paper, the authors apply their results to the study of families whose fibers have log canonical singularities. They show that, under mild hypotheses, the condition that fibers are Cohen-Macaulay is a closed condition. This application will help describe the geometry of the moduli spaces of higher dimensional algebraic varieties.
Reviewer: Karl Schwede (Ann Arbor)
MSC:
| 14B05 | Singularities in algebraic geometry |
| 14E30 | Minimal model program (Mori theory, extremal rays) |
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
| 14B07 | Deformations of singularities |
| 32S35 | Mixed Hodge theory of singular varieties (complex-analytic aspects) |
| 14F18 | Multiplier ideals |
Citations:
Zbl 0465.14009; Zbl 0428.32017; Zbl 0782.00075; Zbl 0541.14002; Zbl 0962.14011; Zbl 1125.14002; Zbl 1198.14003; Zbl 1210.14019References:
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