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Differential forms on log canonical spaces. (English) Zbl 1258.14021
This monumental paper is concerned with differential forms on singular varieties. The classes of singularities treated are those that naturally arise in the Minimal Model Program, that is Kawamata log terminal singularities (klt) and divisorial log terminal singularities (dlt). The main question is whether a $$p$$-form on the smooth locus of a variety $$X$$ of dimension $$n$$ extends regularly to any resolution of singularities for $$p\leq n$$. This can be rephrased as follows. Let $$f:Y\to X$$ be a resolution of singularities then the question is whether the push forward sheaf $$f_*\Omega^p_Y$$ is reflexive for $$p\leq n$$. The authors are able to prove that this is the case for klt pairs via a more general result established for log canonical pairs. With the machinery involved in the proof and much more they also provide a theory of differential forms on dlt varieties. Aside this huge work applications are given to vanishing theorems and Lipman–Zaroski conjecture on the ssmoothness of varieties with locally free tangent sheaf.

##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14B05 Singularities in algebraic geometry 32S20 Global theory of complex singularities; cohomological properties
##### Keywords:
differential forms; log pairs; log canonical; extension results
Full Text:
##### References:
 [1] D. Barlet, Le faisceau $$$\backslash$$omega \^{$$\backslash$$cdot }_{X}$ sur un espace analytique X de dimension pure, in Fonctions de plusieurs variables complexes III (Sém. François Norguet, 1975–1977), Lecture Notes in Math., vol. 670, pp. 187–204, Springer, Berlin, 1978. MR0521919 (80i:32037). [2] M. C. Beltrametti and A. J. Sommese, The Adjunction Theory of Complex Projective Varieties, de Gruyter Expositions in Mathematics, vol. 16, de Gruyter, Berlin, 1995. 96f:14004. · Zbl 0845.14003 [3] C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, J. Am. Math. Soc., 23 (2010), 405–468. doi: 10.1090/S0894-0347-09-00649-3 . · Zbl 1210.14019 · doi:10.1090/S0894-0347-09-00649-3 [4] F. Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble), 54 (2004), 499–630. MR2097416 (2006c:14013). · Zbl 1062.14014 · doi:10.5802/aif.2027 [5] J. A. Carlson, Polyhedral resolutions of algebraic varieties, Trans. Am. Math. Soc., 292 (1985), 595–612. MR808740 (87i:14008). · Zbl 0602.14012 · doi:10.1090/S0002-9947-1985-0808740-3 [6] A. Corti et al., Flips for 3-Folds and 4-Folds, Oxford Lecture Series in Mathematics and Its Applications, vol. 35, Oxford University Press, Oxford, 2007. MR2352762 (2008j:14031). · Zbl 05175029 [7] P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163, Springer, Berlin, 1970. 54 #5232. [8] P. Du Bois, Complexe de de Rham filtré d’une variété singulière, Bull. Soc. Math. Fr., 109 (1981), 41–81. MR613848 (82j:14006). · Zbl 0465.14009 [9] P. Du Bois and P. Jarraud, Une propriété de commutation au changement de base des images directes supérieures du faisceau structural, C. R. Acad. Sci. Paris Sér. A, 279 (1974), 745–747. MR0376678 (51 #12853). · Zbl 0302.14004 [10] A. J. de Jong and J. Starr, Cubic fourfolds and spaces of rational curves, Ill. J. Math., 48 (2004), 415–450. MR2085418 (2006e:14007). · Zbl 1081.14007 [11] H. Esnault and E. Viehweg, Revêtements cycliques, in Algebraic threefolds (Varenna, 1981), Lecture Notes in Mathematics, vol. 947, pp. 241–250. Springer, Berlin, 1982. MR0672621 (84m:14015). [12] H. Esnault and E. Viehweg, Effective bounds for semipositive sheaves and for the height of points on curves over complex function fields, Compos. Math., 76 (1990), 69–85. Algebraic geometry (Berlin, 1988). MR1078858 (91m:14038). · Zbl 0742.14020 [13] H. Esnault and E. Viehweg, Lectures on Vanishing Theorems, DMV Seminar, vol. 20, Birkhäuser, Basel, 1992. MR1193913 (94a:14017). [14] B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure, and A. Vistoli, Fundamental Algebraic Geometry, Mathematical Surveys and Monographs, vol. 123, American Mathematical Society, Providence, 2005. Grothendieck’s FGA explained. MR2222646 (2007f:14001). · Zbl 1085.14001 [15] H. Flenner, Extendability of differential forms on nonisolated singularities, Invent. Math., 94 (1988), 317–326. MR958835 (89j:14001). · Zbl 0658.14009 · doi:10.1007/BF01394328 [16] J. Fogarty, Invariant Theory, W. A. Benjamin, Inc., New York, 1969. MR0240104 (39 #1458). [17] R. Godement, Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1973. Troisième édition revue et corrigée, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII, Actualités Scientifiques et Industrielles, No. 1252. MR0345092 (49 #9831). [18] H. Grauert, Über die Deformation isolierter Singularitäten analytischer Mengen, Invent. Math., 15 (1972), 171–198. MR0293127 (45 #2206). · Zbl 0237.32011 · doi:10.1007/BF01404124 [19] H. Grauert and O. Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math., 11 (1970), 263–292. MR0302938 (46 #2081). · Zbl 0202.07602 · doi:10.1007/BF01403182 [20] D. Greb, S. Kebekus, and S. J. Kovács, Extension theorems for differential forms, and Bogomolov-Sommese vanishing on log canonical varieties, Compos. Math., 146 (2010), 193–219. doi: 10.1112/S0010437X09004321 . · Zbl 1194.14056 · doi:10.1112/S0010437X09004321 [21] D. Greb, S. Kebekus, S. J. Kovács, and T. Peternell, Differential forms on log canonical varieties, Extended version of the present paper, including more detailed proofs and color figures. arXiv:1003.2913 , March 2010. [22] G.-M. Greuel, C. Lossen, and E. Shustin, Introduction to Singularities and Deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR2290112 (2008b:32013). · Zbl 1125.32013 [23] G.-M. Greuel, Dualität in der lokalen Kohomologie isolierter Singularitäten, Math. Ann., 250 (1980), 157–173. MR582515 (82e:32009) · Zbl 0417.14003 [24] A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math., 4 (1960), 228. MR0217083 (36 #177a). [25] A. Grothendieck, Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, vol. 224. Springer, Berlin, 1971, Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud, MR0354651 (50 #7129). [26] F. Guillén, V. Navarro Aznar, P. Pascual Gainza, and F. Puerta, Hyperrésolutions cubiques et descente cohomologique, Lecture Notes in Mathematics, vol. 1335, Springer, Berlin, 1988, Papers from the Seminar on Hodge-Deligne Theory held in Barcelona, 1982. MR972983 (90a:14024). · Zbl 0638.00011 [27] C. D. Hacon and S. J. Kovács, Classification of Higher Dimensional Algebraic Varieties, Oberwolfach Seminars, Birkhäuser, Boston, 2010. [28] C. D. Hacon and J. McKernan, On Shokurov’s rational connectedness conjecture, Duke Math. J., 138 (2007), 119–136. MR2309156 (2008f:14030). · Zbl 1128.14028 · doi:10.1215/S0012-7094-07-13813-4 [29] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer, New York, 1977. MR0463157 (57 #3116). [30] H. Hauser and G. Müller, The trivial locus of an analytic map germ, Ann. Inst. Fourier (Grenoble), 39 (1989), 831–844. MR1036334 (91m:32035). · Zbl 0678.32013 · doi:10.5802/aif.1191 [31] P. Heinzner, Geometric invariant theory on Stein spaces, Math. Ann., 289 (1991), 631–662. MR1103041 (92j:32116). · Zbl 0728.32010 · doi:10.1007/BF01446594 [32] H. Hironaka, On resolution of singularities (characteristic zero), in Proc. Int. Cong. Math., 1962, pp. 507–521. [33] M. Hochster, The Zariski-Lipman conjecture for homogeneous complete intersections, Proc. Am. Math. Soc., 49 (1975), 261–262. MR0360585 (50 #13033). · Zbl 0311.13007 [34] H. Holmann, Quotienten komplexer Räume, Math. Ann., 142 (1960/1961), 407–440. MR0120665 (22 #11414). · Zbl 0097.28602 · doi:10.1007/BF01450934 [35] D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Aspects of Mathematics, vol. E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR1450870 (98g:14012). · Zbl 0872.14002 [36] K. Jabbusch and S. Kebekus, Families over special base manifolds and a conjecture of Campana, Math. Z., to appear. doi: 10.1007/s00209-010-0758-6 , arXiv:0905.1746 , May 2009. · Zbl 1238.14024 [37] K. Jabbusch and S. Kebekus, Positive sheaves of differentials on coarse moduli spaces, Ann. Inst. Fourier (Grenoble), to appear. arXiv:0904.2445 , April 2009. · Zbl 1253.14009 [38] Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. Math., 127 (1988), 93–163. MR924674 (89d:14023). · Zbl 0651.14005 · doi:10.2307/1971417 [39] S. Kebekus and S. J. Kovács, The structure of surfaces mapping to the moduli stack of canonically polarized varieties, preprint (July 2007). arXiv:0707.2054 . · Zbl 1208.14027 [40] S. Kebekus and S. J. Kovács, Families of canonically polarized varieties over surfaces, Invent. Math., 172 (2008), 657–682. doi: 10.1007/s00222-008-0128-8 . MR2393082 · Zbl 1140.14031 · doi:10.1007/s00222-008-0128-8 [41] S. Kebekus and S. J. Kovács, The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties, Duke Math. J., 155 (2010), 1–33. MR2730371. arXiv:0812.2305 · Zbl 1208.14027 · doi:10.1215/00127094-2010-049 [42] J. Kollár, Algebraic groups acting on schemes, Undated, unfinished manuscript. Available on the author’s website at www.math.princeton.edu/$$\sim$$kollar . [43] J. Kollár, Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics, vol. 32, Springer, Berlin, 1996. MR1440180 (98c:14001). [44] J. Kollár, Lectures on Resolution of Singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, 2007. MR2289519. · Zbl 1113.14013 [45] J. Kollár and S. J. Kovács, Log canonical singularities are Du Bois, J. Am. Math. Soc., 23 (2010), 791–813. doi: 10.1090/S0894-0347-10-00663-6 . · Zbl 1202.14003 · doi:10.1090/S0894-0347-10-00663-6 [46] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. 2000b:14018. · Zbl 0926.14003 [47] J. Kollár et al., Flips and Abundance for Algebraic Threefolds, Astérisque, No. 211, Société Mathématique de France, Paris, 1992, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991 (1992). MR1225842 (94f:14013). [48] S. J. Kovács and K. Schwede, Hodge theory meets the minimal model program: a survey of log canonical and Du Bois singularities, in Topology of Stratified, pp. 51–94, 2001. · Zbl 1248.14019 [49] H. B. Laufer, Taut two-dimensional singularities, Math. Ann., 205 (1973), 131–164. MR0333238 (48 #11563). · Zbl 0281.32010 · doi:10.1007/BF01350842 [50] J. Lipman, Free derivation modules on algebraic varieties, Am. J. Math., 87 (1965), 874–898. MR0186672 (32 #4130). · Zbl 0146.17301 · doi:10.2307/2373252 [51] S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 18 (1964), 449–474. MR0173265 (30 #3478). · Zbl 0128.17101 [52] S. Mac Lane, Homology, Classics in Mathematics, Springer, Berlin, 1995, Reprint of the 1975 edition. MR1344215 (96d:18001). · Zbl 0818.18001 [53] H. Maschke, Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind, Math. Ann., 52 (1899), 363–368. MR1511061. · JFM 30.0131.01 · doi:10.1007/BF01476165 [54] Y. Namikawa, Extension of 2-forms and symplectic varieties, J. Reine Angew. Math., 539 (2001), 123–147. MR1863856 (2002i:32011). · Zbl 0996.53050 [55] C. Okonek, M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, 1980. MR561910 (81b:14001). [56] C. A. M. Peters and J. H. M. Steenbrink, Mixed Hodge Structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 52, Springer, Berlin, 2008. MR2393625. [57] D. Prill, Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J., 34 (1967), 375–386. MR0210944 (35 #1829). · Zbl 0179.12301 · doi:10.1215/S0012-7094-67-03441-2 [58] M. Reid, Canonical 3-folds, in A. Beauville (ed.) Algebraic Geometry (Angers, 1979), Sijthoff & Noordhoff, Alphen aan den Rijn, 1980. [59] M. Reid, Young person’s guide to canonical singularities, in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, pp. 345–414, Amer. Math. Soc, Providence, 1987. MR927963 (89b:14016). [60] G. Scheja and U. Storch, Differentielle Eigenschaften der Lokalisierungen analytischer Algebren, Math. Ann., 197 (1972), 137–170. MR0306172 (46 #5299). · Zbl 0229.14002 · doi:10.1007/BF01419591 [61] A. Seidenberg, The hyperplane sections of normal varieties, Trans. Am. Math. Soc., 69 (1950), 357–386. MR0037548 (12,279a). · Zbl 0040.23501 [62] I. R. Shafarevich, Basic Algebraic Geometry. 1, 2nd edn., Springer, Berlin, 1994. Varieties in projective space, Translated from the 1988 Russian edition and with notes by Miles Reid. MR1328833 (95m:14001). [63] J. Steenbrink and D. van Straten, Extendability of holomorphic differential forms near isolated hypersurface singularities, Abh. Math. Semin. Univ. Hamb., 55 (1985), 97–110. MR831521 (87j:32025). · Zbl 0584.32018 · doi:10.1007/BF02941491 [64] J. H. M. Steenbrink, Vanishing theorems on singular spaces, Astérisque, 130 (1985), 330–341, Differential systems and singularities (Luminy, 1983). MR804061 (87j:14026). [65] E. Szabó, Divisorial log terminal singularities, J. Math. Sci. Univ. Tokyo, 1 (1994), 631–639. MR1322695 (96f:14019). · Zbl 0835.14001 [66] B. Teissier, The hunting of invariants in the geometry of discriminants, in Real and Complex Singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pp. 565–678, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977. MR0568901 (58 #27964). [67] J.-L. Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math., 36 (1976), 295–312. MR0481096 (58 #1242) · Zbl 0333.32010 · doi:10.1007/BF01390015 [68] E. Viehweg, Compactifications of smooth families and of moduli spaces of polarized manifolds, Ann. Math., 172 (2010), 809–910. arXiv:math/0605093 . MR2680483. · Zbl 1238.14009 · doi:10.4007/annals.2010.172.809 [69] E. Viehweg and K. Zuo, Base spaces of non-isotrivial families of smooth minimal models, in Complex Geometry (Göttingen, 2000), pp. 279–328, Springer, Berlin, 2002. MR1922109 (2003h:14019). · Zbl 1006.14004 [70] J. M. Wahl, A characterization of quasihomogeneous Gorenstein surface singularities, Compos. Math., 55 (1985), 269–288. MR799816 (87e:32013) · Zbl 0587.14024
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