Twisted cotangent sheaves and a Kobayashi-Ochiai theorem for foliations

Höring, Andreas

Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2465-2480.

Soit $X$ une variété projective normale et $A$ un diviseur de Cartier ample sur $X$. Supposons que $X$ n’est pas l’espace projectif. Nous montrons que le faisceau cotangent tordu ${\Omega }_X\otimes A$ est génériquement nef par rapport à la polarisation $A$. Comme conséquence nous obtenons un théorème de Kobayashi-Ochiai pour les feuilletages  : si $ℱ⊊T_X$ est un feuilletage tel que $detℱ\equiv i_{ℱ}A$, alors $i_{ℱ}$ est au plus le rang de $ℱ$.

Let $X$ be a normal projective variety, and let $A$ be an ample Cartier divisor on $X$. Suppose that $X$ is not the projective space. We prove that the twisted cotangent sheaf ${\Omega }_X\otimes A$ is generically nef with respect to the polarisation $A$. As an application we prove a Kobayashi-Ochiai theorem for foliations: if $ℱ⊊T_X$ is a foliation such that $detℱ\equiv i_{ℱ}A$, then $i_{ℱ}$ is at most the rank of $ℱ$.

Classification : 14F10, 37F75, 14M22, 14E30, 14J40
Keywords: Cotangent sheaf, foliations, Kobayashi-Ochiai theorem
Mot clés : faisceau cotangent, feuilletages, théorème de Kobayashi-Ochiai

[1] Andreatta, M. Some remarks on the study of good contractions, Manuscripta Math., Volume 87 (1995) no. 3, pp. 359-367 | DOI | MR | Zbl

[2] Andreatta, Marco Minimal model program with scaling and adjunction theory, Internat. J. Math., Volume 24 (2013) no. 2, pp. 1350007, 13 | DOI | MR | Zbl

[3] Araujo, Carolina; Druel, Stéphane On codimension 1 del Pezzo foliations on varieties with mild singularities (2012) (arXiv 1210.4013, preprint, to appear in Mathematische Annalen)

[4] Araujo, Carolina; Druel, Stéphane On Fano foliations, Adv. Math., Volume 238 (2013), pp. 70-118 | DOI | MR | Zbl

[5] Araujo, Carolina; Druel, Stéphane; Kovács, Sándor J. Cohomological characterizations of projective spaces and hyperquadrics, Invent. Math., Volume 174 (2008) no. 2, pp. 233-253 | DOI | MR | Zbl

[6] Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 4, Springer-Verlag, Berlin, 2004, pp. xii+436 | MR | Zbl

[7] Beltrametti, Mauro C.; Sommese, Andrew J. The adjunction theory of complex projective varieties, de Gruyter Expositions in Mathematics, 16, Walter de Gruyter & Co., Berlin, 1995, pp. xxii+398 | MR | Zbl

[8] Bogomolov, Fedor A.; McQuillan, Michael L. Rational curves on foliated varieties (February 2001), pp. 1-29 (Prépublications de l’IHES)

[9] Debarre, Olivier Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001, pp. xiv+233 | MR | Zbl

[10] Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom., Volume 3 (1994) no. 2, pp. 295-345 | MR | Zbl

[11] Druel, Stéphane Caractérisation de l’espace projectif, Manuscripta Math., Volume 115 (2004) no. 1, pp. 19-30 | DOI | MR | Zbl

[12] Flenner, Hubert Restrictions of semistable bundles on projective varieties, Comment. Math. Helv., Volume 59 (1984) no. 4, pp. 635-650 | DOI | EuDML | MR | Zbl

[13] Fujita, Takao Remarks on quasi-polarized varieties, Nagoya Math. J., Volume 115 (1989), pp. 105-123 | MR | Zbl

[14] Fujiwara, Tsuyoshi Varieties of small Kodaira dimension whose cotangent bundles are semiample, Compositio Math., Volume 84 (1992) no. 1, pp. 43-52 | EuDML | Numdam | MR | Zbl

[15] Fulton, William Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2, Springer-Verlag, Berlin, 1998, pp. xiv+470 | MR | Zbl

[16] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, pp. xvi+496 (Graduate Texts in Mathematics, No. 52) | MR | Zbl

[17] Höring, Andreas The sectional genus of quasi-polarised varieties, Arch. Math. (Basel), Volume 95 (2010) no. 2, pp. 125-133 | DOI | MR | Zbl

[18] Höring, Andreas On a conjecture of Beltrametti and Sommese, J. Algebraic Geom., Volume 21 (2012) no. 4, pp. 721-751 | DOI | MR | Zbl

[19] Höring, Andreas; Novelli, Carla Mori contractions of maximal length, Publ. Res. Inst. Math. Sci., Volume 49 (2013) no. 1, pp. 215-228 | DOI | MR | Zbl

[20] Kebekus, Stefan; Solá Conde, Luis; Toma, Matei Rationally connected foliations after Bogomolov and McQuillan, J. Algebraic Geom., Volume 16 (2007) no. 1, pp. 65-81 | DOI | MR | Zbl

[21] Kobayashi, Shoshichi; Ochiai, Takushiro Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ., Volume 13 (1973), pp. 31-47 | MR | Zbl

[22] Kollár, János Singularities of the minimal model program, Cambridge Tracts in Mathematics, 200, Cambridge University Press, Cambridge, 2013, pp. x+370 (With a collaboration of Sándor Kovács) | MR | Zbl

[23] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998, pp. viii+254 (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) | MR | Zbl

[24] Lazarsfeld, Robert Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete, 49, Springer-Verlag, Berlin, 2004, pp. xviii+385 (Positivity for vector bundles, and multiplier ideals) | MR | Zbl

[25] Mehta, V. B.; Ramanathan, A. Semistable sheaves on projective varieties and their restriction to curves, Math. Ann., Volume 258 (1981/82) no. 3, pp. 213-224 | DOI | EuDML | MR | Zbl

[26] Miyaoka, Yoichi; Peternell, Thomas Geometry of higher-dimensional algebraic varieties, DMV Seminar, 26, Birkhäuser Verlag, Basel, 1997, pp. vi+217 | MR | Zbl

[27] Paris, Matthieu Caractérisations des espaces projectifs et des quadriques (2010) (arXiv)

[28] Wahl, J. M. A cohomological characterization of ${\mathbf{P}}^n$, Invent. Math., Volume 72 (1983) no. 2, pp. 315-322 | DOI | EuDML | MR | Zbl