##
**Rationally connected foliations after Bogomolov and McQuillan.**
*(English)*
Zbl 1120.14011

The main strategy of Mori theory is to relate the existence and properties of rational curves on a projective manifold \(X\) to the positivity properties of the anticanonical bundle \(-K_X\). For example, it is a well-known theorem that a projective manifold whose anticanonical bundle is ample, is rationally connected [cf. J. Kollár, Y. Miyaoka and S. Mori, J. Algebr. Geom. 1, No. 3, 429–448 (1992; Zbl 0780.14026)]. Initiated by Y. Miyaoka’s landmark paper [in: Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 1, Proc. Symp. Pure Math. 46, No. 1, 245–268 (1987; Zbl 0659.14008)], algebraic geometers want to refine this analysis by shifting the focus from the anticanonical bundle to subsheaves of the tangent bundle.

More precisely one can ask: given a saturated subsheaf \(F \subset T_X\) that satisfies a certain positivity property, what can we say about the existence of rational curves \(C\) on \(X\) that point in the directions of \(F\), i.e. such that the tangent sheaf \(T_C\) is included in \(F| _C\)?

In the article under review a fundamental result concerning this problem is established. The main theorem states the following:

Let \(X\) be a normal complex projective variety, and let \(C\) be a projective curve that is contained in the smooth locus of \(X\). Let \(F \subset T_X\) be an integrable subsheaf which is locally free along \(C\). Suppose now that \(F| _C\) is an ample vector bundle, then a \(F\)-leaf that meets \(C\) in a general point is algebraic and its closure is rationally connected.

This result can be seen as a foliated version of the classical theorem that a complex projective manifold is rationally connected if and only if it contains a very free rational curve. Note furthermore that the main theorem (as the title of the paper says) has been stated in a slightly more general form in a preprint by F. Bogomolov and M. McQuillan [Rational curves on foliated varieties, IHES preprint, 2001]. The authors of the article under review follow the basic approach of the Bogomolov-McQuillan paper, but they give simplified and much more accessible proofs of the technical key points. Using classical techniques from foliation theory, the following corollary is shown:

Let \(X\) be a normal complex projective variety, and let \(T_{X/Q}\) be the relative tangent sheaf of the rationally connected quotient, that is the unique almost holomorphic fibration \(X \rightarrow Q\) such that the general fibre is rationally connected and the base \(Q\) is not uniruled. Let now \(C\) be a general complete intersection curve on \(X\) and suppose that the restriction \(T_X| _C\) contains an ample subsheaf \(F_C\). Then \(F_C\) is contained in the relative tangent sheaf \(T_{X/Q}\) at a general point of the curve \(C\).

This corollary is a refined version of Miyaoka’s famous characterisation of non-uniruled varieties as the varieties whose cotangent bundle is nef on general complete intersection curves.

More precisely one can ask: given a saturated subsheaf \(F \subset T_X\) that satisfies a certain positivity property, what can we say about the existence of rational curves \(C\) on \(X\) that point in the directions of \(F\), i.e. such that the tangent sheaf \(T_C\) is included in \(F| _C\)?

In the article under review a fundamental result concerning this problem is established. The main theorem states the following:

Let \(X\) be a normal complex projective variety, and let \(C\) be a projective curve that is contained in the smooth locus of \(X\). Let \(F \subset T_X\) be an integrable subsheaf which is locally free along \(C\). Suppose now that \(F| _C\) is an ample vector bundle, then a \(F\)-leaf that meets \(C\) in a general point is algebraic and its closure is rationally connected.

This result can be seen as a foliated version of the classical theorem that a complex projective manifold is rationally connected if and only if it contains a very free rational curve. Note furthermore that the main theorem (as the title of the paper says) has been stated in a slightly more general form in a preprint by F. Bogomolov and M. McQuillan [Rational curves on foliated varieties, IHES preprint, 2001]. The authors of the article under review follow the basic approach of the Bogomolov-McQuillan paper, but they give simplified and much more accessible proofs of the technical key points. Using classical techniques from foliation theory, the following corollary is shown:

Let \(X\) be a normal complex projective variety, and let \(T_{X/Q}\) be the relative tangent sheaf of the rationally connected quotient, that is the unique almost holomorphic fibration \(X \rightarrow Q\) such that the general fibre is rationally connected and the base \(Q\) is not uniruled. Let now \(C\) be a general complete intersection curve on \(X\) and suppose that the restriction \(T_X| _C\) contains an ample subsheaf \(F_C\). Then \(F_C\) is contained in the relative tangent sheaf \(T_{X/Q}\) at a general point of the curve \(C\).

This corollary is a refined version of Miyaoka’s famous characterisation of non-uniruled varieties as the varieties whose cotangent bundle is nef on general complete intersection curves.

Reviewer: Andreas Höring (Paris)

### MSC:

14E30 | Minimal model program (Mori theory, extremal rays) |

14J45 | Fano varieties |

14J40 | \(n\)-folds (\(n>4\)) |

14F17 | Vanishing theorems in algebraic geometry |

57R30 | Foliations in differential topology; geometric theory |

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\textit{S. Kebekus} et al., J. Algebr. Geom. 16, No. 1, 65--81 (2007; Zbl 1120.14011)

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