## 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.

### 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

### Citations:

Zbl 0780.14026; Zbl 0659.14008
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### References:

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