This paper is concerned with sufficient criteria to guarantee that a given foliation on a normal variety has algebraic and rationally connected leaves. Following ideas from a preprint of Bogomolov and McQuillan and using the recent work of Graber, Harris, and Starr, we give a clean, short and simple proof of previous results. Apart from a new vanishing theorem for vector bundles in positive characteristic, our proof employs only standard techniques of Mori theory and does not make any reference to the more involved properties of foliations in characteristic p.We also give a new sufficient condition to ensure that all leaves are algebraic.The results are then applied to show that Q-Fano varieties with unstable tangent bundles always admit a sequence of partial rational quotients naturally associated to the Harder-Narasimhan filtration. Since this paper strives for simplicity, not completeness, we do not consider this extra complication here.On the other hand, if F is nonsingular, we can use the Reeb stability theorem to prove the algebraicity of all leaves.Theorem 2. In the setup of Theorem 1, if F is regular, then all leaves are rationally connected submanifolds.In fact, a stronger statement holds; see Theorem 28 in Section 5.As an immediate corollary to Theorem 1, we prove a refinement of Miyaoka's characterization of uniruled varieties. Before stating this corollary, we need to introduce the following notation.Notation 3. Let X be a normal projective variety, and let q : X Q be the rationally connected quotient, defined through the maximally rationally connected fibration ("MRC-fibration") of a desingularization of X; cf. [Kol96, IV, 5.3, 5.5]. Further, suppose that C ⊂ X is a subvariety which is not contained in the singular locus of X, and not contained in the indeterminacy locus of q, and that F ⊂ T X | C is a subsheaf of the restriction of the tangent sheaf to C. We say that F is vertical with respect to the rationally connected quotient, if F is contained in T X|Q at the general point of C.Notation 4. If X is normal, we consider general complete intersection curves in the sense of Mehta-Ramanathan, C ⊂ X. These are reduced, irreducible curves of the form C = H 1 ∩ · · · ∩ H dim X−1 , where the H i ∈ |m i · L i | are general, the L i ∈ Pic(X) are ample and the m i ∈ N large enough, so that the Harder-Narasimhan filtration of T X commutes with restriction to C. If the L i are chosen, we also call C a general complete intersection curve with respect to (L 1 , . . . , L dim X−1 ).We refer to [Fle84] and [Lan04] for a discussion and an explicit bound for the m i .Corollary 5. Let X be a normal complex-projective variety and C ⊂ X a general complete intersection curve. Assume that the restriction T X | C contains an ample locally free subsheaf F C . Then F C is vertical with respect to the rationally connected quotient of X.This statement first appeared implicitly in [Kol92, chap. 9]; but see Remark 23. To our best knowledge, the argument presented here gives the first complete proof of this important result.1.1...

Both authors were supported in full or in part by the priority program "Globale Methoden in der komplexen Geometrie" of the Deutsche Forschungsgemeinschaft, DFG.

In this paper we study smooth projective varieties and polarized pairs with an action of a one dimensional complex torus. As a main tool, we define birational geometric counterparts of these actions, that, under certain assumptions, encode the information necessary to reconstruct them. In particular, we consider some cases of actions of low complexity—measured in terms of two invariants of the action, called bandwidth and bordism rank—and discuss how they are determined by well known birational transformations, namely Atiyah flips and Cremona transformations.

In this paper we classify varieties of Picard number two having two projective bundle structures of any relative dimension, under the assumption that these structures are mutually uniform. As an application we prove the Campana–Peternell conjecture for varieties of Picard number one admitting C ∗ \mathbb {C}^* -actions of a certain kind.

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