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Codimension one regular foliations on rationally connected threefolds. (English) Zbl 1516.14093

A conjecture of Touzet states that a smooth holomorphic foliation on a projective rationally connected manifold is algebraically integrable, i.e., every leaf of the foliation is an algebraic variety. This conjecture is known to be true for surfaces, but in dimensions \(\geq 3\) it is largely open. Thanks to a theorem of Druel it is also known (in all dimensions) on Fano manifolds, i.e., manifolds with ample anti-canonical bundle.
While every Fano manifold is rationally connected, and while the canonical bundle of a rationally connected manifold is never pseudo-effective, typically speaking rationally connected manifolds are not Fano (or even weak Fano).
The author’s main theorem is to prove Touzet’s conjecture for rationally connected threefolds with nef anti-canonical bundle. We recall that a line bundle is nef if it has non-negative degree when restricted to any curve, and that any ample line bundle is nef. Thus the author’s main result provides a generalisation of Druel’s theorem in the threefold setting. The proof proceeds by a careful analysis and classification of the possible steps of the \(K_{\mathcal F}\) minimal model program starting from a smooth foliation. We remark that currently a complete minimal model program for foliations is only known in dimension \(\leq 3\).

MSC:

14M22 Rationally connected varieties
37F75 Dynamical aspects of holomorphic foliations and vector fields

Citations:

Zbl 0893.32019
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References:

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