We shall discuss the analogue of Kodaira-Enriques classification for foliated surfaces. In particular we shall explain what is a minimal model in this context and which foliations do not admit one. We shall also survey (following McQuillen, as Brunella does in his book) Zariski decomposition, numerical dimension and abundance for foliations.
In this talk we present results of Marco Brunella's paper "A positivity property for one-dimensional holomorphic foliations on compact Kahler manifolds". The main theorem says that if such a foliation is not a foliation by ratitonal curves, then its canonical bundle is pseudo-effective: admits a (singular) Herrmitian metric with non-negative curvature in the sense of currents. This result has the following purely complex-geometric application for compact connected non-projective Kahler threefolds: if the canonical bundle of the threefold is not pseudo-effective, then the threefold is covered by rational curves. The proof of the main theorem is based on Brunella - Ilyashenko construction of universal covering tube of foliations, important Brunella's results about its geometry and a deep theorem of P.Dingoyan (main technical part).
We discuss the proofs modulo technical details.
We will deal with codimension one holomorphic foliations with ample normal bundle. The Conjecture claims that every leaf accumulates to the singular set of the foliation for any compact connected complex manifold of dimension > 2. In the case of projective spaces (X = CP^n, n=>3) conjecture has been proved by Lins Neto. Later (2010) conjecture has been proved for complex torus of dimension n=>3 by Brunella. We will show that in the case if Conjecture does not hold then there exists a nonempty compact subset which is invariant by foliation and disjoint from Sing(F) and such set cannot be a sufficiently smooth real hypersurface. Then we will discuss a gap between this result and Conjecture.
Consider a one-dimensional foliations on a complex manifold M. Its hyperbolic leaves inherit the standard Poincar\'e metric from their universal cover, the unit disc. Extend this metric to parabolic leaves by zero. It turns out that this leafwise metric cannot vary arbitrarily from leaf to leaf: its variation should be plurisubharmonic, in some assumptions on M and on the foliation.
First case is of compact complex algebraic surface and a foliation with nef canonical bundle (nef foliation), the second case is of a compact connected Kahler manifold and a foliation with at least some hyperbolic leaves. This allows to show that the set of entire curves tangent to a foliation is small, except for some special cases.
The classical Castelnuovo-de Franchis lemma says that if connected compact Kaehler has two holomorphic 1-forms, linearly independent over \C, but pointwise collinear, then there exists a holomorphic map onto a hyperbolic curve such that these 1-forms are pullbacks.
We shall study the structure of holomorphic 1-forms on compact threefolds of positive algebraic dimension, obtain a description of integrable ones and use this result to extend Castelnuovo-de Franchis lemma to non-Kaehler threefolds.
We will deal with some dynamical properties of holomorphic foliations of codimension one on complex tori. Such foliations essentially fall into three categories: linear foliations, turbulent foliations and foliations with ample normal bundle. We will define these classes and discuss their dynamical properties. Namely, we will show that in the least explicit case - the case of a foliation with ample normal bundle - every leaf accumulates to the singular set of the foliation.
A classical problem in the theory of complex manifolds concerns the existence of complex structures on the six-dimensional sphere S^6. Using octonions one can construct almost-complex structures on S^6, but they are not integrable.
Let X be a compact complex manifold of dimension 3 with the integral homology of S^6 and let Aut(X) be its holomorphic automorphism group. It is known that the automorphism group of a compact complex manifold is a finite dimensional complex Lie group. From the results of Campana, Demailly, Peternell (CDP) and Huckleberry, Kebekus, Peternell (HKP), 1998, it follows that Aut(X) is at most 2-dimensional. I will talk about the work of HKP and about a paper of Marco Brunella, 1999, in wich he proves that Aut(X) can't be isomorphic to the complex affine group.
Harvey and Lawson in 1983 defined a Kahler rank -- measure of kaehliarity of a compact complex surfaces. In particular surface has Kahler rank 1 if it admits non-negative closed (1,1)-form, whose zero locus is contained in a curve. We discuss the paper by Chiose and Toma where authors give partial classification of surfaces of Kahler rank 1. Relying on this classification they prove Harvey and Lawson conjecture -- show that Kahler rank is bimeromorphic invariant.
A locally conformally Kaehler (LCK) manifold is a manifold which is covered by a Kaehler manifold, with the desk transform map acting by (non-isometric) homotheties. A Kato surface is one which admits a spherical shell, that is, a hypersurface S which is CR-equivalent to a 3-sphere with the standard CR-structure, in such a way that a complement of S is connected. Using a clever application of Kaehler potential, Brunella proves that all Kato surfaces are locally conformally Kaehler. This is a landmark paper in the study of LCK metrics on surfaces.
Using machinery developed in his own earlier papers and a work by Chiose-Toma, Brunella completed the classification of complex surfaces admitting positive, exact current with zero-foliation of rank 1. As an application of this result, one can obtain a classification of dimension 2 complex subvarieties of a manifold equipped with a semi-positive, exact form.
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