We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation, and certain logarithmic foliations have stable tangent sheaves.
In this paper, we give the explicit construction of certain components of the space of holomorphic foliations of codimension one, in complex projective spaces. These components are associated to some algebraic representations of the affine Lie algebra aff(C). Some of them, the so-called exceptional or Klein-Lie components, are rigid in the sense that all generic foliations in the component are equivalent (Example 1). In particular, we obtain rigid foliations of all degrees. Some generalizations and open problems are given at the end of §1.
The 7-dimensional link K of a weighted homogeneous hypersurface on the round 9-sphere in C 5 has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-calibrated G 2structure ϕ induced by the Calabi-Yau 3-orbifold basic geometry. We distinguish these pairs (K, ϕ) by the Crowley-Nordström Z 48 -valued ν invariant, for which we prove odd parity and provide an algorithmic formula.We describe moreover a natural Yang-Mills theory on such spaces, with many important features of the torsionfree case, such as a Chern-Simons formalism and topological energy bounds. In fact, compatible G 2 -instantons on holomorphic Sasakian bundles over K are exactly the transversely Hermitian Yang-Mills connections. As a proof of principle, we obtain G 2 -instantons over the Fermat quintic link from stable bundles over the smooth projective Fermat quintic, thus relating in a concrete example the Donaldson-Thomas theory of the quintic threefold with a conjectural G 2 -instanton count.
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