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Register a new userCodimension one holomorphic distributions on the projective three-space
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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 Grothendiecks 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.
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We show that codimension one distributions with at most isolated singularities on certain smooth projective threefolds with Picard rank one have stable tangent sheaves. The ideas in the proof of this fact are then applied to the characterization of certain irreducible components of the moduli space of stable rank 2 reflexive sheaves on $mathbb{P}^3$, and to the construction of stable rank 2 reflexive sheaves with prescribed Chern classes on general threefolds. We also prove that if $mathscr{G}$ is a subfoliation of a codimension one distribution $mathscr{F}$ with isolated singularities, then $Sing(mathscr{G})$ is a curve. As a consequence, we give a criterion to decide whether $mathscr{G}$ is globally given as the intersection of $mathscr{F}$ with another codimension one distribution. Turning our attention to codimension one distributions with non isolated singularities, we determine the number of connected components of the pure 1-dimensional component of the singular scheme.
This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds threedimensional, with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on smooth weighted projective complete intersection Fano manifolds threefold. We also prove that a codimension one locally free distribution with trivial canonical bundle on any Fano threefold, with Picard number equal to one, has a tangent sheaf which either splits or it is stable.
We study foliations by curves on the three-dimensional projective space with no isolated singularities, which is equivalent to assuming that the conormal sheaf is locally free. We provide a classification of such foliations by curves up to degree 3, also describing the possible singular schemes. In particular, we prove that foliations by curves of degree 1 or 2 are either contained on a pencil of planes or legendrian, and are given by the complete intersection of two codimension one distributions. We prove that the conormal sheaf of a foliation by curves of degree 3 with reduced singular scheme either splits as a sum of line bundles or is an instanton bundle. For degree larger than 3, we focus on two classes of foliations by curves, namely legendrian foliations and those whose conormal sheaf is a twisted null correlation bundle. We give characterizations of such foliations, describe their singular schemes and their moduli spaces.
For every integer $g ,geq, 2$ we show the existence of a compact Riemann surface $Sigma$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${mathcal O}^{oplus 2}_{Sigma}$ admits holomorphic connections with $text{SL}(2,{mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. The construction carries over to all very stable and compatible real holomorphic structures for the topologically trivial rank two bundle over $Sigma$ and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.
The space of holomorphic foliations of codimension one and degree $dgeq 2$ in $mathbb{P}^n$ ($ngeq 3$) has an irreducible component whose general element can be written as a pullback $F^*mathcal{F}$, where $mathcal{F}$ is a general foliation of degree $d$ in $mathbb{P}^2$ and $F:mathbb{P}^ndashrightarrow mathbb{P}^2$ is a general rational linear map. We give a polynomial formula for the degrees of such components.
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