arXiv subjects
Publications and source records attributed to Miguel Rodríguez Peña.
We prove a holomorphic residue localization formula for odd holomorphic vector fields on compact complex supermanifolds whose fermionic and bosonic dimensions coincide. Under isolated non-degeneracy hypotheses on the reduced zero set, we give an explicit local residue formula.
We study Poisson-flat connections with logarithmic poles along a simple normal crossings divisor on a holomorphic Poisson manifold, where flatness is required only along the symplectic foliation. After identifying the relevant logarithmic cotangent Poisson Lie algebroid, we define an Euler--Poisson principal part and a residue theory adapted to the canonical logarithmic Hamiltonian generators. Under a precise nonresonance hypothesis, we establish a Poisson Poincaré--Dulac theorem: any logarithmic Poisson-flat connection with prescribed principal part is holomorphically gauge equivalent to a pure Euler--Poisson normal form with constant commuting residues, and this normal form is unique up to Casimir-valued gauge transformations lying in the common centralizer of the residues. To encode both leafwise transport and boundary winding, we construct a twisted leafwise fundamental groupoid via the real oriented blow-up and a 2-pushout that adjoins canonical tangential meridians. In the nonresonant regime, the normal form yields a meridional character determined by the residues, and hence a logarithmic Riemann--Hilbert correspondence in the Poisson setting for this groupoid. Finally, we illustrate the theory with rank-two Poisson modules (Poisson triples) and an explicit family of examples.
We determine the number of singularities - counted whit multiplicities - of generic distributions of dimension and codimension one on smooth complete intersections in compact toric orbifolds with isolated singularities. We also present some applications of this results. First, we analyze the case of regular distributions. As a second application, we establish a Poincaré-type inequality that relates the multidegree of a foliation to the multidegrees of an invariant smooth complete intersection curve.
In this work, we study inequalities and enumerative formulas for flags of Pfaff systems on $\mathbb{P}^n_{\mathbb{C}}$. More specifically, we find the number of independent Pfaff systems that leave invariant a one-dimensional holomorphic foliation and deduce inequalities relating the degrees in the flags, which can be interpreted as the Poincaré problem for flags. Moreover, restricting to a flag of specific holomorphic foliations/distributions, we obtain inequalities involving the degrees. As a consequence, we prove stability results for the tangent sheaf of some rank two holomorphic foliations/distributions.
The number of singularities, counted with multiplicity, of a generic codimension one holomorphic distribution on a compact toric orbifold is determined. As a consequence, we give the classification of regular distributions on rational normal scrolls and weighted projective spaces. Under certain conditions, we also prove that the singular set of a codimension one holomorphic foliation on a toric orbifold admits at least one irreducible component of codimension two, and we give a Darboux-Jouanolou type integrability theorem for codimension one holomorphic foliations. We illustrate our results with several examples.
We give an optimal upper bound of the degree of quasi-smooth hypersurfaces which are invariant by a one-dimensional holomorphic foliation on a compact toric orbifold, i.e. on a complete simplicial toric variety. This bound depends only on the degree of the foliation and of the degrees of the toric homogeneous coordinates.