arXiv subjects

Gilcione Nonato Costa

Publications and source records attributed to Gilcione Nonato Costa.

5 recordsLinked to original sources

arXiv2026

A lower bound for the Milnor number of vector fields

We study holomorphic vector fields whose singular locus contains a smooth positive-dimensional local complete-intersection component. We obtain an exact global formula for its Milnor-Poincaré-Hopf contribution and prove a sharp local lower bound, valid under holomorphic perturbations, in terms of the embedded contribution associated with the component. As an application to gradient vector fields, we derive a lower bound for the total Milnor number of the isolated critical points arising near a smooth positive-dimensional critical component; for a morsification, this becomes a lower bound for the number of Morse critical points converging to that component. Explicit families show that the bounds are sharp and exhibit the redistribution of singularities between a fixed neighbourhood of the component and the hyperplane at infinity in projective compactifications.

math.AG↗

arXiv2025

A higher-dimensional Van den Essen type formula for projective foliations and applications

Let $F$ be a one-dimensional holomorphic foliation on $\mathbb{P}^n$ such that $W\subset Sing(F)$, where $W$ is a smooth complete intersection variety. We determine and compute the variation of the Milnor number $ μ(F, W)$ under blowups, which depends on the vanishing order of the pullback foliation along the exceptional divisor, as well as on numerical and topological invariants of $W$. This represents a higher-dimensional version of Van den Essen's formula for projective foliations of dimension one. As an application, we obtain a lower bound for the Milnor number of the foliation. Also, we use this formula to show that for a foliation on $\mathbb{P}^n$ that is singular along a smooth curve, there exists a finite number of blow-ups with centers on smooth curves such that the induced foliation has multiplicity equal to 1 and that for generic points of the curves in the final stage, the singularities are elementary. Moreover, we obtain a bound on the maximum number of blow-ups needed to resolve the foliation, depending on the numerical and topological invariants of the curve.

math.AG↗

arXiv2021

On the Milnor number of non-isolated singularities of holomorphic foliations and its topological invariance

We define the Milnor number -- as the intersection number of two holomorphic sections -- of a one-dimensional holomorphic foliation $\mathscr{F}$ with respect to a compact connected component $C$ of its singular set. Under certain conditions, we prove that the Milnor number of $\mathscr{F}$ on a three-dimensional manifold with respect to $C$ is invariant by $C^1$ topological equivalences.

math.CV↗

arXiv2017

On foliations by curves with singularities of positive dimension

We present enumerative results for holomorphic foliations by curves on $\mathbb{P}^n$, $n\geq 3$, with singularities of positive dimension. Some of the results presented improve previous ones due to Corrêa--Fernández-Pérez--Nonato Costa--Vidal Martins and Nonato Costa. We also present an enumerative result bounding the number of isolated singularities in a projective subvariety invariant by a holomorphic foliation by curves on $\mathbb{P}^n$ with a singularity of positive dimension.

math.AG↗

arXiv2012

Foliations by Curves with Curves as Singularities

Let $\mathcal F$ be a holomorphic one-dimensional foliation on $\mathbb{P}^n$ such that the components of its singular locus $Σ$ are curves $C_i$ and points $p_j$. We determine the number of $p_j$, counted with multiplicities, in terms of invariants of $\mathcal F$ and $C_i$, assuming that $\mathcal F$ is special along the $C_i$. Allowing just one nonzero dimensional component on $Σ$, we also prove results on when the foliation happens to be determined by its singular locus.

math.AG↗