arXiv subjects

Nivaldo G. Grulha Jr.

Publications and source records attributed to Nivaldo G. Grulha Jr..

7 recordsLinked to original sources

arXiv2020

Gutierrez-Sotomayor Flows on Singular Surfaces

In this work we address the realizability of a Lyapunov graph labeled with GS singularities, namely regular, cone, Whitney, double crossing and triple crossing singularities, as continuous flow on a singular closed $2$-manifold $\mathbf{M}$. Furthermore, the Euler characteristic is computed with respect to the types of GS singularities of the flow on $\mathbf{M}$. Locally, a complete classification theorem for minimal isolating blocks of GS singularities is presented in terms of the branched one manifolds that make up the boundary.

math.DS↗

arXiv2020

The geometrical information encoded by the Euler obstruction of a map

In this work we investigate the topological information captured by the Euler obstruction of a map, $f:(X,0)\to (\mathbb{C}^{2},0)$, with $(X,0)$ a germ of a complex $d$-equidimensional singular space, with $d > 2$, and its relation with the local Euler obstruction of the coordinate functions and, consequently, with the Brasselet number. Nevertheless, under some technical conditions on the departure variety we relate the Chern number of a special collection related to the map-germ $f$ at the origin with the number of cusps of a generic perturbation of $f$ on a stabilization of $(X,f)$.

math.GT↗

arXiv2020

The Effect of Singularization on the Euler Characteristic

In this work, singular surfaces are obtained from smooth orientable closed surfaces by applying three basic simple loop operations, collapsing operation, zipping operation and double loop identification, each of which produces different singular surfaces. A formula that provides the Euler characteristic of the singularized surface is proved. Also, we introduce a new definition of genus for singularized surfaces which generalizes the classical definition of genus in the smooth case. A theorem relating the Euler characteristic to the genus of the singularized surface is proved.

math.GT↗

arXiv2020

Real and complex integral closure, Lipschitz equisingularity and applications on square matrices

Recently the authors investigated the Lipschitz triviality of simple germs of matrices. In this work, we improve some previous results and we present an extension of an integral closure result for the real setting. These tools are applied to investigate classes of square matrices singularities classified by Bruce and Tari.

math.AG↗

arXiv2017

The local Euler obstruction and topology of the stabilization of associated determinantal varieties

This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern--Schwartz--MacPherson class of such varieties. In the second part we compute the Euler characteristic of the stabilization of an essentially isolated determinantal singularity (EIDS). The formula is given in terms of the local Euler obstruction and Gaffney's $m_{d}$ multiplicity.

math.AG↗

arXiv2011

The Multiplicity Polar Theorem, collections of 1-forms and Chern numbers

In this work we show how the Multiplicity Polar Theorem can be used to calculate Chern numbers for a collection of 1-forms.

math.CV↗

arXiv2011

Lê-Greuel type formula for the Euler obstruction and applications

The Euler obstruction of a function can be viewed as a generalization of the Milnor number for functions defined on singular spaces. In this work, using the Euler obstruction of a function, we give a version of the Lê-Greuel formula for two germs of analytic functions with isolated singularity at the origin on a singular space. Using this formula and results of Loeser, we also present an integral formula for the Euler obstruction of a function, generalizing a formula of Kennedy.

math.AG↗