The Bruce-Roberts Numbers of 1-Forms on an ICIS
We relate the Bruce-Roberts numbers of a 1-form with respect to an ICIS to other invariants as the GSV-index, Tjurina and Milnor numbers.
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arXiv subjects
João Nivaldo Tomazella
Publications and source records attributed to João Nivaldo Tomazella.
On invariants of a map germ from n-space to 2n-space
We consider $\mathcal{A}$-finite map germs $f$ from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{2n},0)$. First, we show that the number of double points that appears in a stabilization of $f$, denoted by $d(f)$, can be calculated as the length of the local ring of the double point set $D^2(f)$ of $f$, given by the Mond's ideal. In the case where $n\leq 3$ and $f$ is quasihomogeneous, we also present a formula to calculate $d(f)$ in terms of the weights and degrees of $f$. Finally, we consider an unfolding $F(x,t) = (f_t(x),t)$ of $f$ and we find a set of invariants whose constancy in the family $f_t$ is equivalent to the Whitney equisingularity of $F$. As an application, we present a formula to calculate the Euler obstruction of the image of $f$.
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The Bruce-Roberts Numbers of a Function on an ICIS
We give formulas for the Bruce-Roberts number $μ_{BR}(f,X)$ and its relative version $μ_{BR}^{-}(f,X)$ of a function $f$ with respect to an ICIS $(X,0)$. We show that $μ_{BR}^{-}(f,X)=μ(f^{-1}(0)\cap X,0)+μ(X,0)-τ(X,0)$, where $μ$ and $τ$ are the Milnor and Tjurina numbers, respectively, of the ICIS. The formula for $μ_{BR}(f,X)$ is more complicated and also involves $μ(f)$ and some lengths in terms of the ideals $I_X$ and $Jf$. We also consider the logarithmic characteristic variety, $LC(X)$, and its relative version, $LC(X)^{-}$. We show that $LC(X)^{-}$ is Cohen-Macaulay and that $LC(X)$ is Cohen-Macaulay at any point not in $X\times\{0\}$. We generalize previous results presented by the authors when $(X,0)$ has codimension one and by Bruce and Roberts when it is weighted homogeneous of any codimension.
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Families of ICIS with constant total Milnor number
We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well known result of Gabrielov, Lazzeri and Lê for hypersurfaces. We use A'Campo's theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety.
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Image Milnor number and $\mathscr{A}_e$-codimension for maps between weighted homogeneous irreducible curves
Let $(X,0)\subset (\mathbb{C}^n,0)$ be an irreducible weighted homogeneous singularity curve and let $f:(X,0)\to(\mathbb{C}^2,0)$ be a map germ finite, one-to-one and weighted homogeneous with the same weights of $(X,0)$. We show that $\mathscr{A}_e$-$codim(X,f)=μ_I(f)$, where $\mathscr{A}_e$-$codim(X,f)$ is the $\mathscr{A}_e$-codimension, i.e., the minimum number of parameters in a versal deformation and $μ_I(f)$ is the image Milnor number, i.e., the number of vanishing cycles in the image of a stabilisation of $f$.
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Invariants of Topological Relative Right Equivalences
The constancy of the Milnor number has several characterizations which were summarized by Greuel in 1986. This paper presents a study of these characterizations in the case of families of functions with isolated singularities defined on an analytic variety.
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