Singularities of maps, characteristic classes, and intersection homology
Morse-Conley Theory, Singular Varieties, and Intersection Homology
| Grant number: | 16/23906-3 |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
| Start date: | May 01, 2017 |
| End date: | January 31, 2018 |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Geometry and Topology |
| Principal Investigator: | João Nivaldo Tomazella |
| Grantee: | Guillermo Penafort Sanchis |
| Host Institution: | Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil |
| Associated research grant: | 14/00304-2 - Singularities of differentiable mappings: theory and applications, AP.TEM |
Abstract This project is concerned with vanishing homology of the image $Z_\epsilon= f_\epsilon(\mathbb C^n)$ of perturbations $f_\epsilon$ of finite map-germs $f\colon (\mathbb C^n,0)\to(\mathbb C^p,0)$. Results in this field run parallel to the ones for Milnor fibrations of germs of complete intersections with isolated singularity (ICIS), but the highly singular nature of $Z_\epsilon$ imposes obstacles that have limited progress. The introduction of \emph{multiple-point spaces} of maps is a means to circumvent these obstacles, because they are significantly less singular and their \emph{alternating homology modules} encode the homology of $Z_\epsilon$. However, our ability to describe the multiple-point spaces depends strongly on the corank of $f$ ($\corank f_x$ is the rank of the kernel of the differential $df_x$), and developing a good scheme-theoretical description of multiple-points in the presence of points of corank $\geq 2$ is one of the hardest problems in the field. Pe\~nafort introduced modified multiple-point spaces that overcome these problems in a wide range of dimensions. The main problems to be dealt with are (A) Mond's conjecture, which is the inequality for maps-germs $\mathbb C^n\to \mathbb C^{n+1}$ corresponding to that of Milnor and Tjurina numbers for ICIS. It is only known to be true for $n=1,2$. Bobadilla, Nu\~no and Pe\~nafort have shown that it can be reduced to a discrete family of examples of increasing order. There is still no such family in the literature, and Or\'efice-Okamoto and Pe\~nafort are currently working on the search for such a family for $n=3$. (B) Connectivity results: determination --in terms of the dimensions of source, target and singular locus-- of a range of degrees where the homology of $Z_\epsilon$ is trivial, in the same vein as Kato-Matsumoto's theorem for complete intersections. This is closely related to that the study of Tjurina modifications of determinantal varieties studied by Nu\~no, Tomazella and Or\'efice-Okamoto; and (C) Extended L\^e's conjecture: Is there any injective map-germ $f\colon (\mathbb C^n,0)\to(\mathbb C^p,0)$ satisfying $\corank f>p-n$? (AU) | |
| News published in Agência FAPESP Newsletter about the scholarship: | |
| More itemsLess items | |
| TITULO | |
| Articles published in other media outlets ( ): | |
| More itemsLess items | |
| VEICULO: TITULO (DATA) | |
| VEICULO: TITULO (DATA) | |