On the topology of determinantal links

Abstract

We study the cohomology of the generic determinantal varieties $M_{m,n}^s=\{\phi \in {\mathbb{C}}^{m\times n}:\mathrm{rank}\phi <s\}$, their polar multiplicities, their sections $D_k\cap M_{m,n}^s$ by generic hyperplanes $D_k$ of various dimension $k$, and the real and complex links of the spaces $(D_k\cap M_{m,n}^s,0)$. Such complex links were shown to provide the basic building blocks in a bouquet decomposition for the (determinantal) smoothings of smoothable isolated determinantal singularities. The detailed vanishing topology of such singularities was still not fully understood beyond isolated complete intersections and a few further special cases. Our results now allow to compute all distinct Betti numbers of any determinantal smoothing.

arXiv comment41 pages, 7 tables

References in corpus (5)

• On the Vanishing Topology of Isolated Cohen-Macaulay Codimension 2 Singularities
• Milnor and Tjurina numbers for smoothings of surface singularities
• Local Euler Obstruction and Chern-Mather classes of Determinantal Varieties
• On Chern classes of tensor products of vector bundles
• Kato-Matsumoto-type results for disentanglements