On the topology of determinantal links

TL;DR

This paper investigates the topology of determinantal varieties, computing Betti numbers of their smoothings, and enhances understanding of their links and cohomology, advancing the study of determinantal singularities.

Contribution

It provides a comprehensive computation of Betti numbers for determinantal smoothings and deepens the understanding of their links and cohomological properties.

Findings

Computed all Betti numbers of determinantal smoothings.

Analyzed the topology of links of determinantal varieties.

Extended understanding beyond isolated complete intersections.

Abstract

We study the cohomology of the generic determinantal varieties $M_{m,n}^s = \{ \varphi \in \mathbb C^{m\times n} : \mathrm{rank} \varphi <s \}$, 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.