TY - UNPB

T1 - On the topology of determinantal links

AU - Zach, Matthias

N1 - 41 pages, 7 tables

PY - 2021/7/5

Y1 - 2021/7/5

N2 - 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 $Dk∩Msm,n$ by generic hyperplanes $Dk$ of various dimension $k$, and the real and complex links of the spaces $(Dk∩Msm,n,0)$. 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.

AB - 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 $Dk∩Msm,n$ by generic hyperplanes $Dk$ of various dimension $k$, and the real and complex links of the spaces $(Dk∩Msm,n,0)$. 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.

M3 - Preprint

BT - On the topology of determinantal links

ER -