The structure of the cohomology ring of the filt schemes

TL;DR

This paper analyzes the cohomology ring structure of quot schemes over smooth projective curves, providing a filtration, a simple associated graded ring, and a combinatorial description of the cohomology ring.

Contribution

It introduces a filtration on the cohomology ring of quot schemes, describes the associated graded ring, and offers a combinatorial description using complete filt schemes and a splitting principle.

Findings

Established a filtration on the cohomology ring of quot schemes.

Derived a simple structure for the associated graded ring.

Provided a combinatorial description of the cohomology ring.

Abstract

Let $C$ be a smooth projective curve over the complex number field $\cnum$. We investigate the structure of the cohomology ring of the quot schemes $\Quot(r,n)$, i.e., the moduli scheme of the quotient sheaves of $\nbigo_C^{\oplus r}$ with length $n$. We obtain a filtration on $H^{\ast}(\Quot(r,n))$, whose associated graded ring has a quite simple structure. As a corollary, we obtain a small generator of the ring. We also obtain a precise combinatorial description of $H^{\ast}(\Quot(r,n))$ itself. For that purpose, we consider the complete filt schemes and we use a `splitting principle'. A complete filt scheme has an easy geometric description: a sequence of bundles of projective spaces. Thus the cohomology ring of the complete filt schemes are easily determined. In the cohomology ring of complete filt schemes, we can do some calculations for the cohomology ring of quot schemes. Keywords: Quot scheme, cohomology ring, torus action, splitting principle, symmetric function.