Given integers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n \ge k \ge d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mi>k</mi> <mo>≥</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$X_{n,k,d}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> be the moduli space of n-tuples of lines <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(\ell _1, \dots , \ell _n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>ℓ</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathbb {C}}^k$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\ell _1 + \cdots + \ell _n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>ℓ</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> has dimension d. We give a quotient presentation of the torus-equivariant cohomology of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$X_{n,k,d}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. The form of this presentation, and in particular the torus parameters appearing therein, will arise from the orbit harmonics method of combinatorial deformation theory.

Equivariant Cohomology of Grassmannian Spanning Lines

期刊论文
发表日期：

2026年5月21日

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Raymond Chou,
Tomoo Matsumura,
Brendon Rhoades

摘要

Given integers $n \ge k \ge d$ $n \geq k \geq d$ , let $X_{n,k,d}$ $X_{n, k, d}$ be the moduli space of n-tuples of lines $(\ell _1, \dots , \ell _n)$ $(ℓ_{1}, \dots, ℓ_{n})$ in ${\mathbb {C}}^k$ $C^{k}$ such that $\ell _1 + \cdots + \ell _n$ $ℓ_{1} + \dots + ℓ_{n}$ has dimension d. We give a quotient presentation of the torus-equivariant cohomology of $X_{n,k,d}$ $X_{n, k, d}$ . The form of this presentation, and in particular the torus parameters appearing therein, will arise from the orbit harmonics method of combinatorial deformation theory.