[Paper Review] Geometric Invariants of Recursive Group Orbit Stratification

Xiping Zhang|arXiv (Cornell University)|Sep 20, 2020

Algebraic Geometry and Number Theory25 references4 citations

TL;DR

This paper introduces the $c_{sm}$ invariant for recursive group orbit stratifications over algebraically closed fields of characteristic 0, providing explicit algebraic formulas for local Euler obstructions and sectional Euler characteristics of orbit strata. The method systematically computes these geometric invariants using Chern classes of projectivized orbits, with applications to ordinary, symmetric, and skew-symmetric rank loci.

ABSTRACT

The local Euler obstructions and the Euler characteristics of linear sections with all hyperplanes on a stratified projective variety are key geometric invariants in the study of singularity theory. Despite their importance, in general it is very hard to compute them. In this paper we consider a special type of singularity: the recursive group orbits. They are the group orbits of a sequence of $G_n$ representations $V_n$ satisfying certain assumptions. We introduce a new intrinsic invariant called the $c_{sm}$ invariant, and use it to give explicit formulas to the local Euler obstructions and the sectional Euler characteristics of such orbits. In particular, the matrix rank loci are examples of recursive group orbits. Thus as applications, we explicitly compute these geometry invariants for ordinary, skew-symmetric and symmetric rank loci. Our method is systematic and algebraic, thus works for algebraically closed field of characteristic $0$. Moreover, in the complex setting we also compute the stalk Euler characteristics of the Intersection Cohomology Sheaf complexes for all three types of rank loci.

Motivation & Objective

• To develop a systematic algebraic method for computing local Euler obstructions and sectional Euler characteristics on singular projective varieties.
• To address the longstanding difficulty in computing geometric invariants like local Euler obstructions and sectional Euler characteristics for singular spaces.
• To extend existing formulas to recursive group orbit stratifications, including matrix rank loci, using intrinsic invariants.
• To compute stalk Euler characteristics of intersection cohomology sheaf complexes for rank loci in the complex setting.
• To establish binomial identities via Schubert calculus that hold over any field of characteristic 0, with enumerative significance.

Proposed method

• Introduce the $c_{sm}$ invariant as an intrinsic invariant derived from Chern classes of projectivized group orbits in recursive group orbit stratifications.
• Use the formula $\mathrm{Eu}_{\bar{\mathcal{O}}_{n,k}}(\mathcal{O}_{n,r}) = \sum_{(\mu_1 > \cdots > \mu_l \geq k)} \mathrm{Sm}_{n\mu_1} \cdots \mathrm{Sm}_{\mu_l k}$ to compute local Euler obstructions via flag decompositions.
• Apply Schubert calculus to compute Euler characteristics of Grassmannian schemes parametrizing linear subspaces in generic hypersurfaces.
• Relate the Euler characteristic of fiber spaces to differences of quadratic hypersurfaces in projective space, using topological invariants.
• Use the Chern class formula $c_{\mathrm{sm}}^{Z^{X}_{d,n}} = \frac{c(S^\vee \otimes Q) \cdot c_{\binom{d+2}{2}}(\mathrm{Sym}^2(S^\vee))}{c(\mathrm{Sym}^2(S^\vee))}$ to compute invariants on zero loci of sections.
• Leverage vanishing theorems in Schubert calculus to show that certain integrals vanish when $2d \geq n$, based on rank conditions on bundles.

Experimental results

Research questions

• RQ1 Can a systematic algebraic method be developed to compute local Euler obstructions for recursive group orbit stratifications?
• RQ2 What is the role of the $c_{sm}$ invariant in simplifying the computation of geometric invariants like local Euler obstructions and sectional Euler characteristics?
• RQ3 How do the stalk Euler characteristics of intersection cohomology sheaves behave for rank loci in the complex setting?
• RQ4 What enumerative invariants underlie the binomial identities derived from Schubert integration in symmetric and skew-symmetric cases?
• RQ5 Can the vanishing of certain Schubert integrals be explained by geometric or representation-theoretic symmetries?

Key findings

• The local Euler obstruction of a recursive group orbit stratum is given by a sum over flags of $c_{sm}$ invariants: $\mathrm{Eu}_{\bar{\mathcal{O}}_{n,k}}(\mathcal{O}_{n,r}) = \sum_{\mu_1 > \cdots > \mu_l \geq k} \mathrm{Sm}_{n\mu_1} \cdots \mathrm{Sm}_{\mu_l k}$.
• For generic quadratic hypersurfaces in $\mathbb{P}^{3r+1}$, the number of $2r$-planes contained in them is exactly $2^{d+1} \binom{A}{A-d-1}$ with $A = 2\lfloor (n+1)/2 \rfloor$.
• The sectional Euler characteristic of a generic linear section of a rank locus is computed via the difference of Euler characteristics of generic quadratic hypersurfaces in $\mathbb{P}^{n-2d}$ and $\mathbb{P}^{n-2d-1}$.
• The formula $\int_{G(d+1,n+1)} \frac{c(S^\vee \otimes Q) \cdot c_{\binom{d+2}{2}}(\mathrm{Sym}^2(S^\vee))}{c(\mathrm{Sym}^2(S^\vee))} = 2^{d+1} \binom{m}{m-d-1}$ holds for $n = 2m$ or $n = 2m+1$, valid over any characteristic 0 field.
• When $2d \geq n$, the integral of the Chern class expression vanishes due to the vanishing of $\Delta_{[d+1]}(c(S^\vee))$ in Schubert calculus, as $\mathrm{rk} Q < d+1$.
• The derived binomial identities are purely algebraic and hold universally over any algebraically closed field of characteristic 0, despite their enumerative appearance.