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Curve neighborhoods and combinatorial property $\mathcal{O}$ for a family of odd symplectic partial flag manifolds

Connor Bean, Bradley Cruikshank, Ryan M. Shifler

TL;DR

This work analyzes curve neighborhoods in the odd symplectic partial flag manifold $\mathrm{IF}(1,2;E)$, giving a complete description of the irreducible components of $\Gamma_d$ for Schubert varieties, and showing the associated lattice structure is distributive. It develops a moment-graph framework and a combinatorial quantum Bruhat graph to formulate and prove a combinatorial version of Conjecture $\mathcal{O}$ for $\mathrm{IF}$, connecting curve neighborhood data with eigenstructure of the quantum product by the first Chern class. The main contributions are explicit curve-neighborhood formulas, a lattice-theoretic analysis, and a graph-theoretic route to property $\mathcal{O}$ in the odd symplectic setting. The results provide new combinatorial tools for understanding quantum cohomology and curve-data in non-homogeneous, odd-symplectic geometries with potential implications for related Gamma conjectures.

Abstract

Let $E$ be an odd dimensional complex vector space and $\mbox{IF}:=\mbox{IF}(1,2;E)$ be the family of odd symplectic partial flag manifold. In this paper we give a full description of the irreducible components of the degree $d$ curve neighborhood of any Schubert variety of $\mbox{IF}$, study their lattice structure, and prove a combinatorial version of Conjecture $\mathcal{O}.$

Curve neighborhoods and combinatorial property $\mathcal{O}$ for a family of odd symplectic partial flag manifolds

TL;DR

This work analyzes curve neighborhoods in the odd symplectic partial flag manifold , giving a complete description of the irreducible components of for Schubert varieties, and showing the associated lattice structure is distributive. It develops a moment-graph framework and a combinatorial quantum Bruhat graph to formulate and prove a combinatorial version of Conjecture for , connecting curve neighborhood data with eigenstructure of the quantum product by the first Chern class. The main contributions are explicit curve-neighborhood formulas, a lattice-theoretic analysis, and a graph-theoretic route to property in the odd symplectic setting. The results provide new combinatorial tools for understanding quantum cohomology and curve-data in non-homogeneous, odd-symplectic geometries with potential implications for related Gamma conjectures.

Abstract

Let be an odd dimensional complex vector space and be the family of odd symplectic partial flag manifold. In this paper we give a full description of the irreducible components of the degree curve neighborhood of any Schubert variety of , study their lattice structure, and prove a combinatorial version of Conjecture

Paper Structure

This paper contains 22 sections, 14 theorems, 36 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview
  3. Broader context
  4. Curve neighborhoods
  5. Lattices
  6. Conjecture $\mathcal{O}$
  7. Sufficient Criterion for Property $\mathcal{O}$ to hold
  8. Combinatorial version of the quantum Bruhat graph and Conjecture $\mathcal{O}$
  9. Organization of the main results
  10. Notations and Definitions
  11. A small family of odd symplectic partial flag manifolds
  12. The odd symplectic group
  13. The Weyl group of the symplectic group and odd symplectic minimal representatives
  14. Schubert Varieties in even and odd symplectic partial flag manifolds
  15. Divisors and the first Chern class
  16. ...and 7 more sections

Key Result

Lemma 1.1

If the following conditions hold for a Fano variety $X$: then Property $\mathcal{O}$ holds for $X$.

Figures (5)

  • Figure 1: The moment graph for $\mathrm{IF}$ when $n=2$.
  • Figure 2: In this figure we calculate a few curve neighborhoods of the Schubert point $(1|2)$ for $n=2$ as an example of Proposition \ref{['prop:moment-odd']}.
  • Figure 3: $M_3$ and $N_5$.
  • Figure 4: The table contains the lattices $(\{ \Gamma_{(d_1,d_2)}(X(a|b))\}_{(d_1,d_2) \geq (0,0)}, \leq )$ in $\mathrm{IF}$ for each possible case except the trivial case when $(a|b)=(\bar{2}|\bar{3})$.
  • Figure 5: Combinatorial quantum Bruhat graph of $\mathrm{IF}$ for $n = 2$. Notice that the edge joining $(1|2)$ and $(\bar{2}|1)$ is not in the moment graph.

Theorems & Definitions (31)

  • Lemma 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5: LMSChLi
  • Example 2.1
  • Remark 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 21 more