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Tangent Weights and Invariant Curves in Type A Bow Varieties

Alexander O. Foster, Yiyan Shou

TL;DR

This work delivers a complete classification of torus-invariant curves on Cherkis bow varieties of type A by developing a rich combinatorial framework that encodes invariant curves as surgeries on butterfly and Young diagrams, or via binary contingency tables. A central technical advance is a cancellation-free tangent-weight formula at fixed points, enabling efficient computation of tangent data and the construction of invariant curves through explicit surgeries, including both compact and noncompact types. The authors prove that invariant pencils through a fixed point are spanned by three curve types (I, II, III), and show how BCT block swaps and Young diagram surgeries generate all such curves; Hanany–Witten transitions are used to connect separated and non-separated bow varieties. The methodology is demonstrated on example bow varieties, illustrating how to recover their 1-skeletons and providing tools with potential impact on equivariant cohomology and stable envelopes in mirror-symmetric settings.

Abstract

This paper provides a complete classification of torus-invariant curves in Cherkis bow varieties of type A. We develop combinatorial codes for compact and noncompact invariant curves involving the butterfly diagrams, Young diagrams, and binary contingency tables. As a key intermediate step, we also develop a novel tangent weight formula. Finally, we apply this new machinery to example bow varieties to demonstrate how to obtain their 1-skeletons (union of fixed points and invariant curves).

Tangent Weights and Invariant Curves in Type A Bow Varieties

TL;DR

This work delivers a complete classification of torus-invariant curves on Cherkis bow varieties of type A by developing a rich combinatorial framework that encodes invariant curves as surgeries on butterfly and Young diagrams, or via binary contingency tables. A central technical advance is a cancellation-free tangent-weight formula at fixed points, enabling efficient computation of tangent data and the construction of invariant curves through explicit surgeries, including both compact and noncompact types. The authors prove that invariant pencils through a fixed point are spanned by three curve types (I, II, III), and show how BCT block swaps and Young diagram surgeries generate all such curves; Hanany–Witten transitions are used to connect separated and non-separated bow varieties. The methodology is demonstrated on example bow varieties, illustrating how to recover their 1-skeletons and providing tools with potential impact on equivariant cohomology and stable envelopes in mirror-symmetric settings.

Abstract

This paper provides a complete classification of torus-invariant curves in Cherkis bow varieties of type A. We develop combinatorial codes for compact and noncompact invariant curves involving the butterfly diagrams, Young diagrams, and binary contingency tables. As a key intermediate step, we also develop a novel tangent weight formula. Finally, we apply this new machinery to example bow varieties to demonstrate how to obtain their 1-skeletons (union of fixed points and invariant curves).
Paper Structure (18 sections, 18 theorems, 20 equations, 23 figures)

This paper contains 18 sections, 18 theorems, 20 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Brane diagrams and bow varieties
  4. Torus fixed points
  5. Fixed point restrictions
  6. Hanany--Witten transition and separated bow varieties
  7. A combinatorial formula for tangent weights
  8. Butterfly surgeries and invariant curves
  9. Torus invariant curves
  10. Butterfly surgery
  11. Botched butterfly surgery
  12. Nonsurgery curves
  13. Young diagram surgery
  14. BCT block swap moves
  15. Classification of invariant curves
  16. ...and 3 more sections

Key Result

Theorem 3.2

Given a separated brane diagram $\mathcal{D}$ and a fixed point $p\in\mathop{\mathrm{\mathcal{C}}}\nolimits(\mathcal{D})^\mathbb T$, we have as elements of $K^0_\mathbb T(p)$.

Figures (23)

  • Figure 1: The first example of a brane diagram given in RS.
  • Figure 2: An example of a tie diagram whose underlying brane diagram is the one displayed in Figure \ref{['fig:brane_dgm']}.
  • Figure 3: The table-with-margins corresponding to the tie diagram in Figure \ref{['fig:tie_dgm']}.
  • Figure 4: The tie diagram from Figure \ref{['fig:tie_dgm']} along with its corresponding butterfly diagram.
  • Figure 5: The butterfly diagram of Figure \ref{['fig:butterfly_dgm']} decorated by elements of $K^0_\mathbb T(p)$.
  • ...and 18 more figures

Theorems & Definitions (27)

  • Remark 2.1
  • Definition 2.2
  • Definition 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Corollary 3.4
  • Proposition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 17 more