---
title: "A proof of the Arnold–Givental conjecture"
authors: ["Shaoyun Bai", "Egor Shelukhin", "Yi Wang", "Guangbo Xu"]
year: 2026
venue: "arXiv:math.SG"
arxiv: "2608.27242"
doi: null
source: "https://arxiv.org/abs/2608.27242"
topics: ["curiosity", "maths-foundations"]
seed_rank: 1481
seed_batch: "curiosity-2026-09-01"
reviewed: "2026-09-01"
pool: "maths-foundations"
relevance_score: 10
lineage: symplectic-topology
cites:
  - title: "A proof of the Arnold–Givental conjecture"
    url: "https://arxiv.org/abs/2608.27242"
    year: 2026
    arxiv: "2608.27242"
    doi: null
see: []
---

# A proof of the Arnold–Givental conjecture

## One-sentence takeaway

After 37 years of partial results, Arnold–Givental is now a theorem in full generality: any Hamiltonian image of a real Lagrangian meets it in at least \(\dim_{\mathbb{F}_2} H_*(L)\) points.

## Why it's lovely

Why you might love this: a folklore symplectic lower bound that resisted every positivity / Calabi–Yau / real-form special case finally yields to Tate-equivariant Floer cohomology — Smith localization lifted to loops. Bai–Shelukhin–Wang–Xu (with Hirschi–Porcelli announcing independently the day before) close a 1989 conjecture by treating \(\mathbb{Z}/2\)-symmetric orbits as the only generators that survive Tate cohomology.

## Key ideas

- Lu's reduction: \(\tau_X\)-symmetric Hamiltonians inject fixed 1-periodic orbits into \(\phi(L)\cap L\).
- Tate-equivariant Floer \(\widehat{HF}^*_{\mathbb{Z}/2}\) over \(\mathbb{F}_2((\theta))\) Novikov; free orbits contribute 0, fixed orbits contribute \(\mathcal{K}\).
- Equivariant PSS injects Tate-Morse of \(X\) into Floer; localization identifies the rank with \(\dim H^*(L;\mathbb{F}_2)\).
- Technical spine: involutive global Kuranishi charts + FOP perturbations that stay transverse under conjugation, and a geometric Tate model via \(\widehat{S}^\infty\).

## Caveats

Over \(\mathbb{F}_2\) only (Givental's \(\mathbb{R}P^n\) already shows char ≠ 2 weakens the count). Does not yet prove heaviness / unobstructedness of \(L\). Concurrent independent announcement by Hirschi–Porcelli; this arXiv is Bai–Shelukhin–Wang–Xu's writeup.

## Implications for Broadside

Pure delight, not a build ticket — but the pattern (equivariant localization that kills free orbits and keeps fixed ones) is the same shape as provenance filters that refuse to merge untracked jumps. Pairs with Floer/Arnold lore already in the library as the closed case.

## Links

- arXiv abs: https://arxiv.org/abs/2608.27242
- PDF: https://arxiv.org/pdf/2608.27242.pdf