AI 中文总结
该研究完整证明了Arnold-Givental猜想，结合积分Floer理论、哈密顿Floer上同调约化及定制的$\mathbb Z/2$-等变Floer理论局部化新思路，验证了闭辛流形中横截相交的不动点集计数下界不等式。
AI 中文摘要
我们完整证明了Arnold-Givental猜想：给定闭辛流形$(X, ω)$，反辛对合$τ_X: X \to X$的不动点集为$L={\rm Fix}(τ_X)$，若哈密顿微分同胚$ϕ: X \to X$满足$ϕ(L)$与$L$横截相交，则有不等式：\n\\[ \\# \big( ϕ(L) \cap L \big) \geq {\mathrm dim}_{{\mathbb F}_2} H_*(L; {\mathbb F}_2).\\]\n证明结合了第一与第四作者的积分Floer理论方法、Lu提出的哈密顿Floer上同调约化技巧，以及为该问题定制的$\mathbb Z/2$-等变Floer理论中与局部化相关的新思路。
英文摘要
We prove the Arnold-Givental conjecture in full generality: given a closed symplectic manifold $(X, ω)$, an anti-symplectic involution $τ_X: X \to X$ with fixed point set $L={\rm Fix}(τ_X)$, and a Hamiltonian diffeomorphism $ϕ: X \to X$ such that $ϕ(L)$ intersects transversely with $L$, the following inequality holds: \[ \# \big( ϕ(L) \cap L \big) \geq {\mathrm dim}_{{\mathbb F}_2} H_*(L; {\mathbb F}_2).\] The proof combines the methods of integral Floer theory of the first and fourth authors, a reduction to Hamiltonian Floer cohomology due to Lu, and a new idea related to localization in a $\mathbb Z/2$-equivariant Floer theory tailored to the problem.
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