The Arnold–Givental Lagrangian intersection estimate for the real part of a weighted projective space with not-all-odd weights

For a weighted projective space $\mathbb{C}P^n(\mathbf{q})$ with its standard symplectic orbifold structure, the real part $\mathbb{R}P^n(\mathbf{q})$ is a Lagrangian suborbifold, and Guangcun Lu proved the Arnold-type Lagrangian intersection estimate (AC2) for the pair $(\mathbb{C}P^n(\mathbf{q}),\mathbb{R}P^n(\mathbf{q}))$ when all weights are odd, while proving the Hamiltonian fixed-point estimate (AC1) for arbitrary weights. Lu explicitly asked whether (AC2) continues to hold when some weights are even — the case in which the $\mathbb{Z}_2$-equivariant minimax proof degenerates because the involution's fixed set and its complement are both infinite-dimensional, or, when all weights are even, because the involution is trivial. This record poses that question: prove or refute that every Hamiltonian time-one map $\phi_1$ of $\mathbb{C}P^n(\mathbf{q})$ satisfies $\sharp(\mathbb{R}P^n(\mathbf{q})\cap\phi_1(\mathbb{R}P^n(\mathbf{q})))\ge CL(\mathbb{R}P^n(\mathbf{q});\mathbb{Z}_2)+1$, with the transverse lower bound $\ge SB(\mathbb{R}P^n(\mathbf{q});\mathbb{Z}_2)$, for every weight vector with not-all-odd weights. The follow-up literature (2008–2026) leaves the question open: available orbifold Floer theory covers only torus fibers in toric orbifolds and global quotients, and the recently introduced dihedral-twisted-sector framework needed here is still foundational. Resolving the question requires Lagrangian Floer or Lusternik–Schnirelman theory on symplectic orbifolds for Lagrangians that meet the orbifold strata, or an explicit disjoining Hamiltonian.

Background

A symplectic orbifold is an orbifold — a space locally modeled on a quotient of Euclidean space by a finite group, with compatible charts — equipped with a closed, nondegenerate 2-form. Weighted projective spaces are the basic examples. For a tuple $\mathbf{q}=(q_1,\dots,q_{n+1})$ of positive integers, the weighted projective space $\mathbb{C}P^n(\mathbf{q})$ is the quotient of $\mathbb{C}^{n+1}\setminus\{0\}$ by the $\mathbb{C}^*$-action $\alpha\cdot z=(\alpha^{q_1}z_1,\dots,\alpha^{q_{n+1}}z_{n+1})$. It is a compact symplectic orbifold: the point $[z]_{\mathbf{q}}$ has structure group $\mathbb{Z}/m(z)\mathbb{Z}$, where $m(z)$ is the greatest common divisor of the weights $q_j$ with $z_j\ne0$, and the symplectic form is obtained by symplectic reduction of the standard structure on $\mathbb{C}^{n+1}$ along the Hamiltonian circle action of weights $\mathbf{q}$. When all weights equal 1 this recovers the smooth manifold $\mathbb{C}P^n$. Hamiltonian maps (time-one maps of time-dependent Hamiltonian vector fields) are defined on symplectic orbifolds exactly as on manifolds.

Arnold's conjecture (in its homological form) predicts for a Hamiltonian time-one map $\phi_1$ of a closed symplectic manifold $(P,\omega)$ that $\sharp\mathrm{Fix}(\phi_1)\ge CL(P)+1$, where $CL$ denotes the cup-length (the maximal number of positive-degree cohomology classes with nonzero cup product), and $\sharp\mathrm{Fix}(\phi_1)\ge SB(P)$ when all fixed points are nondegenerate, where $SB$ is the sum of Betti numbers. For a closed Lagrangian submanifold $L$ — an $n$-dimensional submanifold of a $2n$-dimensional symplectic manifold on which the symplectic form vanishes — the companion Lagrangian intersection conjecture (AC2) predicts $\sharp(L\cap\phi_1(L))\ge CL(L)+1,$ and $\sharp(L\cap\phi_1(L))\ge SB(L)$ when $L$ and $\phi_1(L)$ intersect transversally. Fortune proved the fixed-point estimate for $\mathbb{C}P^n$; Givental and, independently with different methods, Chang–Jiang proved (AC2) for $(\mathbb{C}P^n,\mathbb{R}P^n)$, giving $\sharp(\mathbb{R}P^n\cap\phi_1(\mathbb{R}P^n))\ge n+1$. Hofer developed a Lusternik–Schnirelman-theoretic approach to Lagrangian intersections valid when $\pi_2(P,L)=0$, and Oh extended Floer-theoretic methods to monotone Lagrangians, including the pair $(\mathbb{C}P^n,\mathbb{R}P^n)$.

Lu carried these questions to weighted projective spaces. Writing $r(\mathbf{q})$ for the number of odd weights, the real part is $\mathbb{R}P^n(\mathbf{q})=(\mathbb{R}^{n+1}\setminus\{0\})/\mathbb{R}^*,$ with the same weighted scalar action; it embeds as a Lagrangian suborbifold of $\mathbb{C}P^n(\mathbf{q})$. It is a smooth manifold (diffeomorphic to $\mathbb{R}P^n$) exactly when all weights are odd, i.e. $r(\mathbf{q})=n+1$. Lu proved: (i) (AC1) holds on $\mathbb{C}P^n(\mathbf{q})$ for arbitrary weights — any Hamiltonian time-one map has at least $n+1$ fixed points; and (ii) (AC2) holds for $(\mathbb{C}P^n(\mathbf{q}),\mathbb{R}P^n(\mathbf{q}))$ when all weights are odd. The proof of (ii) adapts the $\mathbb{Z}_2$-equivariant minimax scheme of Chang–Jiang on a space of loops/discs, using that the relevant involution acts freely off its fixed set. When some weight is even ($0\le r(\mathbf{q})\le n$), the equivariant argument degenerates: for $1\le r(\mathbf{q})\le n$ the induced $\mathbb{Z}_2$-action has the nontrivial fixed-point set $\{u : u_i=0 \text{ whenever } q_i\notin 2\mathbb{Z}\}$, and both this fixed set and its complement are infinite-dimensional, so the equivariant index arguments collapse; for $r(\mathbf{q})=0$ (all weights even) the involution is trivial. Lu left the question open in all these cases. The expected bounds are computable from the topology of the real part (with $\mathbb{Z}_2$ coefficients): topologically $\mathbb{R}P^n(\mathbf{q})$ is the $(n+1-r(\mathbf{q}))$-fold unreduced suspension of $\mathbb{R}P^{r(\mathbf{q})-1}$, so $CL=1$ and $SB=2$ for $r(\mathbf{q})=0$ (where $\mathbb{R}P^n(\mathbf{q})\cong S^n$), $CL=0$ and $SB=1$ for $r(\mathbf{q})=1$, and $CL=1$ and $SB=r(\mathbf{q})$ for $2\le r(\mathbf{q})\le n$. Note an additional structural difficulty: whenever some weight $q_i$ is even, the real part contains orbifold points of $\mathbb{C}P^n(\mathbf{q})$ (for instance the coordinate point with only $z_i\ne0$, with structure group $\mathbb{Z}/q_i$), so the Lagrangian meets the orbifold strata and is not contained in the smooth locus; even a Hofer-type theorem for Lagrangians inside the smooth locus of a symplectic orbifold — a generalization Lu suggested as a program — would not settle the present question.

Problem Statement

Let $\mathbf{q}=(q_1,\dots,q_{n+1})$ be a tuple of positive integers with $r(\mathbf{q})=\#\{i : q_i \text{ odd}\}\le n$, i.e. with at least one even weight. Equip the weighted projective space $\mathbb{C}P^n(\mathbf{q})$ with its standard reduced symplectic orbifold structure and let $\mathbb{R}P^n(\mathbf{q})\subset\mathbb{C}P^n(\mathbf{q})$ be the real Lagrangian suborbifold. Let $\phi_1$ be the time-one map of the Hamiltonian vector field of a time-dependent Hamiltonian on the orbifold $\mathbb{C}P^n(\mathbf{q})$. Does the Arnold-type Lagrangian intersection estimate (AC2) hold for $(\mathbb{C}P^n(\mathbf{q}),\mathbb{R}P^n(\mathbf{q}))$ in these cases? That is, is it true that $\sharp\bigl(\mathbb{R}P^n(\mathbf{q})\cap\phi_1(\mathbb{R}P^n(\mathbf{q}))\bigr)\ge CL(\mathbb{R}P^n(\mathbf{q});\mathbb{Z}_2)+1,$ with the stronger bound $\sharp\bigl(\mathbb{R}P^n(\mathbf{q})\cap\phi_1(\mathbb{R}P^n(\mathbf{q}))\bigr)\ge SB(\mathbb{R}P^n(\mathbf{q});\mathbb{Z}_2)$ when the two real parts intersect transversally? Concretely, the first bound demands at least two intersection points for $r(\mathbf{q})=0$ (where $\mathbb{R}P^n(\mathbf{q})\cong S^n$) and for $2\le r(\mathbf{q})\le n$, and at least one — i.e. non-displaceability — for $r(\mathbf{q})=1$; the transverse bound demands 2, 1, and $r(\mathbf{q})$ points in the three ranges respectively. The problem is to settle this question of Lu (2008), Section 2.3: either prove the estimate for all weight vectors with $0\le r(\mathbf{q})\le n$ (with the transverse clause under transversality), or refute it by exhibiting a weight vector with an even weight and an explicit Hamiltonian time-one map of $\mathbb{C}P^n(\mathbf{q})$ whose intersection count violates the stated bound. This is question (i) of the source; the companion methodological program of the source — extending Hofer's Lusternik–Schnirelman theory and Oh's monotone Lagrangian Floer theory to closed symplectic orbifolds — is the natural route suggested by the source, since $\mathbb{R}P^n(\mathbf{q})$ meets the orbifold strata of $\mathbb{C}P^n(\mathbf{q})$ whenever some weight is even.

References

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