One-sentence Summary

The authors, affiliated with the University of British Columbia, University of New South Wales, Google DeepMind, and Stanford University, establish that for strictly monotonic degree classes, the motivic class of the space of genus-zero based maps to the complete flag variety $\mathrm{Fl}_{n+1}$Fln+1​ equals that of $\mathrm{GL}_n \times \mathbb{A}^{D - n^2}$GLn​×AD−n2 in the Grothendieck group of varieties, leveraging a novel iterative strategy combining human insight with AI-assisted proof scaffolding; this result implies matching weight polynomials and suggests deep connections between algebraic loop spaces and the rational homotopy type of $U(n)$U(n), though full homotopy equivalence fails in general.

Key Contributions

• The paper studies the algebraic double loop space of the complete flag variety $\mathrm{Fl}_{n+1} = \mathrm{GL}_{n+1}/B$Fln+1​=GLn+1​/B, focusing on the space $\Omega_{\beta}^{2}(\mathrm{Fl}_{n+1})$Ωβ2​(Fln+1​) of degree $\beta$β based maps from $\mathbb{P}^1$P1 to $\mathrm{Fl}_{n+1}$Fln+1​, which serves as an algebraic analog of the topological double loop space and is of interest in both algebraic geometry and homotopy theory.

• Under the condition that $\beta = (d_1, \ldots, d_n)$β=(d1​,…,dn​) is strictly monotonic, the main result establishes an equality in the Grothendieck group of varieties: $[\Omega_{\beta}^{2}(\mathrm{Fl}_{n+1})] = [\mathrm{GL}_n \times \mathbb{A}^{D - n^2}]$[Ωβ2​(Fln+1​)]=[GLn​×AD−n2], where $D = \sum_{k=1}^n 2d_k$D=∑k=1n​2dk​, providing a precise motivic description of this moduli space.

• This motivic equality implies that the weight polynomial of $\Omega_{\beta}^{2}(\mathrm{Fl}_{n+1})$Ωβ2​(Fln+1​) matches that of $\mathrm{GL}_n \times \mathbb{A}^{D - n^2}$GLn​×AD−n2, a variety with the rational homotopy type of $U(n)$U(n), and supports a conjecture that the rational cohomology rings of $\Omega_{\beta}^{2}(\mathrm{Fl}_{n+1})$Ωβ2​(Fln+1​) and $U(n)$U(n) are isomorphic for strictly monotonic $\beta$β.

Introduction

The authors study the space of genus zero, based holomorphic maps from the projective line to the complete flag variety $\mathrm{Fl}_{n+1} = \mathrm{GL}_{n+1}/B$Fln+1​=GLn+1​/B, parameterized by a homology class $\beta = (d_1, \ldots, d_n)$β=(d1​,…,dn​) with strictly monotonic degrees. This space, denoted $\Omega_\beta^2(\mathrm{Fl}_{n+1})$Ωβ2​(Fln+1​), serves as an algebraic analog of the topological double loop space, which is known to have rational homotopy type equivalent to $U(n)$U(n). Understanding its structure is key to probing how algebraic constructions approximate topological invariants. Prior work established this homotopy equivalence in limiting cases—such as minimal $\beta$β or as $\beta \to \infty$β→∞—but failed to capture the full picture for intermediate classes, where the homotopy type may deviate from $U(n)$U(n). The main contribution is a precise formula in the Grothendieck group of varieties: for strictly monotonic $\beta$β, the class $[\Omega_\beta^2(\mathrm{Fl}_{n+1})]$[Ωβ2​(Fln+1​)] equals $[\mathrm{GL}_n \times \mathbb{A}^{D - n^2}]$[GLn​×AD−n2], where $D = \sum_{k=1}^n 2d_k$D=∑k=1n​2dk​. This implies matching point counts over finite fields and suggests deep cohomological agreement, supporting the conjecture that the rational cohomology ring of $\Omega_\beta^2(\mathrm{Fl}_{n+1})$Ωβ2​(Fln+1​) matches that of $U(n)$U(n) for all such $\beta$β. The proof emerged from a novel human-AI collaboration, where AI tools helped generate and refine proof strategies through iterative scaffolding of subproblems, though the final argument is fully human-authored.

Method

The authors leverage a tower of fibrations to analyze the space of degree-d maps from $\mathbb{P}^1$P1 to the flag variety $\mathrm{Fl}_{n+1}$Fln+1​, parameterized by a sequence of degrees $d_n < \cdots < d_1$dn​<⋯<d1​. The framework begins by defining partial flag quotients $\mathrm{Fl}_{n+1,k}$Fln+1,k​, which are moduli spaces of sequences of vector bundle quotients $\mathbb{K}^{n+1} \to V_n \to \cdots \to V_k$Kn+1→Vn​→⋯→Vk​ with $\dim V_a = a$dimVa​=a. Maps from $\mathbb{P}^1$P1 to these spaces are considered, with the space $\Omega_{d_n,\ldots,d_k}^2(\mathrm{Fl}_{n+1,k})$Ωdn​,…,dk​2​(Fln+1,k​) consisting of degree-$d_i$di​ maps $f: \mathbb{P}^1 \to \mathrm{Fl}_{n+1,k}$f:P1→Fln+1,k​ such that $f^*(\mathcal{E}_i) = d_i$f∗(Ei​)=di​ and $f([1:0])$f([1:0]) is the standard partial flag. The maps $\pi_k: \Omega_{d_n,\ldots,d_k}^2(\mathrm{Fl}_{n+1,k}) \to \Omega_{d_n,\ldots,d_{k+1}}^2(\mathrm{Fl}_{n+1,k+1})$πk​:Ωdn​,…,dk​2​(Fln+1,k​)→Ωdn​,…,dk+1​2​(Fln+1,k+1​) are induced by the natural forgetful maps $\mathrm{Fl}_{n+1,k} \to \mathrm{Fl}_{n+1,k+1}$Fln+1,k​→Fln+1,k+1​, which omit the $k$k-th quotient.

[[IMG:|Framework diagram]]

The key insight is that the fiber of $\pi_k$πk​ over a point $f$f corresponds to the space of nowhere vanishing sections of a twisted vector bundle. Specifically, Lemma 2.2 establishes that $\pi_k^{-1}(f) \cong N_{v_{k+1}}(E_{k+1}(d_k - d_{k+1}))$πk−1​(f)≅Nvk+1​​(Ek+1​(dk​−dk+1​)), where $E_{k+1} = f^*(\mathcal{E}_{k+1})$Ek+1​=f∗(Ek+1​) and $v_{k+1}$vk+1​ is the image of the standard basis vector $e_{k+1}$ek+1​ in the fiber over $[1:0]$[1:0]. This identification arises because specifying a refinement of the flag up to $E_k$Ek​ is equivalent to choosing a rank-1 subbundle $R \subset E_{k+1}$R⊂Ek+1​ with $\deg(R) = d_{k+1} - d_k$deg(R)=dk+1​−dk​ and $R|_{[1:0]} = \mathrm{span}(v_{k+1})$R∣[1:0]​=span(vk+1​), which corresponds to a nowhere vanishing section of $E_{k+1}(d_k - d_{k+1})$Ek+1​(dk​−dk+1​) based at $v_{k+1}$vk+1​.

To compute the motivic class of these fibers, Proposition 2.3 provides a formula for the class of the space of based nowhere vanishing sections $N_p(F)$Np​(F) of a vector bundle $F$F of rank $r$r and degree $d$d on $\mathbb{P}^1$P1 satisfying $H^1(F(-2)) = 0$H1(F(−2))=0. The result is $[N_p(F)] = \mathbb{L}^{d-r+1}(\mathbb{L}^{r-1} - 1)$[Np​(F)]=Ld−r+1(Lr−1−1), where $\mathbb{L} = [\mathbb{A}^1]$L=[A1]. Applying this to $F = E_{k+1}(d_k - d_{k+1})$F=Ek+1​(dk​−dk+1​), which has rank $k+1$k+1 and degree $d_{k+1} + (k+1)(d_k - d_{k+1})$dk+1​+(k+1)(dk​−dk+1​), yields the fiber class $[\pi_k^{-1}(f)] = \mathbb{L}^{(k+1)d_k - k d_{k+1} - k}(\mathbb{L}^k - 1)$[πk−1​(f)]=L(k+1)dk​−kdk+1​−k(Lk−1), which is independent of $f$f under the strict monotonicity assumption.

The map $\pi_k$πk​ is not a Zariski locally trivial fibration in general, as the fibers may not be isomorphic as varieties. However, the authors introduce the concept of a motivically trivial fibration, where the base admits a locally closed stratification such that the restriction of the map to each stratum is a Zariski locally trivial fiber bundle with a fiber class independent of the stratum. Proposition 2.7 asserts that $\pi_k$πk​ is such a motivically trivial fibration. The proof, detailed in Section 2.3, constructs this stratification by partitioning the base $\Omega_{d_n,\ldots,d_{k+1}}^2(\mathrm{Fl}_{n+1,k+1})$Ωdn​,…,dk+1​2​(Fln+1,k+1​) based on the splitting type of $E_{k+1}$Ek+1​ and the depth of the basepoint $v_{k+1}$vk+1​ in the fiber $E_{k+1}|_{[1:0]}$Ek+1​∣[1:0]​. For each stratum, the map is shown to be locally trivial with fiber $N_{u_\ell}(F_{\mathbf{r}}(d_k - d_{k+1}))$Nuℓ​​(Fr​(dk​−dk+1​)), where $F_{\mathbf{r}}$Fr​ is the splitting type and $u_\ell$uℓ​ is a reference vector of depth $\ell$ℓ. The automorphism group of $F_{\mathbf{r}}$Fr​ acts transitively on vectors of a given depth, and the stabilizer subgroup is special, ensuring the associated principal bundle is Zariski locally trivial. This allows the authors to compute the motivic class of the total space by multiplying the class of the base by the fiber class at each step of the tower.