AI 中文总结
该研究将多辛约化理论扩展到相对多辛几何，引入相关代数和矩映射，证明约化定理，还证明了动力学约化等的相对类似物及相关变分公式，最后得出相对局部化约束等成果。
AI 中文摘要
我们将Blacker多辛约化理论（其本身是Marsden-Weinstein-Meyer向多元辛流形的约化扩展）扩展到相对多元辛 几何的设定中，其中光滑映射\(F:M→N\)在\(F\)的映射锥复 形中携带一个闭的非退化相对\((k + 1)\)-形式\(\varpi = (\omega,\eta)\)。我们引入相对哈密顿对的李代数和相关的相对矩映射\(\mu\in\Omega^{k - 1}(F,\mathfrak{g}^*)\)，并证明一个相对多辛约化定理：对于一个闭的等变水平\(\phi\in\Omega^{k - 1}(F,\mathfrak{g}^*)\)，\(\mu\)的水平对下降为一个约化的光滑映射\(F_\phi:M_\phi→N_\phi\)，它携带一个唯一的约化闭相对形式\(\varpi_\phi\)；值得注意的是，平凡化分量的水平性由水平的相对闭性强制。我们进一步证明了动力学约化、分裂矩映射\(\mu=\nu\cdot\kappa\)结构理论（其中分裂数据是相对Cartan模型中的一步上同调，使得分裂相对哈密顿\(G\)-空间携带规范的相对同伦矩映射）以及Duistermaat-Heckman型变分公式的相对类似物：相对于合适的共轭分布，约化相对类满足\(\partial_\lambda[\varpi_\psi]=\langle c,\lambda\rangle\cdot[\kappa_\psi]\)，其中\(c\)是通过映射锥的自然\(\Omega(N)\)-模结构作用的目标模型丛的陈类。最后，我们将精确驻相近似转移到目标分量并证明一个真正的相对局部化约束：源上拉回的分裂包的定点贡献恒等地抵消。
英文摘要
We extend the multisymplectic reduction theory of Blacker -- itself the extension of Marsden--Weinstein--Meyer reduction to $k$-plectic manifolds -- to the setting of \emph{relative} multisymplectic geometry, in which a smooth map $F\colon M\to N$ carries a closed nondegenerate relative $(k{+}1)$-form $\varpi=(ω,η)$ in the mapping-cone complex of $F$. We introduce the Leibniz algebra of relative Hamiltonian pairs and the associated relative moment maps $μ\inΩ^{k-1}(F, \mathfrak{g}^{*})$, and prove a relative multisymplectic reduction theorem: for a closed equivariant level $ϕ\inΩ^{k-1}(F, \mathfrak{g}^{*})$, the level pair of $μ$ descends to a reduced smooth map $F_ϕ\colon M_ϕ\to N_ϕ$ carrying a unique reduced closed relative form $\varpi_ϕ$; remarkably, the horizontality of the trivializing component is forced by the relative closedness of the level. We further prove relative analogues of the reduction of dynamics, of the structure theory of split moment maps $μ=ν\cdotκ$ -- for which the splitting datum is a one-step cocycle in the relative Cartan model, so that split relative Hamiltonian $G$-spaces carry canonical relative homotopy moment maps -- and of the Duistermaat--Heckman-type variation formula: with respect to suitable conjugate distributions, the reduced relative class satisfies $\partial_λ[\varpi_ψ]=\langle c, λ\rangle \cdot[κ_ψ]$, where $c$ is the Chern form of the target model bundle acting through the natural $Ω(N)$-module structure of the mapping cone. Finally, we transport the exact stationary phase approximation to the target component and prove a genuinely relative localization constraint: the fixed-point contributions of the pulled-back split package on the source cancel identically.
