广义抛物Hitchin对的无穷小变形
Infinitesimal Deformations of Generalized Parabolic Hitchin Pairs
中文总结 AI 辅助
该研究建立不可约结点曲线上广义抛物Hitchin对的无穷小变形理论，构造对应变形复形，证明特定1-稳定广义抛物Hitchin对的粗模空间是局部完全交且带自然泊松结构。
中文摘要 AI 辅助
我们在不可约结点曲线上建立广义抛物Hitchin对（GPHs）的无穷小变形理论。对任意广义抛物Hitchin对$(E,\phi,F(E))$，我们构造显式的三项变形复形$\mathcal{C}_{(E_\bullet,\phi)}$，其第一上同调$\mathbb{H}^1(\mathcal{C}_{(E_\bullet,\phi)})$参数化一阶变形。作为主要应用，我们证明秩为$n$、次数为$d$且满足$\gcd(n,d)=1$的1-稳定广义抛物Hitchin对的粗模空间$\mathcal{M}_{\mathrm{GPH}}$是局部完全交，在余维1处光滑，因此是正规簇；我们还证明该模空间带有自然的泊松结构。
英文摘要
We develop the infinitesimal deformation theory of generalized parabolic Hitchin pairs (GPHs) on irreducible nodal curves. For any GPH $(E,ϕ,F(E))$ we associate an explicit three-term deformation complex $\mathcal{C}_{(E_\bullet,ϕ)}$ whose first hypercohomology $\mathbb{H}^1(\mathcal{C}_{(E_\bullet,ϕ)})$ parameterizes first-order deformations. As the main application, we prove that the coarse moduli space $\mathcal{M}_{\mathrm{GPH}}$ of $1$-stable generalized parabolic Hitchin pairs of rank $n$ and degree $d$ with $\gcd(n,d)=1$ is a local complete intersection, smooth in codimension one, and hence a normal variety. We further show that this moduli space carries a natural Poisson structure.
