Tannakian classification of equivariant principal bundles on toric varieties (Q2210564)

Let \(k\) be a field and \(X\) a \(k\)-variety (i.e., a separated integral finite type \(k\)-scheme). Let \(G\) and \(\Gamma\) be affine algebraic groups over \(k\). Assume that \(\Gamma\) is acting on \(X\) algebraically from the left. Let \(\mathfrak{Pbun}_G^{\Gamma}(X)\) be the category of \(\Gamma\)-equivariant pricipal \(G\)-bundles on \(X\), and let \(\mathfrak{Vect}^{\Gamma}(X)\) be the category of \(\Gamma\)-equivariant vector bundles on \(X\). Let \(G\text{-Mod}\) be the category of finite dimensional \(k\)-linear representations of \(G\), and let \(\mathfrak{Nor}^{\Gamma}(X)\) be the full subcategory of the functor category \(\mathrm{Fun}\big(G\text{-Mod}, \mathfrak{Vect}^{\Gamma}(X)\big)\), with objects satisfying Nori's axioms F1--F4 as in [\textit{M. V. Nori}, Compos. Math. 33, 29--41 (1976; Zbl 0337.14016)]. In \S2, Generalizing Nori's construction [loc. cit.] of principal \(G\)-bundles in \(\Gamma\)-equivariant setup, the authors constructs an equivalence of categories \[\mathbf{N}_0^{\Gamma} : \mathfrak{Pbun}_G^{\Gamma}(X) \stackrel{\simeq}{\longrightarrow} \mathfrak{Nor}^{\Gamma}(X). \] From \S3 onwards, the authors focus on the case when \(X\) is a toric variety over \(k\). A fan \(\Sigma\) in a lattice \(N\) is a collection of (strongly convex rational polyhedral) cones in \(N_{\mathbb R}\) satisfying certain axioms: face of a cone in \(\Sigma\) is again a cone in \(\Sigma\), and intersection of any two cones \(\sigma_1, \sigma_2 \in \Sigma\) is a face of both \(\sigma_1\) and \(\sigma_2\). Given a lattice \(N\) and a fan \(\Sigma\), each cone \(\sigma \in \Sigma\) determines an affine \(k\)-scheme \(X_{\sigma}\), and the axioms of cones in \(\Sigma\) allow them to glue together appropriately to give a \(k\)-scheme \(X_{\Sigma}\), which is a toric \(k\)-variety under the action of the torus \(T = N\otimes\mathbb{G}_m\); see e.g. [\textit{W. Fulton}, Introduction to toric varieties. The 1989 William H. Roever lectures in geometry. Princeton, NJ: Princeton University Press (1993; Zbl 0813.14039)]. Let \(\mathfrak{Cvec}(\Sigma)\) be the category of compatible \(\Sigma\)-filtered finite dimensional \(k\)-vector spaces (see \S 3 for relevant definitions), and let \(\mathfrak{Vec}^T(X)\) be the category of \(T\)-equivariant vector bundles on the toric \(k\)-variety \(X = X_{\Sigma}\). Fix a closed point \(x_0 \in X\) in the open \(T\)-orbit \(\mathcal{O} \subset X\). For a \(T\)-equivariant vector bundle \(\mathcal{E}\) on \(X\), its fiber \(\mathcal{E}_{x_0}\) at \(x_0\) admits a structure of a compatible \(\Sigma\)-filtered \(k\)-vector space. This gives a functor \[\mathbf{F} : \mathfrak{Vec}^T(X) \longrightarrow \mathfrak{Cvec}(\Sigma), \] which is an exact equivalence of categories; see [\textit{A. A. Klyachko}, Math. USSR, Izv. 35, No. 2, 337--375 (1990; Zbl 0706.14010); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 5, 1001--1039 (1989)]. In \S3, it is shown that the functor \(\mathbf{F}\) respects tensor product structures, and its quasi-inverse (as constructed by Klyachko in [loc. cit.]) is also an additive tensor functor. Let \(\mathfrak{Cnor}(\Sigma)\) be the category whose objects are functors \(\mathbf{M} : G\text{-mod} \longrightarrow \mathfrak{Cvec}(\Sigma)\) satisfying Nori's axioms F1--F4, and morphisms are natural transformations of functors. Then using equivalences \(\mathbf{N}_0^{\Gamma}\) and \(\mathbf{F}\), as mentioned above, the authors deduce an equivalence of categories \[\mathfrak{Pbun}_G^T(X) \stackrel{\simeq}{\longrightarrow} \mathfrak{Cnor}(\Sigma), \] which sends an object \(\mathcal{P} \in \mathfrak{Pbun}_G^T(X)\) to the functor \(\mathbf{F}\circ\mathbf{N}_0^{\Gamma}(\mathcal{P}) \in \mathfrak{Cnor}(\Sigma)\). Henceforth we assume that the base field \(k\) has characteristic \(0\), and \(G\) is a reductive affine algebraic group over \(k\). Let \(\mathfrak{Fvec}(\Sigma)\) be the category of \(\Sigma\)-filtered finite dimensional \(k\)-vector spaces, and let \(\mathbf{O} : \mathfrak{Fvec}(\Sigma) \longrightarrow \mathfrak{Vec}\) be the forgetful functor into the category of \(k\)-vector spaces. Given a functor \(\mathbf{E} : G\text{-Mod} \longrightarrow \mathfrak{Vec}^{\Gamma}(X)\) in the category \(\mathfrak{Nor}^{\Gamma}(X)\), consider the functors \[\mathbf{E}_{\#} := \mathbf{F}\circ\mathbf{E} : G\text{-Mod} \longrightarrow \mathfrak{Cvec}(\Sigma), \ \ \text{and} \ \ E_{\#} := \mathbf{O}\circ\mathbf{E}_{\#} : G\text{-Mod} \longrightarrow \mathfrak{Vec}.\] Note that, \(\mathbf{E}_{\#} \in \mathfrak{Cnor}(\Sigma)\) and the fiber of \(\mathbf{E}_{\#}(V)\) at \(x_0\) is \(E_{\#}(V)\). Let \(\overline{G\text{-Mod}}\) be the category of all (possibly infinite dimensional) \(k\)-linear representations of \(G\). Similarly let \(\overline{\mathfrak{Cvec}}\) and \(\overline{\mathfrak{Vec}}\) be the categories of all compatible \(\Sigma\)-filtered \(k\)-vector spaces, and the category of all \(k\)-vector spaces, respectively. Then in \S5 it is shown that the above functors \(\mathbf{E}_{\#}\) and \(E_{\#}\) uniquely extends to the functors \[\overline{\mathbf{E}}_{\#} : \overline{G\text{-Mod}} \longrightarrow \overline{\mathfrak{Cvec}(\Sigma)}, \ \ \text{and} \ \ \overline{E}_{\#} : \overline{G\text{-Mod}} \longrightarrow \overline{\mathfrak{Vec}} \] which preserves direct limits and tensor products, and satisfies Nori's axioms F2-F4. Consequently, they give rise to functors (also denoted by the same symbols) \(\overline{\mathbf{E}}_{\#}\) and \(\overline{E}_{\#}\) from the category of affine \(G\)-schemes to the category of \(\Sigma\)-filtered \(k\)-algebras and the category of \(k\)-algebras, respectively. It is shown that \(\overline{E}_{\#}(k[G])\) is the algebra of regular functions of the fiber (at \(x_0\)) of the principal \(G\)-bundle \(\overline{\mathbf{E}}(k[G])\). Also the \(\Sigma\)-filtered \(k\)-algebra \(\overline{\mathbf{E}}_{\#}(k[G])\) is compatible, and for any \(n\)-dimensional cone in \(\Sigma\), any action of \(T\) on \(\overline{\mathbf{E}}_{\#}(k[G])\), which is compatible with the \(\Sigma\)-filtration, commutes with the \(G\)-action. Henceforth we further assume that \(k\) is algebraically closed. Let \(\mathfrak{Calg}_G(\Sigma)\) be the category of compatible \(\Sigma\)-filtered \(k\)-algebras \(B\) which admits a \(G\)-action such that \(B\) is \(G\)-equivariantly isomorphic to the algebra \(k[G]\), and for every top-dimensional cone in the fan \(\Sigma\), \(B\) admits a compatible \(T\)-action which commutes with the \(G\)-action on \(B\). The morphisms of objects in \(\mathfrak{Calg}_G(\Sigma)\) are \(G\)-equivariant isomorphisms of compatible \(\Sigma\)-filtered \(k\)-algebras. In \S6 the authors show that the association \(\mathbf{E} \mapsto \overline{\mathbf{E}}_{\#}(k[G])\) gives a faithful functor \[\mathbf{A} : \mathfrak{Nor}^T(X) \longrightarrow \mathfrak{Calg}_G(\Sigma),\] which is an equivalence of categories when all maximal cones in \(\Sigma\) are top-dimensional. As a result, they deduce that for a toric variety \(X = X_{\Sigma}\) for which all maximal conses in \(\Sigma\) are top-dimensional, there is an equivalence of categories between \(\mathfrak{Pbun}_G^T(X)\) and \(\mathfrak{Calg}_G(\Sigma)\). Finally, with the above assumption on \(X\), in \S 7 the authors show that if \(H\) is a reductive algebraic subgroup of \(G\), then a \(T\)-equivariant principal \(G\)-bundle \(E_G\) on \(X = X_{\Sigma}\) admits a \(T\)-equivariant reduction of its structure group to \(H\) if and only if there is a \(\Sigma\)-filtered \(k\)-algebra \(R \in \mathfrak{Calg}_H(\Sigma)\) such that \((R\otimes k[G])^H \cong \overline{\mathbf{E}}_{\#}(k[G])\), where \(\mathbf{E} = \mathbf{N}_0^T(E_G)\).