Abstract
Let $X$ be a smooth complex projective irreducible curve of genus $g \geq 3$. Let $G$ be the simple complex exceptional Lie group $F_4$ or $E_6$ and let $M(G)$ be the moduli space of principal $G$-bundles. In this work we describe the group of automorphisms of $M(G)$. In particular, we prove that the only automorphisms of $M(F_4)$ are those induced by the automorphisms of the base curve $X$ by pull-back and that the automorphisms of $M(E_6)$ are combinations of the action of the automorphisms of $X$ by pull-back, the action of the only nontrivial outer involution of $E_6$ on $M(E_6)$ by taking the dual and the action of the third torsion of the Picard group of $X$ by tensor product. We also prove a Torelli type theorem for the moduli spaces of principal $F_4$ and $E_6$-bundles, which we use as an auxiliary result in the proof of the main theorems, but which is interesting in itself. We finally draw some conclusions about the way we can see the natural map $M(F_4) \rightarrow M(E_6)$ induced by the inclusion of groups $F_4 \hookrightarrow E_6$.
Highlights
Let X be a smooth complex projective irreducible curve of genus g ≥ 3 and let G be a complex reductive Lie group
We study the case in which the structure group G is F4 or E6, both simple complex Lie groups of exceptional type
We prove that the automorphisms of M (F4) are exactly those induced by the automorphisms of the curve X and that every automorphism of M (E6) is a combination of an automorphism of X, the action on M (E6) of an outer automorphism of E6 and the tensor product by an element of the third torsion of the Picard group of X (the last automorphisms are induced by the action of H1(X, Z(E6)) on M (E6), where Z(E6) ∼= Z3 is the center of E6)
Summary
Let X be a smooth complex projective irreducible curve of genus g ≥ 3 and let G be a complex reductive Lie group. This description of the groups of automorphisms will help us to obtain some consequences about how we can see M (F4) included in M (E6) (Propositions 8 and 9) To do all this we first prove a Torelli theorem for M (F4) and M (E6) (Theorems 1 and 2), which says that the moduli space determines the base curve, in the sense that if X and X are complex algebraic curves of genus g and g , respectively, with g, g ≥ 3, G is F4 or E6, M (G) is the moduli space of principal G-bundles over X and M (G) is the moduli space of principal G-bundles over X , if M (G) ∼= M (G), the corresponding base curves are isomorphic, X ∼= X.
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