Erratum to: Holonomic D-modules on abelian varieties

Abstract

Proof. — Since f is smooth, f ∗N = OA ⊗f −1OB f −1N is a holonomic DA-module, and so there is a surjective morphism f ∗N →M to a nontrivial simple holonomic DAmodule M. We will prove the assertion by showing that it is an isomorphism. The support X = SuppN is an irreducible subvariety of B. As N is holonomic, there is a dense Zariski-open subset U ⊆ B such that X ∩ U is nonsingular and such that the restriction N |U is the direct image of a holomorphic vector bundle with integrable connection (E ,∇) on X ∩ U [HTT08, Proposition 3.1.6]. This means that f ∗N is supported on f −1(X), and that its restriction to f −1(U) is the direct image of (f ∗E , f ∗∇). We observe that, on the fibers of f over points of X ∩ U, the latter is a trivial bundle of rank n = rkE . Since M is a quotient of f ∗N , its restriction to f −1(U) is also the direct image of a holomorphic vector bundle with integrable connection on f −1(X ∩ U); as a quotient of (f ∗E , f ∗∇), the restriction of this bundle to the fibers of f must be trivial of some rank k ≤ n. Now let r = dim A−dim B be the relative dimension of f . By adjunction [HTT08, Corollary 3.2.15], the surjective morphism