Erratum to: Holonomic D-modules on abelian varieties

Christian Schnell

doi:10.1007/s10240-015-0079-8

Schnell, Christian

2015-11-16 00:00:00

ERRATUM TO: HOLONOMIC D-MODULES ON ABELIAN VARIETIES by CHRISTIAN SCHNELL Erratum to: Publ. Math. (2015) 121:1–55 DOI 10.1007/s10240-014-0061-x Thomas Krämer and Claude Sabbah pointed out to me that the published proof of Lemma 20.2 only works in the regular case. The purpose of this note is to give a correct proof for the general case. Here is the statement again. Lemma. — Let f :A → B be a surjective morphism of abelian varieties, with connected ﬁbers. If N is a nontrivial simple holonomic D -module, then f N is a simple holonomic D -module. B A ∗ −1 Proof. — Since f is smooth, f N = O ⊗ −1 f N is a holonomic D -module, A f O A and so there is a surjective morphism f N → M to a nontrivial simple holonomic D - module M. We will prove the assertion by showing that it is an isomorphism. The support X = Supp N is an irreducible subvariety of B. As N is holonomic, there is a dense Zariski-open subset U ⊆ Bsuch thatX ∩ U is nonsingular and such that the restriction N | 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 sup- −1 −1 ∗ ∗ ported on f (X), and that its restriction to f (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 = rk E . ∗ −1 Since M is a quotient of f N , its restriction to f (U) is also the direct image of a −1 holomorphic vector bundle with integrable connection on f (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 f N → M gives rise to a nontrivial morphism −r N → H f M which must be injective because N is simple. Over U, the left-hand side is a vector bundle of rank n and the right-hand side a vector bundle of rank k ≤ n; this is only possible if The online version of the original article can be found under doi:10.1007/s10240-014-0061-x. DOI 10.1007/s10240-015-0079-8 362 CHRISTIAN SCHNELL ∗ −1 k = n.But then f N → M is an isomorphism over f (U), and since M is simple, we obtain a short exact sequence 0 → K → f N → M → 0 −1 where K is a holonomic D -module whose support is contained in f (X \ X ∩ U). + ∗ Now recall that f N = f N [r]. By adjunction [HTT08, Corollary 3.2.15], the inclusion K → f N determines a morphism f K [r]→ N , which factors as f K [r]→ H f K → N . + + Since the support of H f K is contained in X \ X ∩ U, the morphism in question must be zero, and so K = 0. We conclude that f N is isomorphic to M. REFERENCES [HTT08] R. HOTTA,K.TAKEUCHI and T. TANISAKI,D-Modules, Perverse Sheaves, and Representation Theory,ProgressinMath- ematics, vol. 236, Birkhäuser Boston Inc., Boston, 2008. C. S. Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA cschnell@math.sunysb.edu publié en ligne le 16 novembre 2015.

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$ERRATUM TO: HOLONOMIC D-MODULES ON ABELIAN VARIETIES by CHRISTIAN SCHNELL Erratum to: Publ. Math. (2015) 121:1–55 DOI 10.1007/s10240-014-0061-x Thomas Krämer and Claude Sabbah pointed out to me that the published proof of Lemma 20.2 only works in the regular case. The purpose of this note is to give a correct proof for the general case. Here is the statement again. Lemma. — Let f :A → B be a surjective morphism of abelian varieties, with connected ﬁbers. If N is a nontrivial simple holonomic D -module, then f N is a simple holonomic D -module. B A ∗ −1 Proof. — Since f is smooth, f N = O ⊗ −1 f N is a holonomic D -module, A f O A and so there is a surjective morphism f N → M to a nontrivial simple holonomic D - module M. We will prove the assertion by showing that it is an isomorphism. The support X = Supp N is an irreducible subvariety of B. As N is holonomic, there is a dense Zariski-open subset U ⊆ Bsuch thatX ∩ U is nonsingular and such that the restriction N | 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 sup- −1 −1 ∗ ∗ ported on f (X), and that its restriction to f (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 = rk E . ∗ −1 Since M is a quotient of f N , its restriction to f (U) is also the direct image of a −1 holomorphic vector bundle with integrable connection on f (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 f N → M gives rise to a nontrivial morphism −r N → H f M which must be injective because N is simple. Over U, the left-hand side is a vector bundle of rank n and the right-hand side a vector bundle of rank k ≤ n; this is only possible if The online version of the original article can be found under doi:10.1007/s10240-014-0061-x. DOI 10.1007/s10240-015-0079-8 362 CHRISTIAN SCHNELL ∗ −1 k = n.But then f N → M is an isomorphism over f (U), and since M is simple, we obtain a short exact sequence 0 → K → f N → M → 0 −1 where K is a holonomic D -module whose support is contained in f (X \ X ∩ U). + ∗ Now recall that f N = f N [r]. By adjunction [HTT08, Corollary 3.2.15], the inclusion K → f N determines a morphism f K [r]→ N , which factors as f K [r]→ H f K → N . + + Since the support of H f K is contained in X \ X ∩ U, the morphism in question must be zero, and so K = 0. We conclude that f N is isomorphic to M. REFERENCES [HTT08] R. HOTTA,K.TAKEUCHI and T. TANISAKI,D-Modules, Perverse Sheaves, and Representation Theory,ProgressinMath- ematics, vol. 236, Birkhäuser Boston Inc., Boston, 2008. C. S. Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA cschnell@math.sunysb.edu publié en ligne le 16 novembre 2015.$

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Erratum to: Holonomic D-modules on abelian varieties

Erratum to: Holonomic D-modules on abelian varieties

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Erratum to: Holonomic D-modules on abelian varieties

Erratum to: Holonomic D-modules on abelian varieties

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