Fourier–Mukai transform for sheaves with connection on complex tori

2026/01/01 by Haohao Liu

paper · doi:10.1017/nmj.2026.10119

crossref issued 2026/01/01 · crossref published 2026/01/01 · crossref published-print 2026/01/01 · crossref published-online 2026/07/07 · crossref created 2026/07/07 · crossref deposited 2026/07/07 · crossref indexed 2026/07/30

Abstract

Abstract Let A , B be dual abelian varieties. Let B ♮ B^\natural upper B Superscript normal ♮ be the universal vectorial extension of B . Laumon and Rothstein independently lift the Fourier–Mukai transform to D A D_A upper D Subscript upper A -modules. They prove that this defines an equivalence of triangulated categories from the derived category D coh b ( D A ) D_\mathrm coh^b(D_A) upper D Subscript coh Superscript b Baseline left parenthesis upper D Subscript upper A Baseline right parenthesis of coherent D A D_A upper D Subscript upper A -modules to the derived category D coh b ( O B ♮ ) D_\mathrm coh^b(O_B^\natural ) upper D Subscript coh Superscript b Baseline left parenthesis upper O Subscript upper B Sub Superscript normal ♮ Subscript Baseline right parenthesis of coherent sheaves on B ♮ B^\natural upper B Superscript normal ♮ . We extend their results to complex tori. As a replacement of coherent algebraic D -modules, we use Kashiwara’s good analytic D -modules. Moreover, to get an equivalence, we have to replace O B ♮ O_B^\natural upper O Subscript upper B Sub Superscript normal ♮ by a commutative O B O_B upper O Subscript upper B -algebra, which is locally a polynomial algebra over O B O_B upper O Subscript upper B . As an application, we recover the Matsushima–Morimoto theorem that on a complex torus, a vector bundle admits a connection if and only if it is translation invariant, and in this case, it admits an integrable connection. Moreover, we use the Laumon–Rothstein transform to show that the derived category D h b ( D A ) D_h^b(D_A) upper D Subscript h Superscript b Baseline left parenthesis upper D Subscript upper A Baseline right parenthesis of holonomic D -modules is a rigid symmetric monoidal triangulated category under the convolution.

Citations

• Vanishing theorems for perverse sheaves on abelian varieties, revisited
• Holonomic D-modules on abelian varieties
• A Remark on Rigidity of BN-sheaves
• Equivalences of monoidal model categories
• Sheaves with connection on abelian varieties
• Higgs bundles and local systems
• Duality between D ( X ) and with its application to picard sheaves
• Sheaves on Manifolds