Fourier–Mukai transform for sheaves with connection on complex tori
Abstract
Let A , B be dual abelian varieties. Let B ♮ Bnatural upper B Superscript normal ♮ be the universal vectorial extension of B . Laumon and Rothstein independently lift the Fourier–Mukai transform to D A DA 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 ) Dcohb(DA) upper D Subscript coh Superscript b Baseline left parenthesis upper D Subscript upper A Baseline right parenthesis of coherent D A DA upper D Subscript upper A -modules to the derived category D coh b ( O B ♮ ) Dcohb(OBnatural ) 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 ♮ Bnatural 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 ♮ OBnatural upper O Subscript upper B Sub Superscript normal ♮ by a commutative O B OB upper O Subscript upper B -algebra, which is locally a polynomial algebra over O B OB 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 ) Dhb(DA) 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.
Related
- Fourier-Mukai transforms by a family of stable sheaves and preservation of stability
- Birational maps of moduli spaces of stable sheaves on a K3 surface induced by Fourier-Mukai transforms
- The quantum connection, Fourier-Laplace transform, and families of A∞ -categories
- A Non Commutative Grauert Theorem and Fourier Mukai Duality for Generalized Complex Tori
- Germwise singular Fourier--Mukai partnerships
BibTeX
@article{liu2026fouriermukai,
title = {{Fourier–Mukai transform for sheaves with connection on complex tori}},
author = {Haohao Liu},
journal = {Nagoya Mathematical Journal},
year = {2026},
month = jul,
doi = {10.1017/nmj.2026.10119},
url = {https://hal.science/hal-05736535v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Fourier–Mukai transform for sheaves with connection on complex tori
AU - Haohao Liu
PY - 2026
DA - 2026/07/07
JO - Nagoya Mathematical Journal
DO - 10.1017/nmj.2026.10119
UR - https://hal.science/hal-05736535v1
AB - 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_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_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.
SN - 0027-7630
ER -