an:07082647
Zbl 1451.14054
Macarro, Luis Narv??ez; Sevenheck, Christian
Tautological systems and free divisors
EN
Adv. Math. 352, 372-405 (2019).
00435690
2019
j
14F10 32S40 32S35 14M17
tautological systems; GKZ-systems; quantum D-modules; toric mirror symmetry; mixed Hodge modules; linear free divisors; Fourier-Laplace transforms; Radon transformation; Spencer complexes; Lie-Rinehart-algebras; Landau-Ginzburg models
The paper under review deals with systems of differential equations defined by certain prehomogeneous vector spaces endowed with actions of algebraic groups admitting open dense orbits. Such \(\mathcal D\)-modules can be considered as examples of the so-called tautological systems studied in many works (see [\textit{M. Kapranov}, in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30--July 4, 1997, and in Kyoto, Japan, July 7--11 1997. Singapore: World Scientific. 236--281 (1998; Zbl 0987.33008); \textit{B. H. Lian} et al., J. Eur. Math. Soc. (JEMS) 15, No. 4, 1457--1483 (2013; Zbl 1272.14033)]).
In fact, the authors investigate the case of reductive groups whose orbits have complements which are linear free divisors satisfying the strongly Koszul condition [\textit{M. Granger} and \textit{M. Schulze}, Publ. Res. Inst. Math. Sci. 46, No. 3, 479--506 (2010; Zbl 1202.14046)]. Under these assumptions it is proved that the associated tautological systems underlie mixed Hodge modules. Moreover, the authors give an explicit representation of the corresponding \(\mathcal D\)-modules similarly to the case of GKZ-systems [\textit{T. Reichelt}, Compos. Math. 150, No. 6, 911--941 (2014; Zbl 1315.14016)].
Aleksandr G. Aleksandrov (Moskva)
Zbl 0987.33008; Zbl 1272.14033; Zbl 1202.14046; Zbl 1315.14016