Period integrals and the Riemann-Hilbert correspondence. (English) Zbl 1387.14042

Huang, An; Lian, Bong H.; Zhu, Xinwen

J. Differ. Geom. 104, No. 2, 325-369 (2016).

A tautological system, introduced in [B. H. Lian et al., J. Eur. Math. Soc. (JEMS) 15, No. 4, 1457–1483 (2013; Zbl 1272.14033)] and [B. H. Lian and S.-T. Yau, Invent. Math. 191, No. 1, 35–89 (2013; Zbl 1276.32004)], arises as a regular holonomic system of partial differential equations that governs the period integrals of a family of complete intersections in a complex manifold \(X\), equipped with a suitable Lie group action. A geometric formula for the holonomic rank of such a system was conjectured in [S. Bloch et al., J. Differ. Geom. 97, No. 1, 11–35 (2014; Zbl 1318.32027)], and was verified for the case of projective homogeneous space under an assumption. In this paper, the authors prove this conjecture in full generality. By means of the Riemann-Hilbert correspondence and Fourier transforms, they also generalize the rank formula to an arbitrary projective manifold with a group action.

Reviewer: Ruifang Song (Mountain View)

MSC:

14D05	Structure of families (Picard-Lefschetz, monodromy, etc.)
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
14D07	Variation of Hodge structures (algebro-geometric aspects)
53D37	Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category

Keywords:

period integrals; Picard-Fuchs systems; family of complete intersections; tautological systems; regular holonomic D-modules; Riemann-Hilbert correspondence; Fourier transform

Citations:

Zbl 1272.14033; Zbl 1276.32004; Zbl 1318.32027