# zbMATH — the first resource for mathematics

Holonomic systems for period mappings. (English) Zbl 1329.57034
Summary: Period mappings were introduced in the sixties to study variation of complex structures of families of algebraic varieties. The theory of tautological systems was introduced recently to understand period integrals of algebraic manifolds. In this paper, we give an explicit construction of a tautological system for each component of a period mapping. We also show that the D-module associated with the tautological system gives rise to many interesting vanishing conditions for period integrals at certain special points of the parameter space.
##### MSC:
 57R35 Differentiable mappings in differential topology 14J10 Families, moduli, classification: algebraic theory 53C29 Issues of holonomy in differential geometry 81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
##### Keywords:
tautological systems
Full Text:
##### References:
 [1] Bloch, S.; Huang, A.; Lian, B. H.; Srinivas, V.; Yau, S.-T., On the holonomic rank problem, J. Differ. Geom., 97, 1, 11-35, (2014) · Zbl 1318.32027 [2] Borel, A., Algebraic D-modules, Perspectives in Mathematics, vol. 2, (1987), Academic Press, Inc. Boston, MA · Zbl 0642.32001 [3] Brylinski, J.-L., Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque, 140-141, 3134, (1986), 251 [4] Griffiths, P., The residue calculus and some transcendental results in algebraic geometry. I, Proc. Natl. Acad. Sci. USA, 55, 5, 1303-1309, (1966) · Zbl 0178.23802 [5] Huang, A.; Lian, B. H.; Zhu, X., Period integrals and the Riemann-Hilbert correspondence · Zbl 1387.14042 [6] Hotta, R., Equivariant D-modules [7] Lian, B. H.; Song, R.; Yau, S.-T., Periodic integrals and tautological systems, J. Eur. Math. Soc., 15, 4, 1457-1483, (2013) · Zbl 1272.14033 [8] Lian, B. H.; Yau, S.-T., Period integrals of CY and general type complete intersections, Invent. Math., 191, 1, 35-89, (2013) · Zbl 1276.32004 [9] Voisin, C., Hodge theory and complex algebraic geometry, vols. I & II, (2002), Cambridge University Press
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.