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Periodic integrals and tautological systems. (English) Zbl 1272.14033
The authors’ purpose of this important paper is to study period integrals and deformations of $$CY$$ complete intersections in homogeneous spaces. They mostly restrict to partial flag varieties. After clear and very intuitive introduction the authors prove that the universal family of $$CY$$ manifolds is deformation complete. Next, they give an explicit construction of $$D$$-modules that governs the period integrals. In order to achieve this construction they introduce a special type of differential systems called tautological. More precisely, for a fixed reductive algebraic group $$G$$, to every $$G$$-variety $$X$$ equipped with a very ample equivariant line bundle $$L$$, they attach a system of differential operators defined on $$H^0(X,L)$$, depending on a group character. They show that the system is regular holonomic when $$X$$ is a homogeneous space. A number of illuminating examples are discussed. In the last section of the paper, they discuss several numerical examples and their solutions.

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14M15 Grassmannians, Schubert varieties, flag manifolds 14J45 Fano varieties 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 33C80 Connections of hypergeometric functions with groups and algebras, and related topics
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##### References:
 [1] Adolphson, A.: Hypergeometric functions and rings generated by monomials. Duke Math. J. 73, 269-290 (1994) · Zbl 0804.33013 · doi:10.1215/S0012-7094-94-07313-4
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