an:06180743
Zbl 1272.14033
Lian, Bong H.; Song, Ruifang; Yau, Shing-Tung
Periodic integrals and tautological systems
EN
J. Eur. Math. Soc. (JEMS) 15, No. 4, 1457-1483 (2013).
00319844
2013
j
14J32 14M15 14J45 34M55 14D05 33C80
Calabi-Yau; period integrals; Picard-Fuchs systems; partial flag varieties
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 \textit{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.
Zbigniew Hajto (Krak??w)