zbMATH — the first resource for mathematics

Period integrals of CY and general type complete intersections. (English) Zbl 1276.32004
The authors show a global Poincaré residue formula to study period integrals of families of complex manifolds. For any compact complex manifold \(X\) endowed with a linear system \(V^*\) of generically smooth CY hypersurfaces, the formula gives an expression for period integrals in terms of a canonical global meromorphic top form on \(X\). For this construction the authors use the notion of a CY principal bundle and a classification of such rank one bundles. They generalize also the construction to CY and general type complete intersections. When \(V\) is an algebraic manifold having a sufficiently large automorphism group \(G\) and \(V^*\) is a linear representation of \(G\), they construct a holonomic \(D\)-module that governs the periodic integrals. The construction is based in part on the theory of tautological systems developed in the paper [B. H. Lian, R. Song and S.-T. Yau, J. Eur. Math. Soc. (JEMS) 15, No. 4, 1457–1483 (2013; Zbl 1272.14033)]. The approach allows the authors to describe explicitly a Picard-Fuchs type system for complete intersection varieties of general type, as well as CY, in any Fano variety. They apply the results to toric manifolds and homogeneous spaces, as special examples and show that the period sheaves are governed by holonomic tautological systems. In the case of \(X\) a general homogeneous manifold, they obtain two different descriptions by using the Borel-Weil theorem and a theorem of Kostant and Lichtenstein, and they enumerate a set of generators for the tautological system. For \(X\) a toric manifold, the tautological systems turn out to be examples of GKZ hypergeometric systems and their extended versions. Moreover, the approach provides a new perspective of old examples such as CY complete intersections in a toric variety or partial flag variety.
Reviewer: Anna Fino (Torino)

32A27 Residues for several complex variables
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
Full Text: DOI arXiv
[1] Audin, M.: The Topology of Torus Actions on Symplectic Manifolds. Birkhäuser, Basel (1991) · Zbl 0726.57029
[2] Batyrev, V.: Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori. Duke Math. J. 69(2), 349–409 (1993) · Zbl 0812.14035 · doi:10.1215/S0012-7094-93-06917-7
[3] Batyrev, V., Cox, D.: On the Hodge structure of projective hypersurfaces in toric varieties. Duke Math. J. 75(2), 293–338 (1994) · Zbl 0851.14021 · doi:10.1215/S0012-7094-94-07509-1
[4] Bott, R.: Homogeneous vector bundles. Ann. Math. 66(2), 203–248 (1957) · Zbl 0094.35701 · doi:10.2307/1969996
[5] Calabi, E.: Métriques Kählériannes et fibrés holomorphes. Ann. Sci. Éc. Norm. Super. 12, 269–294 (1979)
[6] Cox, D.: The homogeneous coordinate ring of a toric variety. J. Algebr. Geom. 4(1), 17–50 (1995) · Zbl 0846.14032
[7] Danilov, V.: The geometry of toric varieties. Russ. Math. Surv. 33, 97–154 (1978) · Zbl 0425.14013 · doi:10.1070/RM1978v033n02ABEH002305
[8] Fulton, W.: Introduction to Toric Varieties. Annals of Math. Studies. Princeton University Press, Princeton (1993) · Zbl 0813.14039
[9] Gel’fand, I., Kapranov, M., Zelevinsky, A.: Hypergeometric functions and toral manifolds. Funct. Anal. Appl. 23, 94–106 (1989). English translation · Zbl 0721.33006 · doi:10.1007/BF01078777
[10] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley-Interscience, New York (1978) · Zbl 0408.14001
[11] Hosono, S., Lian, B.H., Yau, S.-T.: GKZ-generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces. Commun. Math. Phys. 182, 535–577 (1996) · Zbl 0870.14028 · doi:10.1007/BF02506417
[12] Hosono, S., Lian, B.H., Yau, S.-T.: Maximal degeneracy points of GKZ systems. J. Am. Math. Soc. 10(2), 427–443 (1997) · Zbl 0874.32007 · doi:10.1090/S0894-0347-97-00230-0
[13] Jaczewski, K.: Generalized Euler sequence and toric varieties. In: Contemporary Math., vol. 162, pp. 227–247 (1994) · Zbl 0837.14042
[14] Kempf, G.: Equations of isotropy. In: Group Actions and Invariant Theory, Montreal, PQ, 1988. CMS Conf. Proc., vol. 10. Am. Math. Soc., Providence (1989) · Zbl 0661.20031
[15] Lian, B.H., Li, S., Yau, S.-T.: Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces. Am. J. Math. (2012, to appear). arXiv:0910.4215 · Zbl 1253.14036
[16] Lian, B.H., Song, R., Yau, S.-T.: Period integrals and tautological systems. arXiv:1105.2984v1 · Zbl 1272.14033
[17] Lichtenstein, W.: A system of quadrics describing the orbit of the highest weight vector. Proc. Am. Math. Soc. 84(4), 605–608 (1982) · Zbl 0501.22017 · doi:10.1090/S0002-9939-1982-0643758-8
[18] Oda, T.: Convex Bodies and Algebraic Geometry, An Introduction to the Theory of Toric Varieties. Springer, Berlin (1985)
[19] Popov, V.L.: Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles. Math. USSR Izv. 8(2), 301–327 (1974) · Zbl 0301.14018 · doi:10.1070/IM1974v008n02ABEH002107
[20] Procesi, C.: Lie Groups: An Approach Through Invariants and Representations. Universitext. Springer, Berlin (2005). 368 pp. · Zbl 1154.22001
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.