Abstract
Given any connected topological space X, assume that there exists an epimorphism \(\phi {:}\; \pi _1(X) \rightarrow {\mathbb {Z}}\). The deck transformation group \({\mathbb {Z}}\) acts on the associated infinite cyclic cover \(X^\phi \) of X, hence on the homology group \(H_i(X^\phi , {\mathbb {C}})\). This action induces a linear automorphism on the torsion part of the homology group as a module over the Laurent ring \({\mathbb {C}}[t,t^{-1}]\), which is a finite dimensional \({\mathbb {C}}\)-vector space. We study the sizes of the Jordan blocks of this linear automorphism. When X is a compact Kähler manifold, we show that all the Jordan blocks are of size one. When X is a smooth complex quasi-projective variety, we give an upper bound on the sizes of the Jordan blocks, which is an analogue of the Monodromy Theorem for the local Milnor fibration.
Similar content being viewed by others
References
Arapura, D.: Geometry of cohomology support loci for local systems. I. J. Algebr. Geom. 6(3), 563–597 (1997)
Budur, N., Wang, B.: Cohomology jump loci of quasi-projective varieties. Ann. Sci. Cole Norm. Sup. 48(1), 227236 (2015)
Budur, N., Wang, B.: Cohomology jump loci of differential graded Lie algebras. Compos. Math. 151(8), 1499–1528 (2015)
Budur, N., Wang, B.: Local Systems on Analytic Germ Complements. ar**v:1508.07867
de Cataldo, M., Migliorini, L.: The decomposition theorem, perverse sheaves and the topology of algebraic maps. Bull. Am. Math. Soc. (N.S.) 46(4), 535–633 (2009)
Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)
Dimca, A.: Singularities and Topology of Hypersurfaces. Springer, New York (1992)
Dimca, A.: Monodromy and Hodge Theory of Regular Functions. New Developments in Singularity Theory (Cambridge, 2000), Nato Science Series II: Mathematics, Physics and Chemistry, vol. 21. Kluwer Academic Publications, Dordrecht (2001)
Dimca, A., Libgober, A.: Regular functions transversal at infinity. Tohoku Math. J. 58(4), 549–564 (2006)
Dimca, A., Némethi, A.: Hypersurface Complements, Alexander Modules and Monodromy. Real and Complex Singularities, Contemporary Mathematics, vol. 354, pp. 19–43. American Mathematical Society, Providence (2004)
Dimca, A., Papadima, S.: Non-abelian cohomology jump loci from an analytic viewpoint. Commun. Contemp. Math. 16(4), 1350025 (2014)
Fernández, M., Gray, A., Morgan, J.: Compact symplectic manifolds with free circle actions, and Massey products. Mich. Math. J. 38(2), 271–283 (1991)
Goldman, W., Millson, J.: The deformation theory of representations of fundamental groups of compact Kähler manifolds. Inst. Hautes Études Sci. Publ. Math. 67, 43–96 (1988)
Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics, No. 52. Springer, New York (1977)
Libgober, A.: Alexander invariants of plane algebraic curves, Singularities, Part 2 (Arcata, Calif., 1981). In: Proceedings of Symposia in Pure Mathematics, vol. 40, pp. 135–143. American Mathematical Society, Providence, RI (1983)
Libgober, A.: Homotopy groups of the complements to singular hypersurfaces, II. Ann. Math. 139(1), 117–144 (1994)
Malgrange, B.: Letter to the editors. Invent. Math. 20, 171–172 (1973)
Maxim, L.: Intersection homology and Alexander modules of hypersurface complements. Comment. Math. Helv. 81(1), 123–155 (2006)
Morgan, J.: The algebraic topology of smooth algebraic varieties. Inst. Hautes Études Sci. Publ. Math. 48, 137–204 (1978)
Papadima, S., Suciu, A.: Algebraic monodromy and obstructions to formality. Forum Math. 22(5), 973–983 (2010)
Qin, L., Wang, B.: A Family of Compact Complex-Symplectic Calabi–Yau Manifolds that are Nonkähler. ar**v:1601.04337
Wang, B.: Torsion points on the cohomology jump loci of compact Kähler manifolds. Math. Res. Lett. 23(2), 545–563 (2016)
Acknowledgements
We thank Lizhen Qin for many helpful discussions. N. Budur and Y. Liu were partially supported by a FWO Grant, a KU Leuven OT Grant, and a Flemish Methusalem Grant. We also would like to thank the anonymous referee for valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Ngaiming Mok.
Rights and permissions
About this article
Cite this article
Budur, N., Liu, Y. & Wang, B. The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties. Math. Ann. 371, 1069–1086 (2018). https://doi.org/10.1007/s00208-017-1541-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-017-1541-3