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Cohomology jump loci of differential graded Lie algebras

Part of: Families, fibrations (Co)homology theory Deformations of analytic structures

Published online by Cambridge University Press:  06 March 2015

Nero Budur  and
Botong Wang
Nero Budur
Affiliation:
KU Leuven, Belgium University of Notre Dame, USA email Nero.Budur@wis.kuleuven.be Current address: KU Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Botong Wang
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN 46556, USA email bwang3@nd.edu
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Abstract

To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman–Millson, Green–Lazarsfeld, Nadel, Simpson, Dimca–Papadima, and of the second author.

Keywords

deformation theorydifferential graded Lie algebracohomology jump locuslocal system

MSC classification

Primary: 14D15: Formal methods; deformations 32G08: Deformations of fiber bundles 14F35: Homotopy theory; fundamental groups
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 8 , August 2015 , pp. 1499 - 1528
Copyright
© The Authors 2015 

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References

Artin, M., On the solutions of analytic equations, Invent. Math. 5 (1968), 277–291.CrossRefGoogle Scholar
Artin, M., Algebraic approximation of structures over complete local rings, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 23–58.CrossRefGoogle Scholar
Budur, N., Complements and higher resonance varieties of hyperplane arrangements, Math. Res. Lett. 18 (2011), 859–873; (Erratum, Math. Res. Lett. 21 (2014)).CrossRefGoogle Scholar
Budur, N. and Wang, B., Cohomology jump loci of quasi-projective varieties, Ann. Sci. Éc. Norm. Supér., to appear, arXiv:1211.3766.Google Scholar
Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D., Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245–274.CrossRefGoogle Scholar
Dimca, A. and Papadima, S., Nonabelian cohomology jump loci from an analytic viewpoint. Commun. Contemp. Math., to appear, arXiv:1206.3773.Google Scholar
Dimca, A., Papadima, S. and Suciu, A., Topology and geometry of cohomology jump loci, Duke Math. J. 148 (2009), 405–457.CrossRefGoogle Scholar
Eisenbud, D., Commutative algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150 (Springer, New York, 1995).CrossRefGoogle Scholar
Eisenbud, D., Popescu, S. and Yuzvinsky, S., Hyperplane arrangement cohomology and monomials in the exterior algebra, Trans. Amer. Math. Soc. 355 (2003), 4365–4383.CrossRefGoogle Scholar
Esnault, H., Schechtman, V. and Viehweg, E., Cohomology of local systems of the complements of hyperplanes, Invent. Math. 109 (1992), 557–561; Erratum, Invent. Math. 112 (1993), 447.CrossRefGoogle Scholar
Fiorenza, D., Iacono, D. and Martinengo, E., Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves, J. Eur. Math. Soc. (JEMS) 14 (2012), 521–540.CrossRefGoogle Scholar
Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, second edition (Springer, Berlin, 1998).CrossRefGoogle Scholar
Green, M. and Lazarsfeld, R., Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), 389–407.CrossRefGoogle Scholar
Green, M. and Lazarsfeld, R., Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc. 4 (1991), 87–103.CrossRefGoogle Scholar
Goldman, W. and Millson, J., Deformations of flat bundles over Kähler manifolds, Publ. Math. Inst. Hautes Études Sci. 67 (1988), 43–96.CrossRefGoogle Scholar
Grzegorczyk, I. and Teixidor i Bigas, M., Brill–Noether theory for stable vector bundles, in Moduli spaces and vector bundles 2950, London Mathematical Society Lecture Note Series, vol. 359 (Cambridge University Press, Cambridge, 2009).Google Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York–Heidelberg, 1977).CrossRefGoogle Scholar
Lazarsfeld, R. and Popa, M., Derivative complex, BGG correspondence, and numerical inequalities for compact Kähler manifolds, Invent. Math. 182 (2010), 605–633.CrossRefGoogle Scholar
Lübke, M. and Okonek, C., Moduli spaces of simple bundles and Hermitian–Einstein connections, Math. Ann. 276 (1987), 663–674.CrossRefGoogle Scholar
Lurie, J., Formal moduli problems, http://www.math.harvard.edu/∼lurie/papers/DAG-X.pdf .Google Scholar
Manetti, M., Lie description of higher obstructions to deforming submanifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 631–659.Google Scholar
Manetti, M., Differential graded Lie algebras and formal deformation theory, in Algebraic geometry—Seattle 2005. Part 2, Proceedings of Symposia in Pure Mathematics, vol. 80, Part 2 (American Mathematical Society, Providence, RI, 2009), 785–810.Google Scholar
Martinengo, E., Local structure of Brill–Noether strata in the moduli space of flat stable bundles, Rend. Semin. Mat. Univ. Padova 121 (2009), 259–280.CrossRefGoogle Scholar
Martinengo, E., Infinitesimal deformations of Hitchin pairs and Hitchin map, Internat. J. Math. 23 (2012), 125–153.CrossRefGoogle Scholar
Nadel, A., Singularities and Kodaira dimension of the moduli space of flat Hermitian–Yang–Mills connections, Compositio Math. 67 (1988), 121–128.Google Scholar
Popa, M. and Schnell, C., Generic vanishing theory via mixed Hodge modules, Forum Math. Sigma 1 (2013), 1–60.CrossRefGoogle Scholar
Pridham, J. P., Unifying derived deformation theories, Adv. Math. 224 (2010), 772–826 (Corrigendum, Adv. Math. 228 (2011), 2554–2556).CrossRefGoogle Scholar
Roberts, P., Multiplicities and Chern classes in local algebra, Cambridge Tracts in Mathematics, vol. 133 (Cambridge University Press, 1998).CrossRefGoogle Scholar
Simpson, C., Higgs bundles and local systems, Publ. Math. Inst. Hautes Études Sci. 75 (1992), 5–95.CrossRefGoogle Scholar
Simpson, C., Moduli of representations of the fundamental group of a smooth variety. II, Publ. Math. Inst. Hautes Études Sci. 80 (1994), 5–79.CrossRefGoogle Scholar
Uhlenbeck, K. and Yau, S.-T., On the existence of Hermitian–Yang–Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986), S257–S293.CrossRefGoogle Scholar
Wang, B., Cohomology jump loci in the moduli spaces of vector bundles, Preprint (2012),arXiv:1210.1487.Google Scholar
Wang, B., Cohomology jump loci of compact Kähler manifolds, Preprint (2013),arXiv:1303.6937v1.Google Scholar
Wang, B., Examples of topological spaces with arbitrary cohomology jump loci, Preprint (2013), arXiv:1304.0239.Google Scholar