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Endomorphisms of Koszul complexes: formality and application to deformation theory

  • Francesca Carocci
  • Marco ManettiEmail author
Article
  • 4 Downloads

Abstract

We study the differential graded Lie algebra of endomorphisms of the Koszul resolution of a regular sequence on a unitary commutative \(\mathbb {K}\,\)-algebra R and we prove that it is homotopy abelian over \(\mathbb {K}\,\) but not over R (except trivial cases). We apply this result to prove an annihilation theorem for obstructions of (derived) deformations of locally complete intersection ideal sheaves on projective schemes.

Keywords

Koszul complex Deformations of coherent sheaves Differential graded Lie algebras 

Mathematics Subject Classification

17B70 14D15 18G50 13D10 

Notes

Acknowledgements

M.M. wishes to acknowledge the support by Italian MIUR under PRIN Project 2015ZWST2C “Moduli spaces and Lie theory”. F.C. thanks the second named author who proposed this project for her master thesis in the far 2014 [2]. F.C. was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1]. The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London

References

  1. 1.
    Artin, M.: Deformations of Singularities. Tata Institute of Fundamental Research, Bombay (1976)zbMATHGoogle Scholar
  2. 2.
    Carocci, F.: Infinitesimal deformations of locally complete intersection ideal sheaves. Master thesis, Università Sapienza di Roma (2014)Google Scholar
  3. 3.
    Drinfeld, V.: Letter to V. Schechtman, Sept. 18, (1988)Google Scholar
  4. 4.
    Eisenbud, D.: Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)zbMATHGoogle Scholar
  5. 5.
    Fantechi, B., Manetti, M.: Obstruction calculus for functors of Artin rings, I. J. Algebra 202, 541–576 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fiorenza, D., Iacono, D., Martinengo, E.: Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves. J. Eur. Math. Soc. (JEMS) 14(2), 521–540 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fiorenza, D., Manetti, M.: Formality of Koszul brackets and deformations of holomorphic Poisson manifolds. Homol Homotopy Appl 14(2), 63–75 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Getzler, E.: Lie theory for nilpotent \(L_{\infty }\)-algebras. Ann. Math. 170(1), 271–301 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Goldman, W.M., Millson, J.J.: The deformation theory of representations of fundamental groups of compact kähler manifolds. Publ. Math. I.H.E.S. 67, 43–96 (1988)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hinich, V.: Descent of Deligne groupoids. Int. Math. Res. Notices 5, 223–239 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hinich, V.: Deformations of homotopy algebras. Commun. Algebra 32(2), 473–494 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hinich, V.: Deformations of sheaves of algebras. Adv. Math. 195, 102–164 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hovey, M.: Model Categories. Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence (1999)zbMATHGoogle Scholar
  14. 14.
    Iacono, D.: On the abstract Bogomolov–Tian–Todorov theorem. Rend. Mat. Appl. 38, 175–198 (2017)MathSciNetGoogle Scholar
  15. 15.
    Iacono, D., Manetti, M.: Semiregularity and obstructions of complete intersections. Adv. Math. 235, 92–125 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kaledin, D.: Some remarks on formality in families. Mosc. Math. J. 7, 643–652 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Katzarkov, L., Kontsevich, M., Pantev, T.: Hodge theoretic aspects of mirror symmetry. From Hodge theory to integrability and TQFT tt*-geometry. In: Proceedings of Symposia in Pure Mathematics, vol 78, pp. 87–174. American Mathematical Society, Providence (2008)Google Scholar
  18. 18.
    Kontsevich, M.: Topics in algebra-deformation theory. Lecture notes of a course given at Berkeley (1994)Google Scholar
  19. 19.
    Lazarev, A.: Models for classifying spaces and derived deformation theory. Proc. Lond. Math. Soc. 109(1), 40–64 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lunardon, L.: Some remarks on Dupont contraction. Rend. Mat. Appl. (7) 39, 79–96 (2018)Google Scholar
  21. 21.
    Lunts, V.: Formality of DG algebras (after Kaledin). J. Algebra 323(4), 878–898 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lurie, J.: Moduli problems for ring spectra. In: Proceedings of the International Congress of Mathematicians, Vol. II, pp. 1099–1125. Hindustan Book Agency, New Delhi (2010)Google Scholar
  23. 23.
    Manetti, M.: Deformation theory via differential graded Lie algebras. In: Seminari di Geometria Algebrica 1998–1999 Scuola Normale Superiore 21–48 (1999)Google Scholar
  24. 24.
    Manetti, M.: Extended deformation functors. Int. Math. Res. Not. 14, 719–756 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Manetti, M.: Lectures on deformations of complex manifolds. Rend. Mat. Appl. (7) 24, 1–183 (2004)MathSciNetzbMATHGoogle Scholar
  26. 26.
    M. Manetti: Differential graded Lie algebras and formal deformation theory. In Algebraic Geometry: Seattle, 2005, vol. 80, pp. 785–810. In: Proceedings of Symposia in Pure Mathematics (2009)Google Scholar
  27. 27.
    Manetti, M.: On some formality criteria for DG-Lie algebras. J. Algebra 438, 90–118 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Meazzini, F.: A DG-Enhancement of \(D(\mathbf{QCoh}(X))\) with applications in deformation theory. Preprint (2018)Google Scholar
  29. 29.
    Navarro Aznar, V.: Sur la théorie de Hodge-Deligne. Invent. Math. 90, 11–76 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Pridham, J.: Unifying derived deformation theories. Adv. Math. 224(3), 772–826 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Schechtman, V.: Local structure of moduli spaces Preprint (1997)Google Scholar
  32. 32.
    Sernesi, E.: Deformations of Algebraic Schemes. Grundlehren der mathematischen Wissenschaften, vol. 334. Springer, New York, Berlin (2006)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics “James Clerk Maxwell building”University of EdinburghEdinburghUK
  2. 2.Dipartimento di Matematica “Guido Castelnuovo”Università degli studi di Roma “La Sapienza”RomeItaly

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