Lie description of higher obstructions to deforming submanifolds

Marco Manetti

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 4, page 631-659
  • ISSN: 0391-173X

Abstract

top
To every morphism χ : L → M of differential graded Lie algebras we associate a functors of artin rings Def χ whose tangent and obstruction spaces are respectively the first and second cohomology group of the suspension of the mapping cone of χ . Such construction applies to Hilbert and Brill-Noether functors and allow to prove with ease that every higher obstruction to deforming a smooth submanifold of a Kähler manifold is annihilated by the semiregularity map.

How to cite

top

Manetti, Marco. "Lie description of higher obstructions to deforming submanifolds." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.4 (2007): 631-659. <http://eudml.org/doc/272283>.

@article{Manetti2007,
abstract = {To every morphism $\chi \colon L\rightarrow M$ of differential graded Lie algebras we associate a functors of artin rings $\operatorname\{Def\}_\chi $ whose tangent and obstruction spaces are respectively the first and second cohomology group of the suspension of the mapping cone of $\chi $. Such construction applies to Hilbert and Brill-Noether functors and allow to prove with ease that every higher obstruction to deforming a smooth submanifold of a Kähler manifold is annihilated by the semiregularity map.},
author = {Manetti, Marco},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {differential graded Lie algebras; deformation functor; tangent and obstruction spaces; compact Kähler manifolds},
language = {eng},
number = {4},
pages = {631-659},
publisher = {Scuola Normale Superiore, Pisa},
title = {Lie description of higher obstructions to deforming submanifolds},
url = {http://eudml.org/doc/272283},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Manetti, Marco
TI - Lie description of higher obstructions to deforming submanifolds
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 4
SP - 631
EP - 659
AB - To every morphism $\chi \colon L\rightarrow M$ of differential graded Lie algebras we associate a functors of artin rings $\operatorname{Def}_\chi $ whose tangent and obstruction spaces are respectively the first and second cohomology group of the suspension of the mapping cone of $\chi $. Such construction applies to Hilbert and Brill-Noether functors and allow to prove with ease that every higher obstruction to deforming a smooth submanifold of a Kähler manifold is annihilated by the semiregularity map.
LA - eng
KW - differential graded Lie algebras; deformation functor; tangent and obstruction spaces; compact Kähler manifolds
UR - http://eudml.org/doc/272283
ER -

References

top
  1. [1] M. Artin, “Deformations of Singularities”, Tata Institute of Fundamental Research, Bombay, 1976. 
  2. [2] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math.128 (1997), 45–88. Zbl0909.14006MR1437495
  3. [3] S. Bloch, Semi-regularity and de Rham cohomology, Invent. Math.17 (1972) 51-66. Zbl0254.14011MR325613
  4. [4] R. O. Buchweitz and H. Flenner, A semiregularity map for modules and applications to deformations, Compositio Math. 137 (2003), 135–210. arXiv:math.AG/9912245 Zbl1085.14503MR1985003
  5. [5] P. Burchard and H. Clemens, Normal differential operators and deformation theory, In: “Recent progress in intersection theory” (Bologna, 1997), Birkhäuser Boston, 2000, 33–84. arXiv:math.AG/9811171 Zbl0971.14019MR1849291
  6. [6] H. Clemens, Geometry of formal Kuranishi theory. Adv. Math.198 (2005), 311–365. Zbl1092.14004MR2183257
  7. [7] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math.29 (1975), 245–274. Zbl0312.55011MR382702
  8. [8] S. K. Donaldson and P. B. Kronheimer, “The Geometry of Four-Manifolds”, Oxford, University Press, 1990. Zbl0904.57001MR1079726
  9. [9] A. Douady, Obstruction primaire à la déformation, Sém. Cartan 13 (1960/61), Exp. 4. Zbl0156.42803
  10. [10] B. Fantechi and M. Manetti, Obstruction calculus for functors of Artin rings, I. J. Algebra202 (1998), 541–576. Zbl0981.13009MR1617687
  11. [11] D. Fiorenza and M. Manetti, L ∞ structures on mapping cones, Algebra Number Theory1 (2007), 301–330. Zbl1166.17010MR2361936
  12. [12] K. Fukaya, Deformation theory, homological algebra and mirror symmetry, In: “Geometry and Physics of Branes” (Como, 2001), Ser. High Energy Phys. Cosmol. Gravit., IOP Bristol, 2003, 121–209. MR1950958
  13. [13] W. M. Goldman and J. J. Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math.67 (1988), 43–96. Zbl0678.53059MR972343
  14. [14] W. M. Goldman and J. J. Millson, The homotopy invariance of the Kuranishi space, Illinois J. Math.34 (1990), 337–367. Zbl0707.32004MR1046568
  15. [15] M. Green and R. Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), 389–407. Zbl0659.14007MR910207
  16. [16] M. Green and R. Lazarsfeld, Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc.4 (1991), 87–103. Zbl0735.14004MR1076513
  17. [17] R. Hartshorne, “Residues and Duality”, Springer-Verlag, L.N.M., Vol. 20, 1966. Zbl0212.26101MR222093
  18. [18] D. Iacono, “Differential Graded Lie Algebras and Deformations of Holomorphic Maps”, PhD thesis, 2006, arXiv:math.AG/0701091 Zbl1277.32011
  19. [19] N. Jacobson, “Lie Algebras”, Wiley & Sons, 1962. Zbl0121.27504
  20. [20] Y. Kawamata, Unobstructed deformations - a remark on a paper of Z. Ran, J. Algebraic Geom.1 (1992), 183–190. Zbl0818.14004MR1144434
  21. [21] S. Kobayashi, “Differential Geometry of Complex Vector Bundles”, Princeton, Univ. Press, 1987. Zbl0708.53002MR909698
  22. [22] K. Kodaira and D. C. Spencer, A theorem of completeness of characteristic systems of complete continuous systems, Amer. J. Math.81 (1959), 477–500. Zbl0097.36501MR112156
  23. [23] J. Kollár, “Rational curves on algebraic varieties”, Springer-Verlag, Ergebnisse Vol. 32, 1996. Zbl0877.14012
  24. [24] M. Kontsevich, Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66 (2003), 157–216. arXiv:q-alg/9709040 Zbl1058.53065MR2062626
  25. [25] M. Manetti, Deformation theory via differential graded Lie algebras, In: “Seminari di Geometria Algebrica 1998-1999”, Scuola Normale Superiore, 1999. arXiv:math.AG/0507284 MR1754793
  26. [26] M. Manetti, Extended deformation functors, Int. Math. Res. Not. 14 (2002), 719–756. arXiv:math.AG/9910071 Zbl1063.58007MR1891232
  27. [27] M. Manetti, Cohomological constraint to deformations of compact Kähler manifolds, Adv. Math.186 (2004), 125–142. Zbl1075.53074MR2065509
  28. [28] M. Manetti, Lectures on deformations on complex manifolds, Rend. Mat. Appl. 24 (2004), 1–183. arXiv:math.AG/0507286 Zbl1066.58010MR2130146
  29. [29] V. P. Palamodov, Deformations of complex spaces, Uspekhi Mat. Nauk. 31:3 (1976), 129–194. Transl. Russian Math. Surveys 31:3 (1976), 129–197. Zbl0347.32009MR508121
  30. [30] Z. Ran, Hodge theory and the Hilbert scheme, J. Differential Geom.37 (1993), 191–198. Zbl0804.14004MR1198605
  31. [31] M. Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc.130 (1968), 208–222. Zbl0167.49503MR217093
  32. [32] M. Schlessinger and J. Stasheff, Deformation Theory and Rational Homotopy Type, preprint, 1979. 
  33. [33] F. Severi, Sul teorema fondamentale dei sistemi continui di curve sopra una superficie algebrica, Ann. Mat. Pura Appl.23 (1944), 149–181. Zbl0061.33303MR15841
  34. [34] D. Tanré, “Homotopie Rationelle: Modèles de Chen, Quillen, Sullivan”, Springer-Verlag, Lecture Notes in Mathematics Vol. 1025, 1983. Zbl0539.55001MR764769
  35. [35] C. Voisin, Sur la stabilité de sous-variétés lagrangiennes des variétés symplectiques holomorphes, In: “Complex Projective Geometry” (Trieste, 1989/Bergen, 1989), London Math. Soc., Lecture Note Ser. Vol. 179, 1992, 294–303. Zbl0765.32012MR1201391

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.