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Support Theorems for Algebraic Maps

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Abstract

We review some recent progress on the topological properties of algebraic maps, with special emphasis on the determination of the "supports" of a map.

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References

  1. Beilinson A.A., Bernstein J., Deligne P.: Faisceaux pervers. Astèrisque 100, 5–171 (1982)

    MathSciNet  Google Scholar 

  2. Brylinski J., Dubson A., Kashiwara M.: Formule de l’dice pour modules holonomes et obstruction d’Euler locale. C. R. Acad. Sci. Paris Ser. I Math. 293, 573–576 (1981)

    MATH  MathSciNet  Google Scholar 

  3. W. Borho, R. MacPherson, Partial resolutions of nilpotent varieties, Analysis and topology on singular spaces, II, III (Luminy, 1981), 23-74, Astérisque, 101-102, Soc. Math. France, Paris, 1983.

  4. J.-P. Brasselet, Le Dung Trang, J. Seade, Euler obstruction and indices of vector fields. Topology 39, 1193-1208 (2000)

  5. M.A. de Cataldo, T. Hausel, L. Migliorini, Topology of Hitchin systems and Hodge theory of character varieties: the case A 1, Ann. of Math. 175 (2012), 13291407

  6. M. A. de Cataldo, L. Migliorini, The Hard Lefschetz Theorem and the topology of semismall maps, Ann. Scient. Éc. Norm. Sup., 4e srie, t. 35, 2002, 759-772.

  7. M. A. de Cataldo, L. Migliorini, The Hodge Theory of Algebraic maps, Ann. Scient. Éc. Norm. Sup., 4e série, t. 38, (2005), 693-750.

  8. M. A. de Cataldo, L. Migliorini, Intersection forms, algebraic maps and motivic decomposition for resolution of threefolds, in Algebraic Cycles and Motives, London Math.Soc. Lecture Note Series, n.343, vol.1, pp.102-137, Cambridge University Press, Cambridge, UK, 2007.

  9. de Cataldo M. A., Migliorini L.: The Chow motive of semismall resolutions. Math.Res.Lett. 11, 151–170 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. M.A. de Cataldo, L. Migliorini: Hodge-theoretic aspects of the decomposition theorem, Algebraic geometry—Seattle 2005. Part 2, 489–504, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.

  11. M.A. de Cataldo, L. Migliorini, The Decomposition Theorem and the topology of algebraic maps, Bull. of the Amer. Math. Soc., 46, n.4, (2009), 535-633.

  12. M.A. de Cataldo, L. Migliorini, The perverse filtration and the Lefschetz Hyperplane Section Theorem, Ann. of Math., 171 n.3 (2010) 2089-2113.

  13. M. A. de Cataldo, L. Migliorini, The projectors of the decomposition theorem are motivated, Preprint, arXiv:1401.3705.

  14. M. A. de Cataldo, L. Migliorini, M. Mustata, Combinatorics and topology of proper toric maps, Preprint, arXiv:1407.3497.

  15. P. H. Chaoudouard, G. Laumon Le lemme fondamental pondéré. II. Énoncés cohomologiques, Ann. of Math. 176 (2012), 1647-1781

  16. Deligne P.: Théorie de Hodge, II. Publ. Math. IHES 40, 5–57 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  17. Deligne P.: Théorie de Hodge, III. Publ. Math. IHES 44, 5–78 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  18. P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274.

  19. S. Gelfand, R. MacPherson, Verma modules and Schubert cells: a dictionary, Dubreil- Malliavin Algebra Seminar, pp. 1–50, Lecture Notes in Math., 924, Springer, Berlin-New York, 1982.

  20. S. Gelfand, Y.I. Manin, Methods of homological algebra, Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.

  21. R. Godement, Topologie algébrique et théorie des faisceaux, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII. Actualités Scientifiques et Industrielles, No. 1252. Hermann, Paris, 1973.

  22. M. Goresky, Introduction to the papers of R. Thom and J. Mather, Bull. of the Amer. Math. Soc., 49, n.4, (2012), 469474.

  23. M. Goresky, R. MacPherson, Intersection homology theory, Topology 19 (1980), no. 2, 135–162.

  24. M. Goresky, R. MacPherson, Intersection homology. II, Invent. Math. 72 (1983), no. 1, 77–129.

  25. M. Goresky, R. MacPherson, Stratified Morse Theory, Ergebnisse der Mathematik, und ihrer Grenzgebiete 3.folge. Band 2, Springer-Verlag, Berlin Heidelberg 1988.

  26. Hitchin N.: Stable bundles and integrable systems. Duke Math. J. 54, 91–114 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  27. L. Illusie, Catégories dérivées et dualité: travaux de J.-L. Verdier, Ens. Math. (2) 36 (1990), 369-391.

  28. B. Iversen, Cohomology of Sheaves, Universitext, Springer-Verlag, Berlin Heidelberg 1986.

  29. S.L. Kleiman, The development of Intersection Homology Theory, Pure and Appl. Math. Quart. 3 no. 1 (2007) Special issue in honor of Robert MacPherson, 225-282.

  30. MacPherson R.: Chern classes for singular varieties. Ann. of Math. 100, 423–432 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  31. R. MacPherson, Global questions in the topology of singular spaces, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 213-235, PWN, Warsaw, 1984.

  32. L. Migliorini and V. Shende, A support theorem for Hilbert schemes of planar curves, J. Eur. Math. Soc. 15, 2353-2367.

  33. L. Migliorini and V. Shende, Higher discriminants and the topology of algebraic maps, arXiv:1307.4059.

  34. L. Migliorini, V. Shende and F. Viviani, A support theorem for Hilbert schemes of planar curves, II, in preparation.

  35. D. Maulik, Z. Yun, Macdonald formula for curves with planar singularities, Jour. Reine Angew. Math. to appear.

  36. B. C. Ngô, Le lemme fondamental pour les algèbres de Lie, Publ. Math. de l’IHES 111, 1-169 (2010).

  37. M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989).

  38. M. Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333.

  39. M. Saito, Decomposition theorem for proper Kähler morphisms, Tohoku Math. J. (2) 42, no. 2, (1990), 127–147.

  40. V. Shende, J. Tsimerman, Equidistribution on the space of rank two vector bundles over the projective line Preprint, arXiv: 1307.8237.

  41. D. Sullivan, Combinatorial invariants of analytic spaces, Proceedings of Liverpool Singularities Symposium I Lecture Notes in Mathematics Volume 192, 1971, pp 165-177

  42. B. Teissier, Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney, Algebraic geometry (La Rabida, 1981), 314-491, Lecture Notes in Math., 961, Springer, Berlin, 1982.

  43. Verdier J.L.: Stratifications de Whitney et Théorèmes de Bertini-Sard,’. Invent. Math. 36, 295–312 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  44. C. Voisin, Hodge theory and complex algebraic geometry, I, II. Cambridge Studies in Advanced Mathematics, 76, 77. Cambridge University Press, Cambridge, 2003.

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Correspondence to Luca Migliorini.

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Partially supported by project PRIN 2012 “Spazi di moduli e teoria di Lie”.

Lecture held in the Seminario Matematico e Fisico di Milano on March 19, 2012.

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Migliorini, L. Support Theorems for Algebraic Maps. Milan J. Math. 83, 21–45 (2015). https://doi.org/10.1007/s00032-015-0237-y

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