Abstract
If A • is a bounded, constructible complex of sheaves on a complex analytic space X, and \({f : X \rightarrow \mathbb{C}}\) and \({g : X \rightarrow \mathbb{C}}\) are complex analytic functions, then the iterated vanishing cycles φ g [−1](φ f [−1]A •) are important for a number of reasons. We give a formula for the stalk cohomology H*(φ g [−1]φ f [−1]A •) x in terms of relative polar curves, algebra, and Morse modules of A •.
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References
Braden T.: On the reducibility of characteristic varieties. Proc. AMS 130, 2037–2043 (2002)
Briançon J., Maisonobe P., Merle M.: Localisation de systèmes différentiels, stratifications de Whitney et condition de Thom. Invent. Math. 117, 531–550 (1994)
Brylinski J.L., Dubson A., Kashiwara M.: Formule de l’indice pour les modules holonomes et obstruction d’Euler locale. C. R. Acad. Sci. Sér. A 293, 573–576 (1981)
Caubel, C.: Sur la topologie d’une famille de pinceaux de germes d’hypersurfaces complexes. PhD thesis, Université Toulouse III (1998)
Caubel C.: Variation of the milnor fibration in pencils of hypersurface singularities. Proc. Lond. Math. Soc. 83(2), 330–350 (2001)
Dimca, A.: Sheaves in Topology. Universitext. Springer, Berlin (2004)
Ginsburg V.: Characteristic varieties and vanishing cycles. Invent. Math. 84, 327–403 (1986)
Goresky, M., MacPherson, R.: Stratified Morse Theory, volume 40 of Proc. Symp. Pure Math. 517–533. AMS (1983)
Guibert G., Loeser F., Merle M.: Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink. Duke Math. J. 132(3), 409–457 (2006)
Hamm H., Lê D.T.: Un théorème de Zariski du type de Lefschetz. Ann. Sci. Éc. Norm. Sup. 6(series 4), 317–366 (1973)
Kashiwara, M., Schapira, P.: Sheaves on manifolds, volume 292 of Grund. math. Wissen. Springer, Berlin (1990)
Kazhdan D., Lusztig G.: A topological approach to Springer’s representations. Adv. Math. 38, 222–228 (1980)
Lê D.T.: Calcul du Nombre de Cycles Évanouissants d’une Hypersurface Complexe. Ann. Inst. Fourier, Grenoble 23, 261–270 (1973)
Lê D.T.: Topological use of polar curves. Proc. Symp. Pure Math. 29, 507–512 (1975)
Lê D.T.: Le concept de singularité isolée de fonction analytique. Adv. Stud. Pure Math. 8, 215–227 (1986)
Lê D.T., Perron B.: Sur la fibre de Milnor d’une singularité isolée en dimension complexe trois. C. R. Acad. Sci. Pairs Sér. A 289, 115–118 (1979)
Massey D.: Numerical invariants of perverse sheaves. Duke Math. J. 73(2), 307–370 (1994)
Massey D.: Lê Cycles and Hypersurface Singularities, volume 1615 of Lecture Notes in Math. Springer, Berlin (1995)
Massey D.: Hypercohomology of milnor fibres. Topology 35, 969–1003 (1996)
Massey D.: A little microlocal morse theory. Math. Ann. 321, 275–294 (2001)
Massey, D.: Numerical Control over Complex Analytic Singularities, volume 778 of Memoirs of the AMS. AMS (2003)
Massey D.: Enriched relative polar curves and discriminants. Contemp. Math. 474, 107–144 (2008)
Sabbah C.: Quelques remarques sur la géométrie des espaces conormaux. Astérisque 130, 161–192 (1985)
Sabbah C.: Proximité évanescente. Compos. Math. 62, 283–328 (1987)
Saito M.: Mixed Hodge Modules. Publ. RIMS Kyoto Univ. 26, 221–333 (1990)
Schürmann J.: Topology of Singular Spaces and Constructible Sheaves, volume 63 of Monografie Matematyczne. Birkhäuser, Basel (2004)
Teissier B.: Cycles évanescents, sections planes et conditions de Whitney. Astérisque 7–8, 285–362 (1973)