Floer cohomology, multiplicity and the log canonical threshold

McLean, Mark¹

¹ Department of Mathematics, SUNY Stony Brook, Stony Brook, NY, United States

Geometry & topology, Tome 23 (2019) no. 2, pp. 957-1056.

Résumé

Let $f$ be a polynomial over the complex numbers with an isolated singularity at $0$. We show that the multiplicity and the log canonical threshold of $f$ at $0$ are invariants of the link of $f$ viewed as a contact submanifold of the sphere.

This is done by first constructing a spectral sequence converging to the fixed-point Floer cohomology of any iterate of the Milnor monodromy map whose $E^1$ page is explicitly described in terms of a log resolution of $f$. This spectral sequence is a generalization of a formula by A’Campo. By looking at this spectral sequence, we get a purely Floer-theoretic description of the multiplicity and log canonical threshold of $f$.

Détail

DOI : 10.2140/gt.2019.23.957

Classification : 14J17, 32S55, 53D10, 53D40
Keywords: log canonical threshold, Floer cohomology, singularity, Zariski conjecture, multiplicity, symplectic geometry, contact geometry

Bibliographie

[1] M Abouzaid, P Seidel, An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010) 627 | DOI

[2] N A’Campo, La fonction zêta d’une monodromie, Comment. Math. Helv. 50 (1975) 233 | DOI

[3] M F Atiyah, Resolution of singularities and division of distributions, Comm. Pure Appl. Math. 23 (1970) 145 | DOI

[4] W M Boothby, H C Wang, On contact manifolds, Ann. of Math. 68 (1958) 721 | DOI

[5] R Bott, The stable homotopy of the classical groups, Proc. Nat. Acad. Sci. U.S.A. 43 (1957) 933 | DOI

[6] F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799 | DOI

[7] F Bourgeois, A Oancea, S1–equivariant symplectic homology and linearized contact homology, preprint (2012)

[8] N Budur, Singularity invariants related to Milnor fibers : survey, from: "Zeta functions in algebra and geometry" (editors A Campillo, G Cardona, A Melle-Hernández, W Veys), Contemp. Math. 566, Amer. Math. Soc. (2012) 161 | DOI

[9] C Caubel, A Némethi, P Popescu-Pampu, Milnor open books and Milnor fillable contact 3–manifolds, Topology 45 (2006) 673 | DOI

[10] K Cieliebak, A Floer, H Hofer, Symplectic homology, II : A general construction, Math. Z. 218 (1995) 103 | DOI

[11] C Conley, E Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984) 207 | DOI

[12] J Denef, F Loeser, Lefschetz numbers of iterates of the monodromy and truncated arcs, Topology 41 (2002) 1031 | DOI

[13] M R Dörner, The space of contact forms adapted to an open book, PhD thesis, Universität zu Köln (2014)

[14] S Dostoglou, D A Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. 139 (1994) 581 | DOI

[15] L Ein, R Lazarsfeld, M Mustaţǎ, Contact loci in arc spaces, Compos. Math. 140 (2004) 1229 | DOI

[16] J W Fish, Target-local Gromov compactness, Geom. Topol. 15 (2011) 765 | DOI

[17] E Giroux, Géométrie de contact : de la dimension trois vers les dimensions supérieures, from: "Proceedings of the International Congress of Mathematicians" (editor L Tatsien), Higher Ed. Press (2002) 405

[18] V Guillemin, S Sternberg, Symplectic techniques in physics, Cambridge Univ. Press (1990)

[19] J Gutt, Generalized Conley–Zehnder index, preprint (2013)

[20] H Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, Ann. of Math. 79 (1964) 109 | DOI

[21] H Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, II, Ann. of Math. 79 (1964) 205 | DOI

[22] M Hutchings, Lecture notes on Morse homology (with an eye towards Floer theory and pseudoholomorphic curves), lecture notes (2002)

[23] J N Mather, Stratifications and mappings, from: "Dynamical systems" (editor M M Peixoto), Academic (1973) 195 | DOI

[24] D Mcduff, D Salamon, Introduction to symplectic topology, Clarendon (1998)

[25] D Mcduff, D Salamon, J–holomorphic curves and symplectic topology, 52, Amer. Math. Soc. (2004)

[26] M Mclean, The growth rate of symplectic homology and affine varieties, Geom. Funct. Anal. 22 (2012) 369 | DOI

[27] M Mclean, Symplectic homology of Lefschetz fibrations and Floer homology of the monodromy map, Selecta Math. 18 (2012) 473 | DOI

[28] M Mclean, Reeb orbits and the minimal discrepancy of an isolated singularity, Invent. Math. 204 (2016) 505 | DOI

[29] J Milnor, Singular points of complex hypersurfaces, 61, Princeton Univ. Press (1968)

[30] M Mustaţă, IMPANGA lecture notes on log canonical thresholds, from: "Contributions to algebraic geometry" (editor P Pragacz), Eur. Math. Soc. (2012) 407 | DOI

[31] J Von Neumann, Continuous geometry, 25, Princeton Univ. Press (1960)

[32] R S Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966) 1 | DOI

[33] D Petersen, A spectral sequence for stratified spaces and configuration spaces of points, preprint (2016)

[34] J Robbin, D Salamon, The Maslov index for paths, Topology 32 (1993) 827 | DOI

[35] D Salamon, E Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992) 1303 | DOI

[36] P Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000) 103 | DOI

[37] P Seidel, More about vanishing cycles and mutation, from: "Symplectic geometry and mirror symmetry" (editors K Fukaya, Y G Oh, K Ono, G Tian), World Sci. (2001) 429 | DOI

[38] P Seidel, Critical points of complex polynomials from the symplectic viewpoint, lecture (2014)

[39] M F Tehrani, M Mclean, A Zinger, Normal crossings singularities for symplectic topology, preprint (2014)

[40] M F Tehrani, M Mclean, A Zinger, The smoothability of normal crossings symplectic varieties, preprint (2014)

[41] I Uljarevic, Floer homology of automorphisms of Liouville domains, preprint (2014)

[42] A Varchenko, Contact structures and isolated singularities, Mosc. Univ. Math. Bull. 35 (1982) 18

[43] C Viterbo, Functors and computations in Floer homology with applications, I, Geom. Funct. Anal. 9 (1999) 985 | DOI