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Séminaire Bourbaki

Luca MIGLIORINI — HOMFLY polynomials from the Hilbert schemes of a planar curve , after D. Maulik, A. Oblomkov, V. Shende...

Institut Henri Poincaré
Amphithéâtre Hermite
11 rue Pierre-et-Marie-Curie, 75005 Paris

Among the most interesting invariants one can associate with a link LS3 is its HOMFLY polynomial P(L,v,s)Z[v±1,(ss1)±1]. A. Oblomkov and V. Shende conjectured that this polynomial can be expressed in algebraic geometric terms when L is obtained as the intersection of a plane curve singularity (C,p)C2 with a small sphere centered at p: if f=0 is the local equation of C, its Hilbert scheme C[n]p is the algebraic variety whose points are the length n subschemes of C supported at p, or, equivalently, the ideals IC[[x,y]] containing f and such that dimC[[x,y]]/I=n. If m:C[n]pZ is the function associating with the ideal I the minimal number m(I) of its generators, they conjecture that the generating function Z(C,v,s)=ns2nC[n]p(1v2)m(I)dχ(I) coincides, up to a renormalization, with P(L,v,s). In the formula the integral is done with respect to the Euler characteristic measure dχ. A more refined version of this surprising identity, involving a colored'' variant of P(L,v,s), was conjectured to hold by E. Diaconescu, Z. Hua and Y. Soibelman. The seminar will illustrate the techniques used by D. Maulik to prove this conjecture.