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

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Amphithéâtre Hermite (Institut Henri Poincaré)
### Amphithéâtre Hermite

#### Institut Henri Poincaré

11 rue Pierre-et-Marie-Curie, 75005 Paris

Description

Among the most interesting invariants one can associate with a link L⊂S3 is its HOMFLY polynomial P(L,v,s)∈Z[v±1,(s−s−1)±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 I⊂C[[x,y]] containing f and such that dimC[[x,y]]/I=n. If m:C[n]p→Z 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)=∑ns2n∫C[n]p(1−v2)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.