Appears in collection : Bourbaki - Mars 2019

Among the most interesting invariants one can associate with a link $\mathcal L\subset S^3$ is its HOMFLY polynomial $\mathbf P(\mathcal L,v,s)\in \mathbf Z[v^{\pm 1},(s−s^{−1})^{\pm 1}$. A. Oblomkov and V. Shende conjectured that this polynomial can be expressed in algebraic geometric terms when $\mathcal L$ is obtained as the intersection of a plane curve singularity $(C,p)\subset \mathbf C^2$ with a small sphere centered at $p$: if $f=0$ is the local equation of $C$, its Hilbert scheme $C_p^{[n]}$ is the algebraic variety whose points are the length $n$ subschemes of $C$ supported at $p$, or, equivalently, the ideals $I\subset C[[x,y]]$ containing $f$ and such that $\dim C[[x,y]]/I=n$. If $m:C_p^{[n]}\rightarrow\mathbf Z$ is the function associating with the ideal $I$ the minimal number $m(I)$ of its generators, they conjecture that the generating function $$Z(C,v,s)=\sum_n s^{2n}\int_{C_p^{[n]}}(1−v^2)^{m(I)}d\chi(I)$$ coincides, up to a renormalization, with $\mathbf P(\mathcal L,v,s)$. In the formula the integral is done with respect to the Euler characteristic measure $d\chi$. A more refined version of this surprising identity, involving a colored variant of $\mathbf P(\mathcal L,v,s)$, 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.

[After D. Maulik, A. Oblomkov, V. Shende...]