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

samedi 30 mars 2019, 14:30 → 15:30 Europe/Paris

Amphithéâtre Hermite (Institut Henri Poincaré)

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