| Résume | I will introduce a construction of a critical cohomological Hall algebra A associated to an arbitrary quiver $Q$, following the work of Kontsevich and Soibelman. This algebra turns out to have several surprising properties: firstly, it is a free commutative algebra. Secondly, the Kac polynomials are the characteristic polynomials of its generators. Finally, despite being defined in terms of vanishing cycles, the underlying vector space of A admits a much simpler description (in fact many such descriptions) in terms of ordinary compactly supported cohomology. I will explain these properties, and why they add up to a new proof of the Kac positivity conjecture. |
