Rational homotopy theory for moduli of stable rational curves

Vladimir Dotsenko

Published 2019-02-17Version 1

We show that rational cohomology algebras of the moduli spaces of stable rational marked curves are Koszul. This answers an open question of Manin. As a consequence, we compute the rational homotopy algebras of those spaces, and use it to deduce exponential growth for the free loop space homology, implying that Gromov's conjecture on closed geodesics holds for the moduli spaces of stable rational marked curves. We also prove the Koszul property for cohomology of a certain class of smooth projective toric varieties, which in particular implies that cohomology algebras of the Losev-Manin moduli spaces are Koszul. We conjecture that our main result generalises to quadratic algebras that arise as cohomology algebras of De Concini-Procesi wonderful models of hyperplane arrangements.