The quantum Euler class and the quantum cohomology of the Grassmannians

The Poincare duality of classical cohomology and the extension of this dual ity to quantum cohomology endows these rings with the structure of a Froben ius algebra. Any such algebra possesses a canonical "characteristic element ;" in the classical case this is the Euler class, and in the quantum case t his is a deformation of the classical Euler class which we call the "quantu m Euler class." We prove that the characteristic element of a Frobenius alg ebra A is a unit if and only if A is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians , and to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler class equals, as [qu antum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] Landau-Ginzbug potential.