  We show that the limit, by rescaling, of the `new supersymmetric index'
attached to the Fourier-Laplace transform of a polarized variation of Hodge
structure on a punctured affine line is equal to the spectral polynomial
attached to the same object. We also extend the definition by Deligne of a
Hodge filtration on the de Rham cohomology of a exponentially twisted polarized
variation of complex Hodge structure and prove a E_1 degeneration property for
it.
