Special geometry on Calabi-Yau moduli spaces and $Q$-invariant Frobenius rings

I will talk about a new approach to computing the volumes of the compact CY threefolds which may be realized as the hypersurface in the weighted projective space given by the quasihomogenious polynomial $W(x)$. As known the volume of CY manifold defines the Kahler potential of the CY moduli space. The main idea of the approach is to use the one to one correspondence between Hodge structure of the middle Cohomology of the CY manifold and Hodge structure of the Invariant Frobenius ring related with the isolated singularity defined by the same polynomial $W(x)$. This correspondence is realized by the Oscillatory integral presentation for the periods of the holomorphic Calabi-Yau 3-form which makes it possible to efficiently compute the periods without using the Picard-Fuchs equations. The use of the approach is demonstrated by computing the volume for the full complex structures moduli space of Fermat threefold