Two-dimensional superconformal field theories (SCFTs) with N=(2, 2) supersymmetries have intimate relations with the geometries and topologies of Calabi-Yau manifolds. The correlation functions of (twisted) chiral ring operators in N=(2, 2) SCFTs turns out to be the metrics of various vector bundles over moduli spaces of Calabi-Yau manifolds. They are determined by a group of Hitchin type integrable equations, the tt*-equations. In this talk, I will discuss how to exactly compute these correlation functions via supersymmetric localization for SCFTs with Calabi-Yau geometric phases, and thus solve the tt*-equations automatically. Especially in the context of Calabi-Yau geometries, I will give an explicit geometric interpretation to the method developed as the Griffiths transversality on the Hodge bundle over Calabi-Yau complex moduli. At last I will apply this method to the case of complete intersections in toric varieties as an example."