We prove several formulas related to Hodge theory and the KodairaSpencerKuranishi deformation theory of Khler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on CalabiYau manifolds and also a construction of global canonical family of holomorphic ( n , 0 ) -forms on the deformation spaces of CalabiYau manifolds. Similar constructions are also applied to the deformation spaces of compact Khler manifolds.

We prove that the GromovHausdorff compactification of the moduli space of KhlerEinstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tians theorem on the existence of KhlerEinstein metrics on smooth Del Pezzo surfaces and classifies all the degenerations of such metrics. The proof is based on a combination of both algebraic and differential geometric techniques.

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We describe the potential functions for N= 4B supersymmetric quantum mechanics with D (2, 1; ) symmetry.

In this article we show how to compute the semiclassical spectral measure associated with the Schr¨odinger operator on Rn, and, by examining the first few terms in the asymptotic expansion of this measure, obtain inverse spectral results in one and two dimensions. (In particular we show that for the Schr¨odinger operator on R2 with a radially symmetric electric potential, V , and magnetic potential, B, both V and B are spectrally determined.) We also show that in one dimension there is a very simple explicit identity relating the spectral measure of the Schr¨odinger operator with its Birkhoff canonical form.