We prove several rigidity theorems for elliptic genera associated to Toeplitz operators on odd dimensional spin manifolds. We also prove an equivariant odd index theorem for Dirac operators with involution parity and the Atiyah-Hirzebruch vanishing theorems for odd dimensional spin manifolds.

We study the rigidity of polyhedral surfaces using variational principles. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fenchel and Nielsen. By studying the derivative of the cosine laws, we discover a uniform approach to several variational principles on polyhedral surfaces with or without boundary. As a consequence, the work of Penner, Bobenko and Springborn and Thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context.

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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 first Chern form of the moduli space of polarized Calabi-Yau manifolds, with the Hodge metric or the Weil-Petersson metric, represent the first Chern class of the canonical extensions of the tangent bundle to the compactification of the moduli space with normal crossing divisors.