Kawakubo and Uchida showed that, if a closed oriented 4k-dimensional manifold M admits a semi-free circle action such that the dimension of the fixed point set is less than 2k, then the signature of M vanishes. In this note, by using G-signature theorem and the rigidity of the signature operator, we generalize this result to more general circle actions. Combining the same idea with the remarkable WittenTaubesBott rigidity theorem, we explore more vanishing results on spin manifolds admitting such circle actions. Our results are closely related to some earlier results of ConnerFloyd, LandweberStong and HirzebruchSlodowy.

In this paper, we generalize the anomaly cancellation formulas given by Alvarez-Gaum and Witten (1983), Liu (1995) and Han and Zhang (2004)[1],[2],[7] to the cases where an auxiliary bundle W and a complex line bundle are involved with no conditions on the first Pontryagin forms being assumed.

In this paper, we give a simple proof of Mirzakhanis recursion formula of Weil-Petersson volumes of moduli spaces of curves using the Witten-Kontsevich theorem. We also briefly describe a very general recursive phenomenon in the intersection theory of moduli spaces of curves. In particular, we present several new recursion formulas for higher degree classes.

Using the celebrated Witten-Kontsevich theorem, we prove a recursive formula of the n -point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for Gromov-Witten invariants. The latter has been proved recently by X. Liu and R. Pandharipande. We also give a combinatorial interpretation of n -point functions in terms of summation over binary trees.

In this note, we explain that Ross-Thomas' result on the weighted Bergman kernels on orbifolds can be directly deduced from our previous result. This result plays an important role in the companion paper to prove an orbifold version of Donaldson Theorem.