-1-Phenomena for the pluri $\chi_y$-genus and elliptic genus

Abstract

Several independent articles have observed that the Hirzebruch $\chi_y$-genus has an important feature, which the author calls -1-phenomenon and tells us that the coefficients of the Taylor expansion of the $\chi_y$-genus at $y=-1$ have explicit expressions. Hirzebruch's original $\chi_y$-genus can be extended towards two directions: the pluri-case and the case of elliptic genus. This paper contains two parts in which we investigate the -1-phenomena in these two generalized cases respectively and show that in each case there exists a -1-phenomenon in a suitable sense. Our main results in the first part have an application, which states that all characteristic numbers (Chern numbers and Pontrjagin numbers) on manifolds can be expressed, in a very explicit way, in terms of some rationally linear combination of indices of some elliptic operators. This gives an analytic interpretation of characteristic numbers and affirmatively answers a question posed by the author several years ago. The second part contains our attempt to generalize this -1-phenomenon to elliptic genus, a modern version of the $\chi_y$-genus. We first extend the elliptic genus of an almost-complex manifold to a twisted version where an extra complex vector bundle is involved, and show that it is a weak Jacobi form under some assumptions. A suitable manipulation on the theory of Jacobi form will produce new modular forms from this weak Jacobi form and thus much arithmetic information related to the underlying manifold can be obtained, in which the -1-phenomenon of the original $\chi_y$-genus is hidden.