Vol. 273, No. 2, 2015

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

Ping Li

Vol. 273 (2015), No. 2, 331–351

Several independent articles have observed that the Hirzebruch χy-genus has an important feature, which we call 1-phenomenon and which tells us that the coefficients of the Taylor expansion of the χy-genus at y = 1 have explicit expressions. Hirzebruch’s original χy-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 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 rational 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 the elliptic genus, a modern version of the χy-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 forms 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 χy-genus is hidden.

$\chi_y$-genus, pluri-genus, elliptic genus, elliptic operator, characteristic number, Jacobi form, modular form
Mathematical Subject Classification 2010
Primary: 58J20
Secondary: 58J26, 11F11, 11F50
Received: 5 January 2014
Revised: 19 July 2014
Accepted: 15 August 2014
Published: 23 December 2014
Ping Li
Department of Mathematics
Tongji University
Shanghai, 200092