Several independent articles have observed that the Hirzebruch
-genus has an important feature,
which we call
-phenomenon
and which tells us that the coefficients of the Taylor expansion of the
-genus at
have explicit expressions.
Hirzebruch’s original
-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
-phenomena
in these two generalized cases and show that in each case there exists a
-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
-phenomenon
to the elliptic genus, a modern version of the
-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
-phenomenon of
the original
-genus
is hidden.
