Subject: re: Nick's posting
Date: Fri, 27 Sep 2002 16:17:46 -0400 (EDT)
From: Jack Morava
don't know if this is really relevant to Nick's posting:]
+++++++++++++++++++
The Hirzebruch genus sends CP_n to something like
1 + y + ... + y^n ,
which corresponds to the formal group
X + Y + (y-1)XY
X,Y |---> ---------------
1 - yXY
with a logarithm
1 - yX
log_y = (1-y)^{-1} log ( ------ )
1 - X
mixing fractional linear transformations with fractional powers
an interesting way. Maybe the families considered by Stong and Kamata
come from more general fractional linear transformations (which send
0 to 0)?
BTW (some) algebraic geometers like to think of CP_n as a sum
1 + L + ... + L^n
of powers of a `Tate motive' L. From the point of view of cobordism
theory, this suggests that they're looking at complex manifolds through
a telescope with a pretty narrow field of view...
___________________________________________
Subject: reply to Bob Stong's FGL query
Date: Sat, 28 Sep 2002 14:28:52 -0400
From: "Peter S. Landweber"
Admittedly this is not the way Bob Stong and I usually correspond. But
I don't want to pass up this opportunity.
Recent mention of these formal group laws spotted by Bob is made by
Philipp Busato, in his paper: Realization of Abel's universal formal
group law, Math. Z. 239, 527--561 (2002); see fleeting mention on page
529.
A rather complete study of these formal group laws (in a wider context,
which includes Abel's fgl and also Euler's fgl which corresponds to the
level 2 elliptic genus) has been made by V. M. Buchstaber and A. N.
Kholodov: Formal groups, functional equations, and generalized
cohomology theories, Math. USSR Sbornik 69, 77--97 (1991).
On page 78, Buchstaber and Kholodov cite a paper by I. M. Krichever
(which I have not looked at): Formal groups and the Atiyah-Hirzebruch
formula, Math. USSR Izvestiya 38 (1974). Apparently the 2-parameter
family of formal groups (whose corresponding complex genera encompass
the Todd genus and Hirzebruch's one-parameter generalization which can
also be specialized to the Euler characteristic and signature) was
examined by Krichever.
Peter
> Subject: for your list
> Date: Thu, 26 Sep 2002 15:50:23 -0400 (Eastern Daylight Time)
> From: "Nicholas J. Kuhn"
>
> My (non email using) colleague Bob Stong has been thinking about formal
> groups. He has the following pretty two parameter family of examples:
>
> F(x,y) = (x+y+axy)/(1+bxy), with a and b in the underlying commutative ring.
>
> This family specializes to the additive fgl if a=b=0, the
> multiplicative fgl if a=1, b=0, and the flg associated with tanh if
> a=0, b=1. It also includes families described by Hirzebruch (chi_y
> genus).
>
> Bob doesn't know of this family appearing in the literature. I am
> curious if anyone reading this knows if it does.
>
> Nick Kuhn
> njk4x@virginia.edu