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