Sur les genres multiplicatifs définis par des intégrales elliptiques. (On multiplicative genera defined by elliptic integrals) (Q580782)

Multiplicative genera were introduced by F. Hirzebruch in the early 1950's. In the context of oriented manifolds, a multiplicative genus is a unital ring homomorphism \(\Omega_*^{SO}\to R\) from Thom's oriented bordism ring \(\Omega_*^{SO}\) to a commutative \({\mathbb{Q}}\)-algebra \({\mathbb{R}}\). The most important examples, with \(R= {\mathbb{Q}}\), are the signature (L-genus) and, in the case of spin manifolds, the spinor index (\(\hat A\)-genus). Since \(\Omega_*^{SO}\otimes {\mathbb{Q}}\) is a polynomial algebra on the bordism classes of the complex projective spaces \({\mathbb{C}}P^{2k}\) \((k>0)\), a genus \(\phi\) is uniquely determined by its logarithm \[ g(x)=\sum_{k\geq 0}\frac{\phi ({\mathbb{C}}P^{2k})}{2k+1}x^{2k+1}=\int^{x}_{0}\sum_{k\geq 0}\phi ({\mathbb{C}}P^{2k})t^{2k} dt. \] The signature and \(\hat A\)-genus have many special properties; for example, if \(\xi^{2m}\to B\) is a complex even-dimensional vector bundle over a compact oriented smooth manifold, they both vanish on the associated projective space bundle \({\mathbb{C}}P(\xi^{2m})\). The author answers the question: which genera \(\phi\) vanish on all \({\mathbb{C}}P(\xi^{2m})?\) His answer is a lovely one, as is his argument. A necessary and sufficient condition is that the logarithm of \(\phi\) be an elliptic integral of the form \[ g(x)=\int^{x}_{0}(1-2\delta t^ 2+\epsilon t^ 4)^{-1/2} dt \] with \(\delta,\epsilon\in {\mathbb{R}}\). Genera having such a logarithm are called elliptic genera. When \(\delta,\epsilon\in {\mathbb{C}}\) and \(1-2\delta t^ 2+\epsilon t^ 4\) has four distinct roots, the inverse of g(x) under composition is the power series expansion of an elliptic function. The proof that an elliptic genus vanishes on \({\mathbb{C}}P(\xi^{2m})\) is a calculation with characteristic classes, the clincher being the fact that the sum of the residues of an elliptic function is zero. This paper has already had a notable impact, and has given rise to several fascinating developments.