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.