We show that for smooth complex projective varieties the most general combinations of Chern numbers that are invariant under the Kequivalence relation consist of the complex elliptic genera. Received October 30, 2000. Supported in part by NSC project 89-2115-M-007-045. 285 Licensed to Yale Univ. Prepared on Thu Jul 2 05:05:02 EDT 2015 for download from IP 130.132.123.28. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf 286 CHIN-LUNG WANG is the quantum minimal model conjecture raised by Ruan [15] (cf. §6). Another one is Totaro's work [16]: complex elliptic genera = complex cobordism ring modulo classical flops, or equivalently Chern numbers invariant under classical flops consist of precisely the complex elliptic genera.We show in this paper a generalization of Totaro's result: complex elliptic genera = complex cobordism ring modulo K-equivalence. A surprising conclusion is that in the complex cobordism ring, the ideal generated by classical flops equals the seemingly much larger ideal generated by all K-equivalent pairs. To summarize our approach, we follow the meta theorem. The target invariants are genera (or Chern numbers), which by definition are certain integrals, so we only need a nice change of variable formula. This is treated in two steps:Inspired by the work of Hirzebruch et. al.[8] on the characterization of the (real) elliptic genera (of Landweber and Stong) as the most general genera that are multiplicative on fiber bundles with fiber P 2n−1 for all n ∈ N, we characterize the complex elliptic genera (studied by Witten, Hirzebruch and subsequently by Krichever, Höhn and Totaro) in a similar flavor via blowingups along smooth centers.Let X be a compact complex manifold or a proper smooth variety (over an algebraically closed field of arbitrary characteristic). For a commutative ring R, an R-genus ϕ is defined by a power series Q(x) ∈ R [[x]] through Hirzebruch's multiplicative sequence K Q (or K ϕ ). As usual we write Q(x) = x/f (x).Theorem A (residue theorem). For any cycle D in X and for any blowing-up φ : Y → X along smooth center Z with exceptional divisor E, one has for any power seriesHere n i 's denote the formal Chern roots of the normal bundle N Z/X and the residue stands for the coefficient of the degree −1 term of a Laurent power series with coefficients in the cohomology ring or the Chow ring of X.Theorem B (characterization of complex elliptic genera). Consider the following sets of power series f (x) = x + · · · ∈ C[[x]] (or C-genera φ's): S 1 : ϕ admits a first step change of variable formula. That is, for each r ∈ N there exists a power series A(t, r) in t that serves as the Jacobian Licensed to Yale Univ. Prepared on Thu Jul 2 05:05:02 EDT 2015 for download from IP 130.132.123.28. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf K-EQUIVALENCE IN BIRATIONAL GEOMETRY 287 factor such that A(0, r) = 1 and X