Algebraic elliptic cohomology and flops II: $SL$-cobordism

Abstract

In this paper, we study the algebraic cobordism spectrum $MSL$ in the motivic stable homotopy category of Voevodsky over an arbitrary perfect field $k$. Using the motivic Adams spectral sequence, we compute the geometric part of the $\hat{I}\cdot$-completion of $MSL$ (modulo the maximal subgroup that is $l$-divisble for all primes $l\ne 2,chark$). As an application, we study the Krichever's elliptic genus with integral coefficients, restricted to $MSL$. We determine its image, and identify its kernel as the ideal generated by differences of $SL$-flops. This was proved by B. Totaro in the complex analytic setting. In the appendix, we prove some convergence properties of the motivic Adams spectral sequence.

v4. Some typos have been corrected. v3. A proof of a lemma of Novikov has been added