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

TL;DR

This paper explores the algebraic cobordism spectrum in motivic homotopy theory, computes its $ ext{eta}$-completion, and investigates the elliptic genus's kernel related to $SL$-flops, extending Totaro's complex results.

Contribution

It computes the $ ext{eta}$-completion of $MSL$ in motivic homotopy and characterizes the kernel of the elliptic genus restricted to $MSL$ as generated by $SL$-flops.

Findings

Computed the geometric part of the $ ext{eta}$-completion of $MSL$.

Identified the kernel of the elliptic genus as generated by differences of $SL$-flops.

Proved convergence properties of the motivic Adams spectral sequence.

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 $\eta$-completion of $MSL$ (modulo the maximal subgroup that is $l$-divisble for all primes $l\neq2, char k$). 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.