Sabrina Pauli: A1-contractibility of affine modifications
Abstract. Koras-Russell threefolds first appeared in Koras and Russell’s monumental work of showing that every Gm-action on C 3 is linearizable [KR97]. They are smooth, complex, topologically contractible, affine varieties of dimension three not isomorphic to C 3 [ML96] which makes them potential counterexamples to the Zariski cancellation problem [Kra96]. In [HKØ16] it is shown that Koras-Russell threefolds of the first and second kind are stably A 1 - contractible and consequently A 1 -contractible after a finite suspension by the projective line P 1 - pointed at infinity. They are the first examples of smooth, complex, affine varieties not isomorphic to affine space which are stably A 1 -contractible. Koras-Russell threefolds of the first kind are in fact A 1 -contractible as shown in [DF15]. We extend the results in [HKØ16] and [DF15] to a bigger family of hypersurfaces in C 4 including Koras-Russell threefolds of the first kind, threefolds with several degenerate fibers as well as hypersurfaces given by (1) x n z = y α1 + t α2 + xq(x, t, y), where n, α1, α2 ≥ 2 and gcd(α1, α2) = 1, and q(x, y, t) ∈ C[x, y, t] with q(0, 0, 0) 6= 0. The techniques in [HKØ16] and [DF15] can also be applied to hypersurfaces in C 4 given by (2) x n z = (x my + z s ) α1 + t α2 + x with n, m, s, α1, α2 ≥ 2 and gcd(sα1, α2) = 1 obtained by an affine modification [KZ99] of a Koras-Russell threefold of the first kind. Additionally, we provide results on a four dimensional example, namely the hypersurface in C 5 given by (3) x n z = u 2 + t 3 + w 5 + x with n ≥ 2. The given examples (1), (2) and (3) are smooth, topologically contractible, complex, affine varieties not isomorphic to affine space which turn out to be stably A 1 -contractible, making them potential counterexamples to the Zariski cancellation problem.