Speaker: Utsav Chowdhury, Indian Statical Institute, Kolkata, India
Date/Time: 16 December 2022, 5:30pm
Gmeet link: meet.google.com/vxv-adfh-onj
Title: Characterisation of the affine plane using A^1 -homotopy theory
Abstract: Characterisation of the affine n-space is one of the major problems in affine algebraic geometry. Miyanishi showed an affine complex surface X is isomorphic to C^2 if O(X) is a U.F.D., O(X)^∗ = C^∗ and X has a non-trivial Ga-action [3, Theorem 1]. Since the orbits of a Ga-action are affine lines, the existence of a non-trivial Ga-action says that there is a non-constant A^1 in X. Ramanujam showed that a smooth complex surface is isomorphic to C^2 if it is topologically contractible and it is simply connected at infinity [5]. Topological contractibility, in particular, path connectedness says that there are non-constant intervals in X. On the other hand, A^1 -homotopy theory has been developed by F.Morel and V.Voevodsky [4] as a connection between algebra and topology. An algebrogeometric analog of topological connectedness is A^1 -connectedness. In this talk, using ghost homotopy techniques [2, Section 3] we will prove that if a surface X is A^1 -connected, then there is an open dense subset such that through every point there is a non-constant A^1 in X.
As a consequence using the algebraic characterization, we will prove that C^2 is the only A^1 -contractible smooth complex surface. This answers the conjecture that appeared in [1, Conjecture 5.2.3]. We will also see some other useful consequences of this result. This is joint work with Biman Roy.
References
[1] A. Asok, P. A. Østvær; A 1 -homotopy theory and contractible varieties: a survey, Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects. Lecture Notes in Mathematics, vol 2292. Springer, Cham. https://doi.org/10.1007/978-3-030-78977-05.
[2] C. Balwe, A. Hogadi and A. Sawant; A 1 -connected components of schemes. Adv Math, Volume 282, 2016.
[3] M. Miyanishi; An algebraic characterization of the affine plane. J. Math. Kyoto Univ. 15-1 (1975) 19-184.