Characterisation of the affine plane using $\mathbb{A}^1$-homotopy theory

Abstract: Characterisation of the affine $n$-space is one of the major problem in affine algebraic geometry. Miyanishi showed an affine complex surface $X$ is isomorphic to $\mathbb{C}^2$ if $\mathscr{O}(X)$ is a U.F.D., $\mathscr{O}(X)^*= \mathbb{C}^∗$ and $X$ has a non-trivial $\mathbb{G}_a$-action [3, Theorem 1]. Since the orbits of a $\mathbb{G}_a$-action are affine lines, existence of a non-trivial $\mathbb{G}_a$-action says that there is a non-constant $\mathbb{A}^1$ in X. Ramanujam showed that a smooth complex surface is isomorphic to $\mathbb{C}^2$ if it is topologically contractible and it is simply connected at infinity [5]. Topological contractibility, in particular pathconnectedness says that there are non-constant intervals in X. On the other hand, $\mathbb{A}^1$-homotopy theory has been developed by F.Morel and V.Voevodsky [4] as a connection between algebra and topology. An algebrogeometric analogue of topological connectedness is $\mathbb{A}^1$-connectedness. In this talk, using ghost homotopy techniques [2, Section 3] we will prove that if a surface $X$ is $\mathbb{A}^1$-connected, then there is an open dense subset such that through every point there is a non-constant $\mathbb{A}^1$ in X. As a consequence using the algebraic characterisation, we will prove that $\mathbb{C}^2$ is the only $\mathbb{A}^1$-contractible smooth complex surface. This answers the conjecture appeared in [1, Conjecture 5.2.3]. We will also see some other useful consequences of this result. This is a 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. 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.

Mathematics

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IIT Bombay Virtual Commutative Algebra Seminar

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