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SUMMARY:Utsav Chowdhury (Indian Statical Institute\, Kolkata\, India)
DTSTART:20221216T120000Z
DTEND:20221216T130000Z
DTSTAMP:20260423T021102Z
UID:VCAS/150
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/VCAS/150/">C
 haracterisation of the affine plane using $\\mathbb{A}^1$-homotopy theory<
 /a>\nby Utsav Chowdhury (Indian Statical Institute\, Kolkata\, India) as p
 art of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nChara
 cterisation of the affine $n$-space is one of the major problem in affine 
 algebraic geometry. Miyanishi showed an affine complex surface $X$ is isom
 orphic 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 li
 nes\, 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 compl
 ex surface is isomorphic to $\\mathbb{C}^2$ if it is topologically contrac
 tible and it is simply connected at infinity [5]. Topological contractibil
 ity\, in particular pathconnectedness says that there are non-constant int
 ervals 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 a
 nd topology. An algebrogeometric analogue of topological connectedness is 
 $\\mathbb{A}^1$-connectedness. In this talk\, using ghost homotopy techniq
 ues [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 po
 int there is a non-constant $\\mathbb{A}^1$ in X. As a consequence using t
 he 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 othe
 r useful consequences of this result. This is a joint work with Biman Roy.
  \n\nReferences\n[1] A. Asok\, P. A. Østvær\; A 1 -homotopy theory and c
 ontractible varieties: a survey\, Homotopy Theory and Arithmetic Geometry 
 – Motivic and Diophantine Aspects. Lecture Notes in Mathematics\, vol 22
 92. Springer\, Cham. https://doi.org/10.1007/978-3-030-78977-05. \n[2] C. 
 Balwe\, A. Hogadi and A. Sawant\; A 1 -connected components of schemes. Ad
 v Math\, Volume 282\, 2016. \n[3] M. Miyanishi\; An algebraic characteriza
 tion of the affine plane. J. Math. Kyoto Univ. 15-1 (1975) 19-184.\n
LOCATION:https://researchseminars.org/talk/VCAS/150/
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