Mathematical Conversations

This will be an elementary/intuitive introduction to the $\pi_1$ of smooth algebraic varieties in $\mathbf A^1$-homotopy theory (over algebraically closed fields of char 0). We will give some flavor on the $\pi_0$ and $\pi_1$. Most of the topologically (or etale) simply-connected smooth varieties have a non trivial $\pi_1$ in this sense. Examples: the projective spaces $\mathbb P^n$ whose $\pi_1$ is the multiplicative group for $n > 1$ and the "simply-connected" simple algebraic groups $G$ whose $\pi_1$ is Milnor $K^M_2$ (for $G$ not $\mathrm{SL}_2$). We will finish with some computations in the case of smooth projective surfaces (joint with Asok), where being 0-connected for those means being rational. To conclude we will dream about a general surgery approach to the classification of higher dimensional smooth proper $\mathbf A^1$-connected varieties, analogue to the classical one in topology, where the $\pi_1$ we discussed plays exactly the role played by the fundamental group in topology.