Additive group actions, formal solutions to PDEs and Bialynicki-Birula decomposition

Abstract: Let $X$ be a smooth projective variety over $\mathbb{C}$ with an action of $(\mathbb{C}, +)$. Assume that $X$ has a unique fixed point $x_0$. Carrell’s conjecture predicts that $X$ is rational. Restriction of orbits to germs at $x_0$ reduces this conjecture to describing solutions of certain systems of PDE in the formal power series ring $k[[t]]$ with $d(t) = -t^2$. This suggests a stronger form of the conjecture: $X$ is a union of affine spaces. This strengthening would give an analogue of Bialynicki-Birula decomposition for $(\mathbb{C}, +)$. In the talk I will explain the beautiful basics on how the $(\mathbb{C}, +)$-actions, differential equations and rationality intertwine and then present the state of the art on the conjecture. This is a work in progress, comments and suggestions are welcome!

commutative algebraalgebraic geometry

Audience: researchers in the topic

Max Planck Institute nonlinear algebra seminar online

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