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

Let X be a smooth projective variety over CC with an action of (CC, +). 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 (CC, +).

In the talk I will explain the beautiful basics on how the (CC, +)-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!