I would like to find a reference to the following statement:

Statement. Let $X$ be a complex projective manifold with an algebraic action of a $k$-dimensional torus $(\mathbb C^*)^k$. Then the cone of curves of $X$ is generated by $(\mathbb C^*)^k$-invariant curves. In particular, each irreducible proper curve in $X$ is linearly equivalent to a sum of such invariant curves with positive coefficients.

(Note, of course, that for toric manifolds - when the action of $(\mathbb C^*)^k$ has an open orbit - this statement is classical.)

PS. I am aware of a proof of this statement - along the lines of the comment below, but would be especially grateful for a precise reference.

  • 3
    $\begingroup$ I suppose, given a generator $C$, one can take the flat limit under each $\mathbb{C}^*$ in turn to obtain a new curve $\hat C$ which should be linearly equivalent and torus invariant. $\endgroup$ – Ruadhaí Dervan May 17 '19 at 11:09
  • $\begingroup$ Thank you for this comment! $\endgroup$ – guest170519 May 17 '19 at 17:48

This follows from a result of Andre Hirschowitz.

Le groupe de Chow equivariant. C. R. Acad. Sci. Paris Ser. I Math. 298 (1984), no. 5, 87--89.

Hirschowitz proved in this paper that for a projective variety with an algebraic action of a connected solvable group $B$, any effective cycle is rationally equivalent to a $B$-invariant effective cycle. The idea is to consider the $B$-action on the Chow variety $Z$ containing a given effective cycle $z$. Applying the Borel fixed point theorem, one finds a $B$-stable cycle $z_0\in \overline{Bz}$.

Of course the Statement follows immediately from Hirschowitz's result.

As it happens, this statement was reproved several times afterwards.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.