# The cone of curves of complex projective manifolds with an algebraic torus action

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.

• 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. – Ruadhaí Dervan May 17 '19 at 11:09
• Thank you for this comment! – guest170519 May 17 '19 at 17:48

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 $$z_0\in \overline{Bz}$$.