Decompositions from torus actions and compactness of (sub-)level sets

Let $T=\mathbb{C}^{*}$ act on a smooth complex quasi-projective variety $X$. Assume that the limit point $\lim_{t\to 0}t\cdot x$ exists for every $x\in X$.

From the induced $U(1)$-action and its (real) moment map one can cook up a Morse function (or, if the fixed points are not isolated, a Morse-Bott function) $f$ on $X$, whose gradient flow $\varphi_s$ is equal to $\varphi_s (x)=e^{s}\cdot x$ for $s\in\mathbb{R}$ (considering $e^s\in T$). So, in particular, the limit of $s\cdot x$ for $s\to -\infty$ exists for every $x\in X$.

Edit: Let $X^{-}$ be the subset of those $x\in X$ for which $$\lim_{t\to \infty} t\cdot x$$ exists. Assume that $X^-$ is a compact subset of $X$.

Q. Under the above hypotheses, when are the sub-level sets $f^{-1}\left((-\infty,c]\right)$ of $f$ compact for every $c\in\mathbb{R}$? Any references about this?

• One sufficient condition would be that $f$ is bounded below, which is true in many standard examples. – Nick L Jun 21 '17 at 13:57
• Thank you for the comment. Do you also have a reference for "bounded below is enough"? – Qfwfq Jun 21 '17 at 14:13
• If $f(x) \geq k$ for all $x \in X$ then $f^{-1}((-\infty, c]) = f^{-1}([k,c])$ which is compact. pre-image of compact set by continuous function is compact. – Nick L Jun 21 '17 at 14:16
• For standard example one can take any linear action of $\mathbb{C}^*$ on $\mathbb{C}^{n}$ (with positive weights!). – Nick L Jun 21 '17 at 14:18
• You must be assuming $f$ proper (pre-image of compact set by continuous function is not compact in general!). When is $f$ proper, in the context of the question? – Qfwfq Jun 21 '17 at 14:42