Google searches for "local ample cone" and "local Kähler cone" yield no results, but maybe there is a different term.

Let $\pi : \hat X \to X$ be a resolution of an isolated singularity on the (complex) projective variety $X$, with exceptional set $E$, which may have components of various dimensions. Assume that we can find a set of generators for the divisor class group Cl$(\hat X)$ consisting of two types of divisors:

1. linear combinations of components of $E$, or
2. divisors which do not intersect any component of $E$.

It seems to me that an ample divisor might be built as follows. Find a divisor $D'$ of type (1) which has positive intersection numbers with all sub-varieties of $E$ (this may not exist). Choose an ample divisor $D$ on the singular space $X$. This pulls back to a divisor $\pi^*D$ which of type (2). Then $D' + n\pi^*D$ should be ample for large enough $n$.

The space of possible $D'$ is what I want to call the "local ample cone", as it consists of divisors in the exceptional set which are 'locally ample'. In the language of Kähler geometry, we are staying close to the wall of the Kähler cone of $\hat X$ at which $E$ shrinks to zero volume, so from this point of view we are considering the local structure of the Kähler cone near this wall. I suppose I have two questions for the experts out there:

Is the above construction sensible?

Does this notion already exist in the literature in some form?

In case it matters, I am particulary interested in the case of Calabi-Yau threefolds (including the condition $h^{0,1} = h^{0,2} = 0$).

Google searches for "local ample cone" and "local Kähler cone" yield no results, but maybe there is a different term.

Let $\pi : \hat X \to X$ be a resolution of an isolated singularity on the (complex) projective variety $X$, with exceptional set $E$, which may have components of various dimensions. Assume that we can find a set of generators for the divisor class group Cl$(\hat X)$ consisting of two types of divisors:

1. linear combinations of components of $E$, or
2. divisors which do not intersect any component of $E$.

It seems to me that an ample divisor might be built as follows. Find a divisor $D'$ of type (1) which has positive intersection numbers with all sub-varieties of $E$ (this may not exist). Choose an ample divisor $D$ on the singular space $X$. This pulls back to a divisor $\pi^*D$ which of type (2). Then $D' + n\pi^*D$ should be ample for large enough $n$.

The space of possible $D'$ is what I want to call the "local ample cone", as it consists of divisors in the exceptional set which are 'locally ample'. In the language of Kähler geometry, we are staying close to the wall of the Kähler cone of $\hat X$ at which $E$ shrinks to zero volume, so from this point of view we are considering the local structure of the Kähler cone near this wall. I suppose I have two questions for the experts out there:

Is the above construction sensible?

Does this notion already exist in the literature in some form?

In case it matters, I am particulary interested in the case of Calabi-Yau threefolds (including the condition $h^{0,1} = h^{0,2} = 0$).