Is there a notion of 'local ample/Kähler cone' for resolved singularities?

Question

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 $h^{0,1} = h^{0,2} = 0$).

Answer 1

I think this should be called the relatively ample cone and its closure the relatively nef cone. Yes, this already exists and is sensible.

This dissertation might be somewhat useful (although things are done for smooth $X$ instead of singular $X$).

UMich dissertation

EDIT: With regards to your two conditions (and the comments above), I think this condition is basically factoriality for the singularity. Suppose $x \in X$ is an isolated singularity as you describe and suppose that we are working with only one singular point.

Claim: Suppose that $X$ is factorial, this means that every divisor through $x$ on $X$ is going to be linearly equivalent to a divisor not passing through $x$.

Proof of claim: Suppose that $D > 0$ is an effective divisor passing through $x \in X$ (the effectivity assumption is harmless since we can write $D = D'-D''$ for effective $D'$ and $D''$). Choose a very ample and sufficiently ample Cartier divisor $A$ such that $\Gamma(X, O_X(A - D))$ is globally generated. Choose a global section $s$ generating the stalk $O_{X,x}(A-D)$ (this is possible since $D$ is Cartier at $x$, since $O_{X,x}$ is a UFD). Consider the inclusion $$\Gamma(X, O_X(A-D)) \subseteq \Gamma(X, O_X(A))$$ and so view $s$ as a global section of $O_X(A)$. It follows that $s$ determines a divisor $S \sim A$. Furthermore, $S$ and $D$ coincide in a neighborhood of $x \in X$ by construction. Further choose $T \sim A$ to be any divisor not passing through $x$ (this is possible since $A$ is very ample). Now then $D - S + T \sim D$, but $D - S + T$ is zero near $x \in X$. This proves the claim.

It follows from the claim that there exists a set of generating divisors of the type you want.

On the other hand, if you have divisors on $\hat{X}$ that satisfies those two conditions, then the images (pushforwards) of those divisors also generate the divisor class group of $X$. In particular, the divisor class group of $X$ is generated by divisors not passing through $x$. In particular, it is generated by Cartier divisors (since $x \in X$ was the only singular point). This implies that $X$ is factorial.