# Augmented Stable Base Locus and Positive Definite Singular Hermitian Metrics

I'm a bit confused about a lemma that I came across in Demailly's lecture notes on hyperbolicity, and how it is related to the notion of the augmented stable base locus.

Namely, consider a big divisor $L$ on a smooth complex projective variety $X$. One defines the stable base locus of $L$ as $B(L) = \bigcap_{m}(Bs(mL))$ and the augmented stable base locus $B_+(L)$ as $\bigcap_A B(L - A)$, where the intersection is over all ample divisors.

Now let $V$ be a subvariety of $X$ of dimension $d$ and define the restricted volume $volL_{X|V} = \limsup_{m \rightarrow \infty} \frac{dim (Im(H^0(L) \rightarrow H^0(L|V))}{m^d/d!}$.

In "Restricted Volumes and Base Loci of Linear Series" (www.math.uic.edu/~mpopa/papers/RVLS.pdf), the authors prove that the irreducible components of $B_+(L)$ are exactly those subvarieties of $X$ with zero restricted volume. They remark that if $L$ is not nef, then it might well happen that $volL_{X|V} < vol(L|V)$. Thus, I'm assuming (although I have never seen an example) that $L|V$ might be big but yet $volL_{X|V} = 0$?

However, I came across the following theorem in Demailly's lecture notes on hyperbolicity "Algebraic criteria for Kobayashi hyperbolicity...", also available online, published well before recent literature on the augmented stable base locus. I'm interested in his Thm 7.2. I will paraphrase the relevant parts of this theorem (as I read it):

i) L admits a singular metric with positive definite curvature current iff L is big

ii) $B_+(L)$ is contained in the degeneration set of the metric on L. Also, suppose $mL = A + E$, for $A$ ample and $E$ effective. L admits a positive definite singular metric with degeneration set supp($E$).

iv) Suppose for each irreducible component of the degeneration set that $L$ restricted to this component is big, then $L$ admits a different metric with degeneration set that is the union of all the degeneration sets of each restriction of $L$ (and thus no longer containing the original component).

If you put this all together, this implies that for an arbitrary big (not necessarily nef) $L$ that $L$ restricted to a component of $B_+(L)$ is not big. For if it were, then part iv) and part ii) would imply that we could modify the metric on L to ensure that this component is not contained in the degeneration set of L and thus not contained in $B_+(L)$.

And indeed, this is how he uses 7.2iv) in his proof of the schneider-tancredi theorem in 13.6 of the same paper. He says that $B_+(L)$ can not contain a component of a certain form because L restricted to this component would be big, and then he cites 7.2iv) to derive a contradiction. Again, there is no nefness hypothesis on $L$ in this theorem.

So my questions for the experts are:

1) am I reading this theorem, and Demailly's proof of Schneider-Tancredi incorrectly

2) if not, is Demailly wrong about 7.2 iv)

3) if not, is this indeed a strengthening of the restricted volume result?

Jordan

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