# What properties of line bundles can be detected cohomologically?

Let $X$ be a proper, finite type scheme over a field $k$. What useful properties of line bundles (e.g. amplitude, nefness) can be detected cohomologially?

For example, in our setting we have the Cartan-Grothendieck-Serre theorem characterizing ample line bundles on $X$ as those line bundles $L$ such that for any coherent sheaf $\mathscr{F}$ on $X$, we have $H^i(X, \mathscr{F} \otimes L^{\otimes n}) = 0$ for all $i > 0$ and $n \gg 0$. Restricting further to projective varieties, we have that a line bundle $L$ is big iff $h^0(X, L^{\otimes n})$ grows like $n^{\dim X}$.

Do other such characterizations hold for nef line bundles? semiample? other useful classes of line bundles which I've forgotten?

• I give Dan Popovici's characterization of big line bundle: Any almost positive current $T$ admits a Lebesgue decomposition into the sum of an absolutely continuous part $T_{ac}$ and a singular part. By taking the supremum over all $T$ in the Chern class of $L$ of the mass of the n-th exterior power of $T_{ac}$. Bigness of $L$ is then equivalent to existence of a possibly singular metric $h$ for which the curvature current is non-negative and its absolutely continuous part has positive n-mass. – user21574 Jul 22 '17 at 2:43
• Dan Popovici, Regularization of currents with mass control and singular Morse inequalities, J. Differential Geom. Volume 80, Number 2 (2008), 281-326. – user21574 Jul 22 '17 at 2:44
• Assume that the curvature current $c(L,h)$ is smooth on the complement of some proper analytic subset$Z$ of $X$ and that $c(L,h)$ is strictly positive on some tubular neighborhood $B$ of $Z$. Then $∫_{X_{reg}(≤1,L)}c(L,h)^n$ exists. If in addition this integral is positive, then $L$ is big, and in particular,$X$ is a Moishezon space – user21574 Jul 22 '17 at 2:51

A generalization of ampleness, called $q$-ampleness, has been studied notably by Totaro, see the second paper in this volume: a line bundle $L$ on $X$ is $q$-ample if for any coherent sheaf $\mathscr{F}$ on $X$, we have $H^i(X, \mathscr{F} \otimes L^{\otimes n}) = 0$ for all $i > q$ and $n \gg 0$. See Totaro's paper and also this paper by J.C. Ottem for some interesting applications of this notion.