7
$\begingroup$

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?

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ 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. $\endgroup$
    – user21574
    Jul 22, 2017 at 2:43
  • 2
    $\begingroup$ Dan Popovici, Regularization of currents with mass control and singular Morse inequalities, J. Differential Geom. Volume 80, Number 2 (2008), 281-326. $\endgroup$
    – user21574
    Jul 22, 2017 at 2:44
  • 3
    $\begingroup$ 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 $\endgroup$
    – user21574
    Jul 22, 2017 at 2:51

1 Answer 1

Reset to default
4
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • 1
    $\begingroup$ Ah, thank you very much! These notions look very interesting. I'll keep them in mind. $\endgroup$
    – Eric Canton
    Jul 1, 2014 at 13:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.