ASYMPTOTIC INVARIANTS OF LINE BUNDLESLAWRENCE EINThis is joint work with Lazarsfeld, Mustaţǎ, Nakamaye and Popa. Let X be asmooth projective variety of dimension n. Denote by N S(X)Q and N S(X)R the NersonSeveri groups of X with rational and real coefficients respectively. In N S(X)R , there
are two natural closed cones. These are the nef cone and the cone of pseudo effectivedivisors. The interior of the nef cone is the ample cone and the interior of the cone ofpseudo effective divisors is the cone of big divisors.
It is classical that ample divisors on X satisfy many beautiful geometric, cohomological and numerical properties. By contrast, examples due to Cutkosky and others have
led to the impression that the linear series associated with big divisors are in general
mired in pathology.
From the recent work of Nakayama, Tsuji, Boucksom-Demailly-Paun-Peternell andourselves (with Lazarsfeld, Mustaţǎ Nakamaye and Popa), it has become apparent that
arbitrary big divisors in fact display a surprising number of properties analogous to thoseof ample line bundles.
If D is a divisor on X with Q-coefficients, then the stable base locus B(D) is defined
to be the common base locus of the complete linear series |mD| for sufficiently divisible
m. Understanding B(D) is difficult in general, since this locus does not depend only on
the numerical class of D. It was discovered by Nakamaye that an upper approximation,
the augmented base locus of D, behaves much better than B(D) itself. This is defined as
B+ (D) := B(D − A), where A is any “small” ample Q-divisor, and is a proper subsetof X if and only if D is big. A similar definition can be given for any R-divisor. Thevariation of B+ is shown to give an interesting decomposition of the cone of big divisors
Big(X); for a detailed study cf. [ELMNP2]. In the case of nef divisors, it was shown in[Na1] that B+ (D) can be described numerically by the vanishing of intersection numbers:[V,(1)
B+ (D) =(Ddim V ·V )=0

Recall that if L is a line bundle on X, then its volume is an asymptotic invariant
naturally associated to the Riemann-Roch problem. Specifically, it measures the rate ofgrowth of the sections of multiples of L:
volX (L) := lim supm→∞h0 (X, Lm ).
mn /n!

More generally, suppose that V is a subvariety of dimension d ≥ 1 in X. The restrictedvolume of L along V measures the rate of growth of the sections of multiples of L on V
1

2L.EINthat can be lifted to X:
dimC Im(H 0 (X, Lm ) → H 0 (V, Lm |V )).
md /d!
m→∞
The definition extends in a natural way to Q-divisors and it is easy to see that if V 6⊆
B+ (D), then volX|V (D) > 0.
volX|V (L) := lim sup

If L is ample, then volX (L) = c1 (L)n . One gets the following geometric interpretation. If D1 , . . . , Dn are general divisors in |mL|, then#(D1 ∩ . . . ∩ Dn )volX (D) =.
mnThe following theorem is the analogous result for big divisors.
Theorem A. Let D be a Q-divisor and V a subvariety of dimension d ≥ 1 such thatV 6⊆ B+ (D). If we consider, for m divisible enough, general divisors D1 , . . . , Dd in |mD|,then we have#(V ∩ D1 ∩ . . . ∩ Dd r B(D))
volX|V (D) = lim.
m→∞md
The following is analogue of Nakamaye’s theorem for big divisors.
Theorem B. If D is a Q-divisor on X, then B+ (D) is the union of all positive dimensional subvarieties V such that volX|V (D) = 0.
References[Bo1] S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. ÉcoleNorm. Sup. (4) 37 (2004), 45–76.
[Bo2] S. Boucksom, On the volume of a line bundle, Internat. J. Math. 13 (2002), 1043–1063.
[BDPP] S. Boucksom, J.-P. Demailly, M. Pǎun and T. Peternell, The pseudo-effective cone of a compactKähler manifold and varieties of negative Kodaira dimension, math.AG/0405285.
[DEL] J.-P. Demailly, L. Ein and R. Lazarsfeld, A subadditivity property of multiplier ideals, Michigan
Math. J. 48 (2000), 137–156.
[Ein] L. Ein, Linear systems with removable base loci, Comm. Algebra 28 (2000), no. 12, 5931–5934.
[EKL] L. Ein, O.Küchle and R. Lazarsfeld, Local positivity of ample line bundles, J. Diff. Geom. 42(1995), 193–219.
[ELMNP1] L. Ein, R. Lazarsfeld, M. Mustaţǎ, M. Nakamaye and M. Popa, Asymptotic invariants of linebundles, preprint math.AG/0505054.
[ELMNP2] L. Ein, R. Lazarsfeld, M. Mustaţǎ, M. Nakamaye and M. Popa, Asymptotic invariants of baseloci, preprint math.AG/0308116.
[ELN] L. Ein, R. Lazarsfeld, M. Nakamaye, Zero estimates, intersection theory, and a theorem of Demailly, in Andreatta and Peternell eds., Higher dimensional complex varieties, de Gruyter (1996),183–208.
[Laz] R. Lazarsfeld, Positivity in algebraic geometry I & II, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48 & 49, Springer-Verlag, Berlin, 2004.
[Na1] M. Nakamaye, Stable base loci of linear series, Math. Ann. 318 (2000), 837–847.
[Na2] M. Nakamaye, Base loci of linear series are numerically determined, Trans. Amer. Math. Soc. 355(2002), 551–566.
[Ts1] H. Tsuji, On the structure of pluricanonical systems of projective varieties of general type, preprint,
1999.
[Ts2] H. Tsuji, Numerical trivial fibrations, preprint math.AG/0001023.

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3[Zar] O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic
surface, Ann. of Math. 76 (1962), 550–615.