Restricted Bergman kernel asymptotics

Abstract

Abstract. In this paper, we investigate a restricted version of Bergman kernels for high powers of a big line bundle over a smooth projective variety. The geometric meaning of the leading term is specified. As a byproduct, we derive some integral representations for the restricted volume.

On the volume of graded linear series and Monge–Ampère mass

“…In the complete case W = R this is proved by [Ber09]. It also holds for the restricted linear series, by [His12]. See also [BD09].…”

“…This formula enables us to investigate the positivity of W locally in X and it might be helpful for lower bound estimate of vol(W ) in the future. Theorem 1.3 is a natural generalization of the results in [Bou02], [Ber09], [BB10], [BEGZ10], and [His12]. Note that the birationality assumption is necessary for general W = R (see Remark 4.3).…”

“…If X is not contained in the augmented base locus B + (L) of Y (for the definition, see [Laz04], Definition 10.3.2), W defines a birational graded linear series. For restricted linear series, the author investigated the asymptotic of related Bergman kernels in [His12] and obtained Theorem 1.3 as a corollary. The point is that in this special case we can prove Question 1.4 thanks to an L 2 -extension theorem for the subvariety X.…”

“…Moreover, thanks to the currently available technology, the proof in the present paper get much simpler than the previous ones, even in the case of complete graded linear series. On the other hand, our approach is rather algebraic relying on the result of [KK09], [LM09], [Jow10], and [DBP12] and does not give much information about Bergman kernel asymptotics as [Ber09] or [His12]. In particular, the following regularity problem is open.…”

Remarks on $L^2$-jet extension and extension of singular Hermitian metric with semipositive curvature

“…In some sense they measure the positivity of line bundles along subvarieties. An analytic description of volumes was first given in [His11]. For a singular Hermitian metric h = e −ϕ on X, let us denote by (dd c ϕ| S ) n−p the non-pluripolar Monge-Ampère product of ϕ| S , a positive measure which coincides with usual Monge-Ampère product (dd c ϕ| S ) n−p when ϕ is smooth.…”