Deligne pairings and the Knudsen-Mumford expansion
| dc.creator | Phong, D. H. | |
| dc.creator | Ross, Julius | |
| dc.creator | Sturm, Jacob | |
| dc.date | 2006-12-19 | |
| dc.date | 2007-07-19 | |
| dc.date.accessioned | 2026-07-07T08:19:09Z | |
| dc.date.available | 2026-07-07T08:19:09Z | |
| dc.description | Let $X\to B$ be a proper flat morphism between smooth quasi-projective varieties of relative dimension $n$, and $L\to X$ a line bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for $\det (π_* L^k)$ in terms of Deligne pairings of $L$ and the relative canonical bundle $K$. This generalizes a theorem of Deligne which holds for families of relative dimension one. As a corollary, we show that when $X$ is smooth (or, more generally, if $X$ fits in a smooth family), the line bundle associated to $X\to B$, which was introduced by the first and third authors, coincides with the CM bundle defined by Paul-Tian. In a second corollary, we establish asymptotics for the K-energy along Bergman rays, generalizing a formula obtained by Paul-Tian. | |
| dc.description | Additional details provided; corollary added | |
| dc.identifier | https://arxiv.org/abs/math/0612555 | |
| dc.identifier | http://arxiv.org/abs/math/0612555 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/134669 | |
| dc.subject | Differential Geometry | |
| dc.subject | Algebraic Geometry | |
| dc.title | Deligne pairings and the Knudsen-Mumford expansion | |
| dc.type | text |