Griffiths extremality, interpolation of norms, and Kähler quantization

Abstract

Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge-Ampère type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kähler geometry, related to the construction of flat maps for the Mabuchi metric.

A Wess--Zumino--Witten type equation in the space of Kähler potentials in terms of Hermitian--Yang--Mills metrics

“…In this final section, we compare results in this paper to the work with Darvas [DW19], where we consider two other families closely related to G v and…”

“…where a norm function U z is called Griffiths negative if log U z (f (z)) is psh for any holomorphic section f : W ⊂ D → H 0 (X, L k ) * . Denote the upper envelopes of F v and F k v by U and U k respectively; then one result in [DW19] is that F S k ((U k z ) * ) converges to U uniformly. The difference between F k v and G k v is obvious, we simply change plurisubharmonicity to subharmonicity.…”

“…The proof of Theorem 1.2 hinges on Theorem 2.1, a result regarding the positivity of direct image bundles. Although Berndtsson's theorem [Bern09] has played a crucial role in approximation theorems similar to Theorem 1.2 (for example [Bern13], [BK12], [DLR18], and [DW19]), when it comes to approximating by Hermitian-Yang-Mills metrics, a subharmonic analogue of Berndtsson's theorem is desired. It is Theorem 2.1, where we prove a version of positivity of direct image bundles for weights that are subharmonic on graphs.…”

“…Section 3 is devoted to Theorem 1.2, and section 4 to Theorem 1.3. In the final section 6, we draw parallels with the paper [DW19].…”

“…This paper grew out of joint work [DW19] with Tamás Darvas, and I am grateful to him for many useful discussions. I am indebted to László Lempert for his critical remarks and suggestions.…”