Search

From the work of Phong and Sturm in 2007, for a polarised projective manifold and an ample test configuration, one can associate the geodesic ray of plurisubharmonic metrics on the polarising line bundle using the solution of the Monge-Amp\`ere equation on an equivariant resolution of singularities of the test configuration. We prove that the Mabuchi chordal distance between the geodesic rays associated with two ample test configurations coincides with the spectral distance between the associated filtrations on the section ring. This gives an algebraic description of the boundary at the infinity of the space of positive metrics, viewed - as it is usually done for spaces of negative curvature - through geodesic rays.

We show that submultiplicative norms on section rings of polarised projective manifolds are asymptotically equivalent to sup-norms associated with metrics on the polarisation. As an application, we establish that over canonically polarised manifolds, convex hull of Narasimhan-Simha pseudonorm over pluricanonical sections is asymptotically equivalent to the sup-norm associated with the supercanonical metric of Tsuji, refining a result of Berman-Demailly. As another application, we deduce that the jumping measures associated with bounded submultiplicative filtrations on section rings converge to the spectral measures of their Bergman geodesic rays, generalizing previous results of Witt Nystr\"om and Hisamoto. We show also that the latter measures can be described using pluripotential theory. More precisely, we establish that Bergman geodesic rays are maximal, i.e. they can be constructed through plurisubharmonic envelopes. As a final application, for non-continuous metrics on ample line bundles over projective manifolds, we prove that a weak version of semiclassical holomorphic extension theorem holds for generic submanifolds. This means that up to a negligible portion, the totality of holomorphic sections over generic submanifolds extends in an effective way to the ambient manifold. As an unexpected byproduct, we show that injective and projective tensor norms on symmetric algebras of finitely dimensional complex normed vector spaces are asymptotically equivalent., Comment: v2. 43 pages; substantial extension of v1. Shifted the paper's main focus towards submultiplicative norms. This is reflected in the change of the presentation and a new title. Added a classification theorem for submultiplicative norms with non-continuous Fubini-Study metric. Added an application of the latter theorem to submultiplicative filtrations, geodesic rays and extension theorem

We study the asymptotics of the $L^2$-optimal holomorphic extensions of holomorphic jets associated with high tensor powers of a positive line bundle along submanifolds. More precisely, for a fixed complex submanifold in a complex manifold, we consider the operator which for a given holomorphic jet along the submanifold of a positive line bundle associates the $L^2$-optimal holomorphic extension of it to the ambient manifold. When the tensor power of the line bundle tends to infinity, we give an explicit asymptotic formula for this extension operator. This is done by a careful study of the Schwartz kernels of the extension operator and related Bergman projectors. It extends our previous results, done for holomorphic sections instead of jets. As an application, we prove the asymptotic isometry between two natural norms on the space of holomorphic jets: one induced from the ambient manifold and another from the submanifold., 42 pages. This article extends our previous results, done for holomorphic sections instead of jets. The proof of the semiclassical extension theorem for holomorphic sections from arXiv:2109.06851 relied on spectral geometry and localisation techniques. The proof of the corresponding result in the current article, done more generally for holomorphic jets, builds upon Taylor-type expansion of holomorphic differential near submanifold from arXiv:2109.06851, on the study of Toeplitz operators with weak exponential decay from arXiv:2201.04102, and on multiplicative defect operator, introduced in the most simple form in arXiv:2201.04102

In a setting of a complex manifold with a fixed positive line bundle and a submanifold, we consider the optimal Ohsawa-Takegoshi extension operator, sending a holomorphic section of the line bundle on the submanifold to the holomorphic extension of it on the ambient manifold with the minimal $L^2$-norm. We show that for a tower of submanifolds in the semiclassical setting, i.e. when we consider a large tensor power of the line bundle, the extension operators satisfy transitivity property modulo some small defect, which can be expressed through Toeplitz type operators. We calculate the first significant term in the asymptotic expansion of this "transitivity defect". As a byproduct, we deduce the composition rules for Toeplitz type operators, the extension and restriction operators, and calculate the second term in the asymptotic expansion of the optimal constant in the semi-classical version of Ohsawa-Takegoshi extension theorem., 52 pages

We study the asymptotics of Ohsawa-Takegoshi extension operator and orthogonal Bergman projector associated with high tensor powers of a positive line bundle. More precisely, for a fixed complex submanifold in a complex manifold, we consider the operator which associates to a given holomorphic section of a positive line bundle over the submanifold the holomorphic extension of it to the ambient manifold with the minimal $L^2$-norm. When the tensor power of the line bundle tends to infinity, we obtain an explicit asymptotic expansion of this operator. This is done by proving an exponential estimate for the Schwartz kernel of the extension operator, and showing that this Schwartz kernel admits a full asymptotic expansion in powers of the line bundle for close parameters. Along the way we prove similar results for the projection onto the holomorphic sections orthogonal to those which vanish along the submanifold. Motivated by possible applications to the invariance of plurigenera problem, our results are proved for manifolds and embeddings of bounded geometry. As an application, we obtain asymptotically optimal $L^{\infty}$-bounds on the holomoprhic extensions from submanifolds, refining previous results of Zhang and Bost., Comment: A version of the extension theorem with respect to the supremum norm is added, see Theorem 1.10. References to earlier works have been added. A minor gap in the proof from Section 4.3 is corrected by a simpler argument

This paper is devoted to the study of the asymptotics of Monge-Amp\`ere volumes of direct images associated with high tensor powers of an ample line bundle. We study the leading term of this asymptotics and provide a classification of bundles saturating the topological bound of Demailly. In the special case of high symmetric powers of ample vector bundles, this provides a characterization of vector bundles admitting projectively flat Hermitian structures.