Recent zbMATH articles in MSC 32L05https://zbmath.org/atom/cc/32L052021-06-15T18:09:00+00:00WerkzeugSpaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry.https://zbmath.org/1460.320442021-06-15T18:09:00+00:00"Boucksom, Sébastien"https://zbmath.org/authors/?q=ai:boucksom.sebastien"Eriksson, Dennis"https://zbmath.org/authors/?q=ai:eriksson.dennisSummary: Let \(L\) be an ample line bundle on a (geometrically reduced) projective variety \(X\) over any complete valued field. Our main result describes the leading asymptotics of the determinant of cohomology of large powers of \(L\), with respect to the supnorm of a continuous metric on the Berkovich analytification of \(L\). As a consequence, we establish in this setting the existence of transfinite diameters and equidistribution of Fekete points, following a strategy going back to Berman, Witt Nyström and the first author for complex manifolds. In the non-Archimedean case, our approach relies on a version of the Knudsen-Mumford expansion for the determinant of cohomology on models over the (possibly non-Noetherian) valuation ring, as a replacement for the asymptotic expansion of Bergman kernels in the complex case, and on the reduced fiber theorem, as a replacement for the Bernstein-Markov inequalities. Along the way, a systematic study of spaces of norms and the associated Fubini-Study type metrics is undertaken.Effective geometric Hermitian positivstellensatz.https://zbmath.org/1460.320332021-06-15T18:09:00+00:00"Tan, Colin"https://zbmath.org/authors/?q=ai:tan.colin"To, Wing-Keung"https://zbmath.org/authors/?q=ai:to.wing-keungSummary: We consider positive Hermitian algebraic functions on holomorphic line bundles over compact complex manifolds. In particular, we consider the tensor product of positive powers of a positive Hermitian algebraic function satisfying the strong global Cauchy-Schwarz condition on a holomorphic line bundle with another fixed positive Hermitian algebraic function on another holomorphic line bundle. Our main result is to give an effective estimate (in terms of certain geometric data) on the smallest power needed to be taken so that the resulting tensor product is a maximal sum of Hermitian squares, or equivalently, the induced Hermitian metric on the resulting line bundle is the pull-back (via some holomorphic map) of the standard Hermitian metric on the universal line bundle over some complex projective space. This result is an effective version of Catlin-D'Angelo's Hermitian Positivstellensatz.Metric properties of parabolic ample bundles.https://zbmath.org/1460.320302021-06-15T18:09:00+00:00"Biswas, Indranil"https://zbmath.org/authors/?q=ai:biswas.indranil"Pingali, Vamsi Pritham"https://zbmath.org/authors/?q=ai:pingali.vamsi-prithamSummary: We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern-Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Griffiths conjecture in the parabolic setting and prove some results that provide evidence in its favor for certain kinds of parabolic bundles. For these kinds of parabolic structures, we prove that the conjecture holds on Riemann surfaces. We also prove that a Berndtsson-type result holds and that there are metrics on stable bundles over surfaces whose Schur forms are positive.Analytic characterization of nef and good line bundles.https://zbmath.org/1460.320322021-06-15T18:09:00+00:00"Kim, Dano"https://zbmath.org/authors/?q=ai:kim.danoSummary: We give an analytic characterization of nef and good line bundles on smooth projective varieties. This supplements \textit{F. Russo}'s paper [Bull. Belg. Math. Soc. - Simon Stevin 16, No. 5, 943--951 (2009; Zbl 1183.14011)] and extends it in the analytic direction of singular hermitian metrics.Flat bundles over some compact complex manifolds.https://zbmath.org/1460.320312021-06-15T18:09:00+00:00"Deng, Fusheng"https://zbmath.org/authors/?q=ai:deng.fusheng"Fornæss, John Erik"https://zbmath.org/authors/?q=ai:fornass.john-erikSummary: We construct examples of flat fiber bundles over the Hopf surface such that the total spaces have no pseudoconvex neighborhood basis, admit a complete Kähler metric, or are hyperconvex but have no nonconstant holomorphic functions. For any compact Riemannian surface of positive genus, we construct a flat \(\mathbb{P}^1\) bundle over it and a Stein domain with real analytic boundary in it whose closure does not have pseudoconvex neighborhood basis. For a compact complex manifold with positive first Betti number, we construct a flat bundle over it such that the total space is hyperconvex but admits no nonconstant holomorphic functions.Les formes automorphes feuilletées.https://zbmath.org/1460.320432021-06-15T18:09:00+00:00"Otal, Jean-Pierre"https://zbmath.org/authors/?q=ai:otal.jean-pierreLet \(X_\Gamma\) be a compact Riemann surface of hyperbolic type; it is realized as the factor-space \(\mathbb{H}/\Gamma\) of the upper half-plane \(\mathbb{H}\) with respect to a Fuchsian group \(\Gamma\). The author considers a family \(\mathcal{B}_s(\Gamma)\) of line bundles over the unit tangent bundle \(T^1X_\Gamma\). It is assumed that the bundles, when restricted to the leaves of an either stable or unstable foliation, depend holomorphically on \(s\). A continuous section of the foliation \(\mathcal{B}_s(\Gamma)\to T^1X_\Gamma\), holomorphic along the leaves of a stable foliation, is called a stable foliated automorphic form of weight \(s\). In a similar way, unstable foliated automorphic forms can be defined.
Let \(0<\Re s<1\). In Section 4, with the help of the eigenfunctions of the Laplace operator on \(X_\Gamma\) and Helgason distributions on \(\partial \mathbb{H}\), two stable foliated automorphic forms \(\mathcal{F}_s\) and \(\mathcal{F}_{1-s}\) of weights \(s\) and \(1-s\) are constructed. Further their properties are studied. The author constructs a natural isomorphism \(\Phi_s: A_s(\Gamma)\to A_{1-s}(\Gamma)\). Here \(A_s(\Gamma)\) is the space of stable foliated automorphic forms of weight \(s\). In Theorem~30, it is shown that the isomorphisms \(\Phi_s\) and \(\Phi_{1-s}\) are inverse to each other.
Reviewer: Samyon R. Nasyrov (Kazan)From Hörmander's \(L^2\)-estimates to partial positivity.https://zbmath.org/1460.320702021-06-15T18:09:00+00:00"Inayama, Takahiro"https://zbmath.org/authors/?q=ai:inayama.takahiroSummary: In this article, using a twisted version of Hörmander's \(L^2\)-estimate, we give new characterizations of notions of partial positivity, which are uniform \(q\)-positivity and RC-positivity. We also discuss the definition of uniform \(q\)-positivity for singular Hermitian metrics.