arXiv subjects

Hossein Raufi

Publications and source records attributed to Hossein Raufi.

6 recordsLinked to original sources

arXiv2022

Chern forms of hermitian metrics with analytic singularities on vector bundles

We define Chern and Segre forms, or rather currents, associated with a Griffiths positive singular hermitian metric $h$ with analytic singularities on a holomorphic vector bundle $E$. The currents are constructed as pushforwards of generalized Monge-Ampère products on the projectivization of $E$. The Chern and Segre currents represent the Chern and Segre classes of $E$, respectively, and coincide with the Chern and Segre forms of $E$ and $h$, where $h$ is smooth. Moreover, our currents coincide with the Chern and Segre forms constructed by the first three authors and Ruppenthal in the cases when these are defined.

math.CV↗

arXiv2017

Chern forms of singular metrics on vector bundles

We study singular hermitian metrics on holomorphic vector bundles, following Berndtsson-P{ă}un. Previous work by Raufi has shown that for such metrics, it is in general not possible to define the curvature as a current with measure coefficients. In this paper we show that despite this, under appropriate codimension restrictions on the singular set of the metric, it is still possible to define Chern forms as closed currents of order 0 with locally finite mass, which represent the Chern classes of the vector bundle.

math.CV↗

arXiv2014

An extension theorem of Ohsawa-Takegoshi type for sections of a vector bundle

Using $L^2$-methods for the $\bar\partial$-equation we prove that the Ohsawa-Takegoshi extension theorem also holds for holomorphic sections of a vector bundle, over compact Kähler manifolds. We then proceed to show that the conditions that are needed are more liberal than the ones one would need if one instead reduced the extension problem to line bundles through the usual algebraic geometric procedure of studying the projective bundle associated with the vector bundle.

math.CV↗

arXiv2014

Singular hermitian metrics on holomorphic vector bundles

We introduce and study a notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and P{ă}un. We define what it means for such a metric to be curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the cuvature $Θ^h$ as a matrix of currents. We then proceed to show that such metrics can be regularised in such a way that the corresponding curvature tensors converge weakly to $Θ^h$. Finally we define what it means for $h$ to be strictly negatively curved in the sense of Nakano and show that it is possible to regularise such metrics with a sequence of smooth, strictly Nakano negative metrics.

math.CV↗

arXiv2013

Log concavity for matrix-valued functions and a matrix-valued Prékopa theorem

We give two different definitions of what it means for a matrix-valued function to be log concave, guided by similar notions in complex differential geometry. After discussing a few simple examples, we proceed to develop some of the basic properties associated with these new concepts. Finally, we prove a matrix-valued Prékopa theorem using a weighted, vector-valued Paley-Wiener theorem, and positivity properties of direct image bundles.

math.CV↗

arXiv2012

The Nakano vanishing theorem and a vanishing theorem of Demailly-Nadel type for holomorphic vector bundles

We prove the classical Nakano vanishing theorem with Hörmander $L^2$-estimates on a compact Kähler manifold using Siu's so called $\partial\dbar$-Bochner-Kodaira method, thereby avoiding the Kähler identities completely. We then introduce singular hermitian metrics on holomorphic vector bundles, and proceed to prove a vanishing theorem of Demailly-Nadel type for these in the special case where the base manifold is a Riemann surface.

math.CV↗