Calabi-Yau metrics on canonical bundles of complex flag manifolds

Abstract

In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat Kähler metrics obtained through the Calabi ansatz technique. We use this approach to provide several explicit examples of noncompact complete Calabi-Yau manifolds, these examples include canonical bundles of non-toric flag manifolds (e.g. Grassmann manifolds and full flag mani-arXiv:1709.07956v2 [math.DG] 16 Dec 2017The idea of the proof of the above theorem is to combine the Calabi ansatz technique [4] with the description of quasi-potentials for invariant Kähler metrics on flag manifolds provided by Azad and Biswas in [3]. By means of this idea we can describe explicit examples of complete Calabi-Yau metrics. In this way, besides recovering many well known results related to X P = CP n (which are obtained from different methods), we also establish a new class of examples in the context of non-toric flag manifolds.This paper is organized as follows: In Section 2, we introduce the basic notation and results concerned with Chern classes of holomorphic line bundles over flag manifolds, then after that we prove Theorem 1. In Section 3, as an application of Theorem 1 we provide explicit examples of complete Calabi-Yau metrics on the total space of the canonical bundle for a huge class of complex flag manifolds, including Grassmann manifolds,

Homogeneous Contact Manifolds and Resolutions of Calabi–Yau Cones

“…In Subsection 3.3, we provide concrete examples which illustrate the content developed in the previous subsections. The main references for this section are [18], [3], [20], [38], [7, Chapter 2], [21, Appendix D.1].…”

“…The examples described in this section provide a huge class of concrete nontrivial examples for the existence part of Conjecture 5.2. Many of the computations which we have done for homogeneous contact manifolds associated to SL(n + 1, C) also can be done for other classical groups, namely, SO(n, C) and Sp(2n, C), see for instance [20] to some computations of the Calabi Ansatz metric in low dimensional cases.…”

General composite non-Abelian strings and flag manifold sigma models

“…Lastly, because it is built up of individual Grassmannian-like terms, it is completely straightforward to write a Kähler potential that generates this Non-Linear Sigma Model, which instantly provides us with the full N = (2, 2) NLSM action. Flag Manifolds are known to be Kähler manifolds (in fact they are Calabi-Yau spaces, see [15]), but the Calabi construction for them yields one metric with no tunable parameters like we have here, thanks to our Ansatz which has this block merger property: it is rigid, where we have a deformable metric.…”

“…These presentations of the Flag manifold Sigma Model have very recently gone under some investigation ( [14], [15] respectively), but do not make any contact with the vortex strings which bear them, and due to this, do not bear the coupling structure derived in this work, a direct consequence of the structure of magnetic flux distribution in four dimensions, and an important tool to observe the "block merging" phenomenon on the worldsheet.…”

Ricci-Flat Metrics on Vector Bundles Over Flag Manifolds

“…Flag manifolds admit Kähler-Einstein metrics, as studied in [14,15]. The authors of [16] use Calabi's ansatz to construct a Ricci-flat metric on the canonical bundle of the flag manifold, equipped with such a Kähler-Einstein metric. An important characteristic of Calabi's ansatz, that we review in Sect.…”