Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpecial Bohr - Sommerfeld geometry on Riemann surfaces: toy problems
83
0
0.0
(
0
)
Authors
Nikolai A. Tyurin
Ask ChatGPT about the research
No Arabic abstract
Special Bohr - Sommerfeld geometry, first formulated for simply connected symplectic manifolds (or for simple connected algebraic varieties), gives rise to some natural problems for the simplest example in non simply connected case. Namely for any algebraic curve one can define a correspondence between holomorphic differentials and certain finite graphs. Here we ask some natural questions appear with this correspondence. It is a partial answer to the question of A. Varchenko about possibility of applications of Special Bohr -Sommerfeld geometry in non simply connected case. The russian version has been translated.
rate research
Read More
In the paper we continue to study Special Bohr-Sommerfeld geometry of compact symplectic manifolds. Using natural deformation parameters we avoid the difficulties appeared in the definition of the moduli space of Special Bohr-Sommerfeld cycles for compact simply connected algebraic varieties. As a byproduct we present certain remarks on the Weinstein structures and Eliashberg conjectures.
The Quillen connection on ${mathcal L} rightarrow {mathcal M}_g$, where ${mathcal L}^*$ is the Hodge line bundle over the moduli stack of smooth complex projective curves curves ${mathcal M}_g$, $g geq 5$, is uniquely determined by the condition that its curvature is the Weil--Petersson form on ${mathcal M}_g$. The bundle of holomorphic connections on ${mathcal L}$ has a unique holomorphic isomorphism with the bundle on ${mathcal M}_g$ given by the moduli stack of projective structures. This isomorphism takes the $C^infty$ section of the first bundle given by the Quillen connection on ${mathcal L}$ to the $C^infty$ section of the second bundle given by the uniformization theorem. Therefore, any one of these two sections determines the other uniquely.
Let $X$ be a compact connected Riemann surface of genus at least two. Let $M_H(r,d)$ denote the moduli space of semistable Higgs bundles on $X$ of rank $r$ and degree $d$. We prove that the compact complex Bohr-Sommerfeld Lagrangians of $M_H(r,d)$ are precisely the irreducible components of the nilpotent cone in $M_H(r,d)$. This generalizes to Higgs $G$-bundles and also to the parabolic Higgs bundles.
An electrical potential U on a bordered Riemann surface X with conductivity function sigma>0 satisfies equation d(sigma d^cU)=0. The problem of effective reconstruction of sigma is studied. We extend to the case of Riemann surfaces the reconstruction scheme given, firstly, by R.Novikov (1988) for simply connected X. We apply for this new kernels for dbar on affine algebraic Riemann surfaces constructed in Henkin, arXiv:0804.3761
We study holomorphic $(n+1)$-chains $E_nto E_{n-1} to >... to E_0$ consisting of holomorphic vector bundles over a compact Riemann surface and homomorphisms between them. A notion of stability depending on $n$ real parameters was introduced in the work of the first two authors and moduli spaces were constructed by the third one. In this paper we study the variation of the moduli spaces with respect to the stability parameters. In particular we characterize a parameter region where the moduli spaces are birationally equivalent. A detailed study is given for the case of 3-chains, generalizing that of 2-chains (triples) in the work of Bradlow, Garcia-Prada and Gothen. Our work is motivated by the study of the topology of moduli spaces of Higgs bundles and their relation to representations of the fundamental group of the surface.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
