Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHiggs bundles and geometric structures on manifolds
225
0
0.0
(
0
)
Authors
Daniele Alessandrini
Ask ChatGPT about the research
No Arabic abstract
Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichmuller theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchins equations. Baraglia has shown in his Ph.D. Thesis that Higgs bundles can also be used to construct geometric structures in some interesting cases. In this paper, we will explain the main ideas behind this theory and we will survey some recent results in this direction, which are joint work with Qiongling Li.
rate research
Read More
We propose and prove the Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. This generalizes the equivariant Verlinde formula for the case of $SU(n)$ proposed previously by the second and third author. We further establish a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. Finally, we prove that these dimensions form a one-parameter family of $1+1$-dimensional TQFT, uniquely classified by the complex Verlinde algebra, which is a one-parameter family of Frobenius algebras. We construct this one-parameter family of Frobenius algebras as a deformation of the classical Verlinde algebra for $G$.
For a simple, simply connected, complex group G, we prove the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over families of smooth projective curves with marked points.
This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic bundles and generalised parabolic $L$-twisted Hitchin pairs on its normalisation as well as on an analogue of the Hitchin map for generalised parabolic $L$-twisted Hitchin pairs.
As a natural extension of the theory of uniform vector bundles on Fano manifolds, we consider uniform principal bundles, and study them by means of the associated flag bundles, as their natural projective geometric realizations. In this paper we develop the necessary background, and prove some theorems that are flag bundle counterparts of some of the central results in the theory of uniform vector bundles.
In this paper we give a splitting criterion for uniform vector bundles on Fano manifolds covered by lines. As a consequence, we classify low rank uniform vector bundles on Hermitian symmetric spaces and Fano bundles of rank two on Grassmannians.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
