Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHitchin Pairs for non-compact real Lie groups
114
0
0.0
(
0
)
Authors
Peter B. Gothen
Ask ChatGPT about the research
No Arabic abstract
Hitchin pairs on Riemann surfaces are generalizations of Higgs bundles, allowing the Higgs field to be twisted by an arbitrary line bundle. We consider this generalization in the context of $G$-Higgs bundles for a real reductive Lie group $G$. We outline the basic theory and review some selected results, including recent results by Nozad and the author arXiv:1602.02712 [math.AG] on Hitchin pairs for the unitary group of indefinite signature $mathrm{U}(p,q)$.
rate research
Read More
We provide a construction of the moduli spaces of framed Hitchin pairs and their master spaces. These objects have come to interest as algebra
Let $X$ be a compact Riemann surface $X$ of genus at--least two. Fix a holomorphic line bundle $L$ over $X$. Let $mathcal M$ be the moduli space of Hitchin pairs $(E ,phiin H^0(End(E)otimes L))$ over $X$ of rank $r$ and fixed determinant of degree $d$. We prove that, for some numerical conditions, $mathcal M$ is irreducible, and that the isomorphism class of the variety $mathcal M$ uniquely determines the isomorphism class of the Riemann surface $X$.
A conjectural recursive relation for the Poincare polynomial of the Hitchin moduli space is derived from wallcrossing in the refined local Donaldson-Thomas theory of a curve. A doubly refined generalization of this theory is also conjectured and shown to similarly determine the Hodge polynomial of the same moduli space.
For $epsilon$-lc Fano type varieties $X$ of dimension $d$ and a given finite set $Gamma$, we show that there exists a positive integer $m_0$ which only depends on $epsilon,d$ and $Gamma$, such that both $|-mK_X-sum_ilceil mb_irceil B_i|$ and $|-mK_X-sum_ilfloor mb_irfloor B_i|$ define birational maps for any $mge m_0$ provided that $B_i$ are pseudo-effective Weil divisors, $b_iinGamma$, and $-(K_X+sum_ib_iB_i)$ is big. When $Gammasubset[0,1]$ satisfies the DCC but is not finite, we construct an example to show that the effective birationality may fail even if $X$ is fixed, $B_i$ are fixed prime divisors, and $(X,B)$ is $epsilon$-lc for some $epsilon>0$.
A 1-truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of $Map_*(BG,BH)$, $Map(BG,BH)$, and $Map(EG, B_GH)^G$ for compact Lie groups $G$ and $H$ with $H$ 1-truncated, showing that they are computed entirely in terms of spaces of homomorphisms from $G$ to $H$. These results generalize the well-known case when $H$ is finite, and the case of $H$ compact abelian due to Lashof, May, and Segal.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
