Topological invariants of parabolic $G$-Higgs bundles

The Beauville-Narasimhan-Ramanan correspondence for twisted Higgs $V$-bundles and components of parabolic $\text{Sp}(2n,\mathbb{R})$-Higgs moduli Spaces

“…For points (V, β, γ) in the moduli space M max par (Sp (2n, R)) for any n ≥ 2, we further fix the parabolic structure on the parabolic bundle V and study the connected component count problem for different values of rational weights. Among other results, we verify the prediction made in [16] on the number of components when the flag α on V is trivial with weight 1 2 : Corollary. 8.1 The number of connected components of the moduli space M max par (Sp (4, R) , α) with fixed trivial filtration and weight 1 2 is…”

“…The deformation complex. In [16], §3, we studied the deformation theory for parabolic G-Higgs bundles, when G is a semisimple reductive Lie group. For H ⊂ G a maximal compact subgroup and g = h ⊕ m a Cartan decomposition of the Lie algebra, the deformation complex of a parabolic G-Higgs bundle (E, Φ) over (X, D) is defined as the complex of sheaves…”

“…In this section we review the correspondence between parabolic Higgs bundles and Higgs V -bundles. Further details may be found in [16] and the references therein. 4.1.…”

“…Theorem 4.2 (Theorem 6.8 in [16]). There is a bijective correspondence between isomorphism classes of holomorphic Higgs V -bundles with good trivialization (E, Θ, Φ) and isomorphism classes of parabolic Higgs bundles F, Θ, Φ .…”

“…In the case when the weights in the parabolic structure are rational numbers, there is a bijection between moduli of semistable parabolic Higgs bundles over X and semistable Higgs bundles over an orbifold M (called a V -surface) with underlying manifold the Riemann surface X. In the construction of the orbifold surface the denominator of the weights is describing the cyclic group action around the marked points of D (see [7], [16], [18] for the detailed construction). For our purposes, we provide here an L-twisted version of this correspondence (Proposition 5.6), for L a line V -bundle over M .…”