scholarly journals Moduli space of branched superminimal immersions of a compact Riemann surface into S4

1999 ◽  
Vol 66 (1) ◽  
pp. 32-50 ◽  
Author(s):  
Bonaventure Loo
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Compact Riemann Surface ◽  
Projective Varieties ◽  
Fibre Products ◽  
Dimension 2 ◽  
Path Connected

AbstractIn this paper we describe the moduli spaces of degree d branched superminimal immersions of a compact Riemann surface of genus g into S4. We prove that when d ≥ max {2g, g + 2}, such spaces have the structure of projectivzed fibre products and are path-connected quasi projective varieties of dimension 2d − g + 4. This generalizes known results for spaces of harmonic 2-spheres in S4.

scholarly journals On moduli spaces of Hitchin pairs

2011 ◽  
Vol 151 (3) ◽  
pp. 441-457 ◽  
Author(s):  
INDRANIL BISWAS ◽  
PETER B. GOTHEN ◽  
MARINA LOGARES
Keyword(s):  
Riemann Surface ◽  
Line Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Isomorphism Class ◽  
Compact Riemann Surface ◽  
List Type ◽  
Holomorphic Line Bundle ◽  
Type Number ◽  
Additional Hypothesis

AbstractLetXbe a compact Riemann surfaceXof genus at–least two. Fix a holomorphic line bundleLoverX. Letbe the moduli space of Hitchin pairs (E, φ ∈H0(End0(E) ⊗L)) overXof rankrand fixed determinant of degreed. The following conditions are imposed:(i)deg(L) ≥ 2g−2,r≥ 2, andL⊗rKX⊗r;(ii)(r, d) = 1; and(iii)ifg= 2 thenr≥ 6, and ifg= 3 thenr≥ 4.We prove that that the isomorphism class of the varietyuniquely determines the isomorphism class of the Riemann surfaceX. Moreover, our analysis shows thatis irreducible (this result holds without the additional hypothesis on the rank for low genus).

scholarly journals INTERSECTION COHOMOLOGY OF RANK 2 CHARACTER VARIETIES OF SURFACE GROUPS

2021 ◽  
pp. 1-40
Author(s):  
Mirko Mauri
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Compact Riemann Surface ◽  
Character Variety ◽  
Degree Zero ◽  
Surface Groups ◽  
Higgs Bundles ◽  
Character Varieties ◽  
Rank 2

Abstract For $G = \mathrm {GL}_2, \mathrm {SL}_2, \mathrm {PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the G-character variety of a compact Riemann surface C and of the moduli space of G-Higgs bundles on C of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.

scholarly journals MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY

2016 ◽  
Vol 4 ◽  
Author(s):  
RENZO CAVALIERI ◽  
SIMON HAMPE ◽  
HANNAH MARKWIG ◽  
DHRUV RANGANATHAN
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Tropical Geometry ◽  
Stable Curves ◽  
Tropical Curves ◽  
Graphic Matroid ◽  
Fibre Products ◽  
Algebraic Counterpart

We study moduli spaces of rational weighted stable tropical curves, and their connections with Hassett spaces. Given a vector $w$ of weights, the moduli space of tropical $w$-stable curves can be given the structure of a balanced fan if and only if $w$ has only heavy and light entries. In this case, the tropical moduli space can be expressed as the Bergman fan of an explicit graphic matroid. The tropical moduli space can be realized as a geometric tropicalization, and as a Berkovich skeleton, its algebraic counterpart. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of unweighted spaces, and explore parallels with the algebraic world.

scholarly journals Stability of the parabolic Poincaré bundle

2018 ◽  
Vol 15 (05) ◽  
pp. 1850081
Author(s):  
Suratno Basu ◽  
Indranil Biswas ◽  
Krishanu Dan
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Compact Riemann Surface ◽  
Stable Bundles ◽  
Parabolic Bundle ◽  
Natural Polarization

Given a compact Riemann surface [Formula: see text] and a moduli space [Formula: see text] of parabolic stable bundles on it of fixed determinant of complete parabolic flags, we prove that the Poincaré parabolic bundle on [Formula: see text] is parabolic stable with respect to a natural polarization on [Formula: see text].

scholarly journals CHEN–RUAN COHOMOLOGY OF SOME MODULI SPACES, II

2010 ◽  
Vol 21 (04) ◽  
pp. 497-522 ◽  
Author(s):  
INDRANIL BISWAS ◽  
MAINAK PODDAR
Keyword(s):  
Riemann Surface ◽  
Line Bundle ◽  
Prime Number ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Line Bundles ◽  
Holomorphic Line Bundle ◽  
Stable Vector Bundles

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let [Formula: see text] denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L⊗r is trivial. This group Γ acts on [Formula: see text] by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold.

scholarly journals A Note on Toric Varieties Associated with Moduli Spaces

2011 ◽  
Vol 54 (3) ◽  
pp. 561-565
Author(s):  
James J. Uren
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Toric Variety ◽  
Toric Varieties ◽  
Pants Decomposition ◽  
Basic Facts ◽  
The Moment ◽  
Moment Polytopes

AbstractIn this note we give a brief review of the construction of a toric variety coming from a genus g ≥ 2 Riemann surface Σg equipped with a trinion, or pair of pants, decomposition. This was outlined by J. Hurtubise and L. C. Jeffrey. A. Tyurin used this construction on a certain collection of trinion decomposed surfaces to produce a variety DMg , the so-called Delzant model of moduli space, for each genus g. We conclude this note with some basic facts about the moment polytopes of the varieties . In particular, we show that the varieties DMg constructed by Tyurin, and claimed to be smooth, are in fact singular for g ≥ 3.

scholarly journals Brauer Group of Moduli of Higgs Bundles and Connections

2018 ◽  
pp. 387-398
Author(s):  
David Baraglia ◽  
Indranil Biswas ◽  
Laura P. Schaposnik
Keyword(s):  
Riemann Surface ◽  
Algebraic Group ◽  
Moduli Spaces ◽  
Compact Riemann Surface ◽  
Brauer Group ◽  
Higgs Bundles ◽  
Smooth Locus

Given a compact Riemann surface X and a semi-simple affine algebraic group G defined over C, there are moduli spaces of Higgs bundles and of connections associated to (X, G). The chapter computes the Brauer group of the smooth locus of these varieties.

scholarly journals Moduli Spaces of Vector Bundles over a Real Curve: ℤ/2-Betti Numbers

2014 ◽  
Vol 66 (5) ◽  
pp. 961-992 ◽  
Author(s):  
Thomas Baird
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Lagrangian Submanifolds ◽  
Betti Numbers ◽  
Stable Bundles ◽  
Complex Curve ◽  
Yang Mills ◽  
Real Curve

AbstractModuli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi–stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah–Bott's “Yang–Mills over a Riemann Surface” to compute ℤ/2–Betti numbers of these spaces.

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