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 Brick Manifolds and Toric Varieties of Brick Polytopes

10.37236/5038 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Laura Escobar
Keyword(s):  
Toric Variety ◽  
Toric Varieties ◽  
Moment Map ◽  
Twisted Product ◽  
General Fiber ◽  
Moment Polytope ◽  
The Moment ◽  
Subword Complex ◽  
Do So

Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We prove that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg, Lange and Thomas. These stories connect in a nice way: we show that the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion's resolutions of Richardon varieties.

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 The vanishing of the chow cohomology ring for affine toric varieties and additional results

2019 ◽  
Author(s):  
◽  
Ryan Matthew Richey
Keyword(s):  
Recent Work ◽  
Moduli Space ◽  
Toric Variety ◽  
Cohomology Ring ◽  
Toric Varieties ◽  
Chow Ring

From the recent work of Edidin and Satriano, given a good moduli space morphism between a smooth Artin stack and its good moduli space X, they prove that the Chow cohomology ring of X embeds into the Chow ring of the stack. In the context of toric varieties, this implies that the Chow cohomology ring of any toric variety embeds into the Chow ring of its canonical toric stack. Furthermore, the authors give a conjectural description of the image of this embedding in terms of strong cycles. One consequence of their conjectural description, and an additional conjecture, is that the Chow cohomology ring of any affine toric variety ought to vanish. We prove this result without any assumption on smoothness. Afterwards, we present a series of results related to their conjectural description, and finally, we provide a conjectural toric description of the image of this embedding for complete toric varieties by utilizing Minkowski weights.

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 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 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.

TORELLI THEOREM FOR MODULI SPACES OF SL(r,ℂ)-CONNECTIONS ON A COMPACT RIEMANN SURFACE

2009 ◽  
Vol 11 (01) ◽  
pp. 1-26
Author(s):  
INDRANIL BISWAS ◽  
VICENTE MUÑOZ
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Complex Structure ◽  
Real Numbers ◽  
Stable Vector Bundles ◽  
Complex Variety ◽  
Projective Curves ◽  
Torelli Theorem

Let X be any compact connected Riemann surface of genus g, with g ≥ 3. For any r ≥ 2, let [Formula: see text] denote the moduli space of holomorphic SL (r,ℂ)-connections over X. It is known that the biholomorphism class of the complex variety [Formula: see text] is independent of the complex structure of X. If g = 3, then we assume that r ≥ 3. We prove that the isomorphism class of the variety [Formula: see text] determines the Riemann surface X uniquely up to an isomorphism. A similar result is proved for the moduli space of holomorphic GL (r,ℂ)-connections on X. We also show that the Torelli theorem remains valid for the moduli spaces of connections, as well as those of stable vector bundles, on geometrically irreducible smooth projective curves defined over the field of real numbers.

HOMOTOPY DECOMPOSITIONS OF GAUGE GROUPS OVER RIEMANN SURFACES AND APPLICATIONS TO MODULI SPACES

2011 ◽  
Vol 22 (12) ◽  
pp. 1711-1719 ◽  
Author(s):  
STEPHEN D. THERIAULT
Keyword(s):  
Riemann Surface ◽  
Gauge Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Vector Bundles ◽  
Homotopy Groups ◽  
Gauge Groups ◽  
Stable Vector Bundles

For a prime p, the gauge group of a principal U(p)-bundle over a compact, orientable Riemann surface is decomposed up to homotopy as a product of spaces, each of which is commonly known. This is used to deduce explicit computations of the homotopy groups of the moduli space of stable vector bundles through a range, answering a question of Daskalopoulos and Uhlenbeck.

scholarly journals An approach of the minimal model program for horospherical varieties via moment polytopes

2015 ◽  
Vol 2015 (708) ◽  
Author(s):  
Boris Pasquier
Keyword(s):  
Minimal Model ◽  
Toric Varieties ◽  
Cartier Divisor ◽  
Model Program ◽  
Minimal Model Program ◽  
Spherical Varieties ◽  
The Family ◽  
Moment Polytope ◽  
The Moment ◽  
Moment Polytopes

AbstractWe describe the minimal model program in the family of ℚ-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of a Cartier divisor of the variety we begin with. In particular, we generalize the results on MMP for toric varieties due to M. Reid, and we complete the results on MMP for spherical varieties due to M. Brion in the case of horospherical varieties.

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