scholarly journals On the moduli spaces of semi-stable plane sheaves of dimension one and multiplicity five

2011 ◽  
Vol 55 (4) ◽  
pp. 1467-1532 ◽  
Author(s):  
Mario Maican
Keyword(s):  
Moduli Spaces ◽  
Stable Plane ◽  
Dimension One

scholarly journals Moduli space holography and the finiteness of flux vacua

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Thomas W. Grimm
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Boundary Data ◽  
Hodge Theory ◽  
Two Dimensional ◽  
Hodge Conjecture ◽  
String Compactifications ◽  
Finiteness Result ◽  
Dimensional Gravity ◽  
Dimension One

Abstract A holographic perspective to study and characterize field spaces that arise in string compactifications is suggested. A concrete correspondence is developed by studying two-dimensional moduli spaces in supersymmetric string compactifications. It is proposed that there exist theories on the boundaries of each moduli space, whose crucial data are given by a Hilbert space, an Sl(2, ℂ)-algebra, and two special operators. This boundary data is motivated by asymptotic Hodge theory and the fact that the physical metric on the moduli space of Calabi-Yau manifolds asymptotes near any infinite distance boundary to a Poincaré metric with Sl(2, ℝ) isometry. The crucial part of the bulk theory on the moduli space is a sigma model for group-valued matter fields. It is discussed how this might be coupled to a two-dimensional gravity theory. The classical bulk-boundary matching is then given by the proof of the famous Sl(2) orbit theorem of Hodge theory, which is reformulated in a more physical language. Applying this correspondence to the flux landscape in Calabi-Yau fourfold compactifications it is shown that there are no infinite tails of self-dual flux vacua near any co-dimension one boundary. This finiteness result is a consequence of the constraints on the near boundary expansion of the bulk solutions that match to the boundary data. It is also pointed out that there is a striking connection of the finiteness result for supersymmetric flux vacua and the Hodge conjecture.

scholarly journals Comparison of Poisson structures on moduli spaces

2021 ◽  
Author(s):  
Indranil Biswas ◽  
Francesco Bottacin ◽  
Tomás L. Gómez
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Poisson Structures ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraic Vector ◽  
Pure Dimension ◽  
Stable Sheaves ◽  
Fixed Rank ◽  
Dimension One

AbstractLet X be a complex irreducible smooth projective curve, and let $${{\mathbb {L}}}$$ L be an algebraic line bundle on X with a nonzero section $$\sigma _0$$ σ 0 . Let $${\mathcal {M}}$$ M denote the moduli space of stable Hitchin pairs $$(E,\, \theta )$$ ( E , θ ) , where E is an algebraic vector bundle on X of fixed rank r and degree $$\delta $$ δ , and $$\theta \, \in \, H^0(X,\, {\mathcal {E}nd}(E)\otimes K_X\otimes {{\mathbb {L}}})$$ θ ∈ H 0 ( X , E n d ( E ) ⊗ K X ⊗ L ) . Associating to every stable Hitchin pair its spectral data, an isomorphism of $${\mathcal {M}}$$ M with a moduli space $${\mathcal {P}}$$ P of stable sheaves of pure dimension one on the total space of $$K_X\otimes {{\mathbb {L}}}$$ K X ⊗ L is obtained. Both the moduli spaces $${\mathcal {P}}$$ P and $${\mathcal {M}}$$ M are equipped with algebraic Poisson structures, which are constructed using $$\sigma _0$$ σ 0 . Here we prove that the above isomorphism between $${\mathcal {P}}$$ P and $${\mathcal {M}}$$ M preserves the Poisson structures.

scholarly journals Boundedness of semi-stable sheaves of small ranks

1980 ◽  
Vol 78 ◽  
pp. 65-94 ◽  
Author(s):  
Masaki Maruyama
Keyword(s):  
Moduli Spaces ◽  
Stable Sheaves ◽  
Dimension One

As for the construction of moduli spaces of stable sheaves, the boundedness of semi-stable sheaves is one of the most important questions which are left unanswered. In the case of dimension one, the boundedness was proved by M. F. Atiyah [1]. When the dimension of the base variety is two and the rank is two, F. Takemoto and D. Mumford showed the boundedness independently ([13]). The author proved in [7] that the boundedness holds for every rank in the case of dimension two, and then D. Gieseker gave another proof of it in [3].

scholarly journals On the existence of connected components of dimension one in the branch locus of moduli spaces of riemann surfaces

2012 ◽  
Vol 111 (1) ◽  
pp. 53 ◽  
Author(s):  
Antonio F. Costa ◽  
Milagros Izquierdo
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Fuchsian Groups ◽  
Connected Components ◽  
Branch Locus ◽  
Isolated Points ◽  
Compact Riemann Surfaces ◽  
Dimension One ◽  
Substantial Interest

Let $g$ be an integer $\geq3$ and let $B_{g}=\{X\in\mathcal{M}_{g}: \mathrm{Aut}(X)\neq Id\}$ be the branch locus of $M_{g}$, where $M_{g}$ denotes the moduli space of compact Riemann surfaces of genus $g$. The structure of $B_{g}$ is of substantial interest because $B_{g}$ corresponds to the singularities of the action of the modular group on the Teichmüller space of surfaces of genus $g$ (see [14]). Kulkarni ([15], see also [13]) proved the existence of isolated points in the branch loci of the moduli spaces of Riemann surfaces. In this work we study the isolated connected components of dimension 1 in such loci. These isolated components of dimension one appear if the genus is $g=p-1$ with $p$ prime $\geq11$. We use uniformization by Fuchsian groups and the equisymmetric stratification of the branch loci.

Geometry and Physics: Volume II

2018 ◽  
Keyword(s):  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Geometric Analysis ◽  
Mathematical Physics ◽  
Poisson Geometry ◽  
Special Holonomy ◽  
Low Dimensional Topology ◽  
Generalized Complex Structures ◽  
Wide Range ◽  
Low Dimensional

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

scholarly journals EXTREMAL CASES OF RAPOPORT–ZINK SPACES

2021 ◽  
pp. 1-56
Author(s):  
Ulrich Görtz ◽  
Xuhua He ◽  
Michael Rapoport
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Complete List ◽  
Qualitative Properties ◽  
Model Case ◽  
Divisible Groups

Abstract We investigate qualitative properties of the underlying scheme of Rapoport–Zink formal moduli spaces of p-divisible groups (resp., shtukas). We single out those cases where the dimension of this underlying scheme is zero (resp., those where the dimension is the maximal possible). The model case for the first alternative is the Lubin–Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases.

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