scholarly journals Cobordism invariants of the moduli space of stable pairs

2016 ◽  
Vol 94 (2) ◽  
pp. 427-446 ◽  
Author(s):  
Junliang Shen
Keyword(s):  
Moduli Space ◽  
Stable Pairs

scholarly journals The KSBA compactification of the moduli space of $$D_{1,6}$$-polarized Enriques surfaces

2021 ◽  
Author(s):  
Luca Schaffler
Keyword(s):  
Finite Group ◽  
Moduli Space ◽  
Blow Up ◽  
Unit Cube ◽  
Finite Group Action ◽  
Stable Pair ◽  
Enriques Surfaces ◽  
Secondary Polytope ◽  
Stable Pairs ◽  
A Finite Group

AbstractWe describe a compactification by stable pairs (also known as KSBA compactification) of the 4-dimensional family of Enriques surfaces which arise as the $${\mathbb {Z}}_2^2$$ Z 2 2 -covers of the blow up of $${\mathbb {P}}^2$$ P 2 at three general points branched along a configuration of three pairs of lines. Up to a finite group action, we show that this compactification is isomorphic to the toric variety associated to the secondary polytope of the unit cube. We relate the KSBA compactification considered to the Baily–Borel compactification of the same family of Enriques surfaces. Part of the KSBA boundary has a toroidal behavior, another part is isomorphic to the Baily–Borel compactification, and what remains is a mixture of these two. We relate the stable pair compactification studied here with Looijenga’s semitoric compactifications.

MODULI OF STABLE PAIRS FOR HOLOMORPHIC BUNDLES OVER RIEMANN SURFACES

1991 ◽  
Vol 02 (05) ◽  
pp. 477-513 ◽  
Author(s):  
STEVEN B. BRADLOW ◽  
GEORGIOS D. DASKALOPOULOS
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Holomorphic Section ◽  
Stable Bundles ◽  
Space Problem ◽  
Yang Mills ◽  
Closed Riemann Surface ◽  
Holomorphic Bundles ◽  
Stable Pairs ◽  
Compact Kähler Manifold

It this paper we study the space of gauge equivalence classes of pairs [Formula: see text] where [Formula: see text] represents a holomorphic structure on a complex bundle, E, over a closed Riemann Surface, and ϕ is a holomorphic section. We define a space of stable pairs and consider the moduli space problem for this space. The space of stable pairs, [Formula: see text], is related to the space of solution to the Vortex (Hermitian-Yang-Mills-Higgs) equation. Using the parameter, τ, which appears in this equation we can define subspaces [Formula: see text] within [Formula: see text]. We show that under suitable restrictions on τ and the degree of E, the space [Formula: see text] is naturally a finite dimensional, Hausdorff, compact Kähler manifold. We show further that there is a natural holomorphic map from this space onto the Seshadri compactification of the moduli space of stable bundles and that this map is generically a fibration.

HECKE CORRESPONDENCE FOR SYMPLECTIC BUNDLES WITH APPLICATION TO THE PICARD BUNDLES

2006 ◽  
Vol 17 (01) ◽  
pp. 45-63 ◽  
Author(s):  
INDRANIL BISWAS ◽  
TOMÁS L. GÓMEZ
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Bilinear Form ◽  
Moduli Space ◽  
Vector Bundles ◽  
Projective Bundle ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Hecke Correspondence ◽  
Stable Pairs

We construct a Hecke correspondence for a moduli space of symplectic vector bundles over a curve. As an application we prove the following. Let X be a complex smooth projective curve of genus g(X) > 2 and L a line bundle over X. Let [Formula: see text] be the moduli space parametrizing stable pairs of the form (E,φ), where E is a vector bundle of rank 2n over X and φ : E ⊗ E → L a skew-symmetric nondegenerate bilinear form on the fibers of E. If deg (E) ≥ 4n(g(X)-1), then there is a projectivized Picard bundle on [Formula: see text], which is a projective bundle whose fiber over any point [Formula: see text] is ℙ(H0(X,E)). We prove that this projective bundle is stable.

MODULI OF STABLE PAIRS FOR HOLOMORPHIC BUNDLES OVER RIEMANN SURFACES II

1993 ◽  
Vol 04 (06) ◽  
pp. 903-925 ◽  
Author(s):  
STEVEN BRADLOW ◽  
GEORGIOS D. DASKALOPOULOS
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Projective Variety ◽  
Algebraic Varieties ◽  
Stable Bundles ◽  
Topological Information ◽  
Holomorphic Bundles ◽  
Stable Pairs

In this paper we continue our investigation of the moduli space of stable pairs introduced in Part I. We obtain certain topological information, and we give a proof that this moduli space admits the structure of a nonsingular projective variety. We show that the natural map from the moduli space of stable pairs onto the Seshadri compactification of stable bundles is a morphism of algebraic varieties.

scholarly journals KSBA compactification of the moduli space of K3 surfaces with a purely non-symplectic automorphism of order four

2021 ◽  
Vol 64 (1) ◽  
pp. 99-127
Author(s):  
Han-Bom Moon ◽  
Luca Schaffler
Keyword(s):  
Finite Group ◽  
Moduli Space ◽  
K3 Surfaces ◽  
Double Cover ◽  
Minimal Resolution ◽  
Finite Group Action ◽  
Git Quotient ◽  
Stable Pairs ◽  
A Finite Group ◽  
Order Four

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and $U(2)\oplus D_4^{\oplus 2}$ lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of $\mathbb {P}^{1}\times \mathbb {P}^{1}$ branched along a specific $(4,\,4)$ curve. We show that, up to a finite group action, this stable pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient $(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$ with the symmetric linearization.

scholarly journals Moduli of stable sheaves supported on curves of genus three contained in a quadric surface

2020 ◽  
Vol 20 (4) ◽  
pp. 507-522
Author(s):  
Mario Maican
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Betti Numbers ◽  
Free Resolutions ◽  
Quadric Surface ◽  
Arithmetic Genus ◽  
Blow Down ◽  
Stable Sheaves ◽  
Stable Pairs ◽  
Smooth Quadric Surface

AbstractWe study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the variation of the moduli spaces of α-semi-stable pairs. We classify the stable sheaves using locally free resolutions or extensions. We give a global description: the moduli space is obtained from a certain flag Hilbert scheme by performing two flips followed by a blow-down.

Topology of the quantum modified moduli space

2001 ◽  
Vol 15 (4) ◽  
pp. 279-289
Author(s):  
S. L. Dubovsky
Keyword(s):  
Moduli Space

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

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