Sklyanin algebras and Hilbert schemes of points

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P².The generic noncommutative plane corresponds to the Sklyanin algebra S=Skl(E,σ) constructed from an automorphism σ of infinite order on an elliptic curve E⊂P². In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1−n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P²?E.     

Weil-Petersson Areas of The Moduli Spaces Of Tori

We study the Teichmüller spaces of torus with one branch point of order v and of torus with a totally geodesic boundary curve of length m, respectively. Applying the obtained results for the corresponding moduli spaces we find that the Weil-Petersson area of the moduli space of torus with one conical point of order v is (π²/6)(1 - l/v²) and that of the moduli space of torus with a totally geodesic boundary curve of length m is π²/6 + m²/24.     

Ортопоперечники классов многомерных периодических функций, мажоранта смещанных модулей непрерывности которых содержит как степенные, так и логарифмические множители

Ortho-widths are studied for classes of multivariate periodic functions in the spaces L ^p , 1 ≤ p ≤ ∞, whose mixed moduli of continuity are bounded by a certain product of power-and logarithmic type functions.     

On the symplectic structure of instanton moduli spaces

We study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of SU(2) instantons on both commutative and non-commutative R⁴. We identify global Darboux coordinates and quadratic Hamiltonians on classical phase spaces for which these quiver varieties are natural completions. We also show that the group of non-commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non-commutative instantons. This paper should be viewed as a step towards extending known results for Calogero–Moser spaces to the instanton moduli spaces.     

Some Inequalities for Derivatives of Trigonometric and Algebraic Polynomials

We investigate inequalities for derivatives of trigonometric and algebraic polynomials in weighted L ^P spaces with weights satisfying the Muckenhoupt A _p condition. The proofs are based on an identity of Balázs and Kilgore [1] for derivatives of trigonometric polynomials. Also an inequality of Brudnyi in terms of rth order moduli of continuity ω_r will be given. We are able to give values to the constants in the inequalities.     

Multi-dimensional volumes and moduli of convexity in banach spaces

Summary Let E be a Banach space. Using the definition for the k-dimensional volume enclosed by k + 1 vectors due to Silverman [16], one can define the modulus of k-rotundity of E. In [22] it was shown that k-uniformly rotund Banach spaces are isomorphic to uniformly rotund spaces and, indeed, have some of the same isometric properties with respect to non-expansive and nearest-point maps. The present paper examines the modulus of k-rotundity more thoroughly. Included are a result on the asymptotic behavior of the moduli for l²; a generalization of Dixmier's Theorem on higher-duals of non-reflexive spaces; and an inequality relating the moduli of E^**/E and those of E. The modulus of2-rotundity is shown to be equivalent to one of the moduli defined by V. D. Milman [13] and a necessary and sufficient condition for an l^p-product of spaces to be 2-uniformly rotund is given.Some of the results of this paper are contained in the Ph. D. dissertation of the first author, written under the direction of the second author.     

Polygons in Minkowski three space and parabolic Higgs bundles of rank 2 on \mathbb{C}{{\mathbb{P}}^1}

Consider the moduli space of parabolic Higgs bundles (E, Φ) of rank two on ??¹ such that the underlying holomorphic vector bundle for the parabolic vector bundle E is trivial. It is equipped with the natural involution defined by $ \left( {E,\varPhi } \right)\mapsto \left( {E,-\varPhi } \right) $ . We study the fixed point locus of this involution. In [GM], this moduli space with involution was identified with the moduli space of hyperpolygons equipped with a certain natural involution. Here we identify the fixed point locus with the moduli spaces of polygons in Minkowski 3-space. This identification yields information on the connected components of the fixed point locus.     

Spin structures on instanton moduli spaces

We determine the necessary and sufficient conditions for the based SU(l)-instanton moduli spaces over CP² and the unbased SU(2)-instanton moduli spaces over S⁴ having spin structures.     

Moduli of Quaternionic Superminimal Immersions of 2-Spheres into Quaternionic Projective Spaces

The aim of this paper is to describe the moduli spaces of degree d quaternionic superminimal maps from 2-spheres to quaternionic projective spaces HPn. We show that such moduli spaces have the structure of projectivized fibre products and are connected quasi-projective varieties of dimension 2nd + 2n + 2. This generalizes known results for spaces of harmonic 2-spheres in S⁴.     

Tilting Theoretical Approach to Moduli Spaces Over Preprojective Algebras

We apply tilting theory over preprojective algebras Λ to the study of moduli spaces of Λ-modules. We define the categories of semistable modules and give equivalences, so-called reflection functors, between them by using tilting modules over Λ. Moreover we prove that the equivalence induces an isomorphism of K-schemes between moduli spaces. In particular, we study the case when the moduli spaces are related to Kleinian singularities, and generalize some results of Crawley-Boevey (Am J Math 122:1027–1037, 2000).     

Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces

In this paper we prove that the moduli spaces of framed vector bundles over a surface X, satisfying certain conditions, admit a family of Poisson structures parametrized by the global sections of a certain line bundle on X. This generalizes to the case of framed vector bundles previous results obtained in [B2] for the moduli space of vector bundles over a Poisson surface. As a corollary of this result we prove that the moduli spaces of framed SU(r) – instantons on S⁴ = ℝ⁴ ∪ {∞} admit a natural holomorphic symplectic structure.     

Arc spaces and DAHA representations

A theorem of Y. Berest, P. Etingof and V. Ginzburg states that finite-dimensional irreducible representations of a type A rational Cherednik algebra are classified by one rational number m/n. Every such representation is a representation of the symmetric group S _n . We compare certain multiplicity spaces in its decomposition into irreducible representations of S _n with the spaces of differential forms on a zero-dimensional moduli space associated with the plane curve singularity x ^m = y ⁿ .     

On a Generalization of Szász–Mirakjan–Kantorovich Operators

In this paper we introduce and study a sequence of positive linear operators acting on suitable spaces of measurable functions on [0,+∞[, including L ^p ([0,+∞[) spaces, 1 ≤ p < +∞, as well as continuous function spaces with polynomial weights. These operators generalize the Szász–Mirakjan–Kantorovich operators and they allow to approximate (or to reconstruct) suitable measurable functions by knowing their mean values on a sequence of subintervals of [0,+∞[ that do not constitute a subdivision of it. We also give some estimates of the rates of convergence by means of suitable moduli of smoothness.     

Real moduli of complex objects: Surfaces and bundles

Here we study the real locus (i.e. the fixed locus by the conjugation) of a few moduli spaces (defined overR) of complex objects (essentially moduli spaces of surfaces of general type or of vector bundles on curves andP ²).     

Double solid twistor spaces: the case of arbitrary signature

In a recent paper ([9]) we constructed a series of new Moishezon twistor spaces which are a kind of variant of the famous LeBrun twistor spaces. In this paper we explicitly give projective models of another series of Moishezon twistor spaces on n CP ² for arbitrary n≥3, which can be regarded as a generalization of the twistor spaces of ‘double solid type’ on 3CP ² studied by Kreußler, Kurke, Poon and the author. Similarly to the twistor spaces of ‘double solid type’ on 3CP ², projective models of the present twistor spaces have a natural structure of double covering of a CP ²-bundle over CP ¹. We explicitly give a defining polynomial of the branch divisor of the double covering, whose restriction to fibers is degree four. If n≥4 these are new twistor spaces, to the best of the author’s knowledge. We also compute the dimension of the moduli space of these twistor spaces. Differently from [9], the present investigation is based on analysis of pluri-(half-)anticanonical systems of the twistor spaces.     

Moduli spaces of convex projective structures on surfaces

We introduce explicit parametrisations of the moduli space of convex projective structures on surfaces, and show that the latter moduli space is identified with the higher Teichmüller space for SL₃(R) defined in [V.V. Fock, A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, math.AG/0311149]. We investigate the cluster structure of this moduli space, and define its quantum version.     

Some geometric properties of integral modules

Day's characterization of those spaces 1^p(X_i) which are uniformly convex, in terms of the moduli of convexity of the X_i, is generalized for arbitrary integral modules on measure spaces (K,m) and simplified when m is finite. For this latter purpose a lemma on the moduli of convexity and of smoothness is proved which incidentally gives a further necessary condition for the existence of integral modules in given direct integrals. Further the notions of strict convexity and smoothness of an integral module are related to those of its components.