Connections and Higgs fields on a principal bundle

Let M be a compact connected Kähler manifold and G a connected linear algebraic group defined over \({\mathbb{C}}\) . A Higgs field on a holomorphic principal G-bundle ε _G over M is a holomorphic section θ of \(\text{ad}(\epsilon_{G})\otimes {\Omega}^{1}_{M}\) such that θ∧ θ = 0. Let L(G) be the Levi quotient of G and (ε _G (L(G)), θ _l ) the Higgs L(G)-bundle associated with (ε _G , θ). The Higgs bundle (ε _G , θ) will be called semistable (respectively, stable) if (ε _G (L(G)), θ _l ) is semistable (respectively, stable). A semistable Higgs G-bundle (ε _G , θ) will be called pseudostable if the adjoint vector bundle ad(ε _G (L(G))) admits a filtration by subbundles, compatible with θ, such that the associated graded object is a polystable Higgs vector bundle. We construct an equivalence of categories between the category of flat G-bundles over M and the category of pseudostable Higgs G-bundles over M with vanishing characteristic classes of degree one and degree two. This equivalence is actually constructed in the more general equivariant set-up where a finite group acts on the Kähler manifold. As an application, we give various equivalent conditions for a holomorphic G-bundle over a complex torus to admit a flat holomorphic connection.     

On semistable principal bundles over a complex projective manifold,II

Let (X, ω) be a compact connected Kähler manifold of complex dimension d and \({E_G\,\longrightarrow\,X}\) a holomorphic principal G–bundle, where G is a connected reductive linear algebraic group defined over \({\mathbb{C}}\). Let Z(G) denote the center of G. We prove that the following three statements are equivalent:

1. (1)
   There is a parabolic subgroup \({P\,\subset\,G}\) and a holomorphic reduction of structure group \({E_P\,\subset\,E_G}\) to P, such that the corresponding L(P)/Z(G)–bundle

   $E_{L(P)/Z(G)}\,:=\,E_P(L(P)/Z(G))\,\longrightarrow\,X$

   admits a unitary flat connection, where L(P) is the Levi quotient of P.
    
2. (2)
   The adjoint vector bundle ad(E _G ) is numerically flat.
    
3. (3)
   The principal G–bundle E _G is pseudostable, and

   $\int\limits_X c_2({\rm ad}(E_G))\omega^{d-2}\,=\,0.$
    

If X is a complex projective manifold, and ω represents a rational cohomology class, then the third statement is equivalent to the statement that E _G is semistable with c ₂(ad(E _G )) = 0.

     

Gradings on Modules Over Lie Algebras of E Types

For any grading by an abelian group G on the exceptional simple Lie algebra \(\mathcal {L}\) of type E ₆ or E ₇ over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of finite-dimensional G-graded simple \(\mathcal {L}\)-modules, as well as necessary and sufficient conditions for a finite-dimensional \(\mathcal {L}\)-module to admit a G-grading compatible with the given G-grading on \(\mathcal {L}\).     

Generators for decompositions of tensor products of modules

If K is a field of finite characteristic p, G is a cyclic group of order q = p ^α , U and W are indecomposable KG-modules, and p ≥ dim U + dim W ? 1, we describe how to find a generator for each of the indecomposable components of the KG-module \({U \otimes W}\).     

Affine ADE bundles over surfaces with $$ p _{g}=0$$

Given any Kodaira curve C in a complex surface X, we construct a simply-laced affine Lie algebra bundle \(\mathcal {E}\) over X. When \( p _{g}(X)=0\), we construct deformations of holomorphic structures on \(\mathcal {E}\) such that the new bundle is trivial over any ADE curve \( C^{\prime }\) inside C and therefore descends to the singular surface obtained by contracting \(C^{\prime }\).     

A Higgs bundle on a Hermitian symmetric space

Let \(M := \Gamma\backslash G/K\) be the quotient of an irreducible Hermitian symmetric space G/K by a torsionfree cocompact lattice \(\Gamma\subset G\) . There is a natural flat principal G-bundle over the compact Kähler manifold M which is constructed from the principal Γ-bundle over M defined by the quotient map \(G/K\longrightarrow M\) . We construct the principal G-Higgs bundle over M corresponding to this flat G-bundle. This principal G-Higgs bundle is rigid if \({\rm dim}_\mathbb{C} M\,\geq\,2\) .     

A unifying approach to the Margolis–Meakin and Birget–Rhodes group expansion

Let G be a group. We show that the Birget–Rhodes prefix expansion \(G^{Pr}\) and the Margolis–Meakin expansion M(X; f) of G with respect to \(f:X\rightarrow G\) can be regarded as inverse subsemigroups of a common E-unitary inverse semigroup P. We construct P as an inverse subsemigroup of an E-unitary inverse monoid \(U/\zeta \) which is a homomorphic image of the free product U of the free semigroup \(X^+\) on X and G. The semigroup P satisfies a universal property with respect to homomorphisms into the permissible hull C(S) of a suitable E-unitary inverse semigroup S, with \(S/\sigma _S=G\), from which the characterizing universal properties of \(G^{Pr}\) and M(X; f) can be recaptured easily.     

An Oka Principle for a Parametric Infinite Transitivity Property

It is an elementary fact that the action by holomorphic automorphisms on \(\mathbb {C}^n\) is infinitely transitive, i.e., m-transitive for any \(m\in \mathbb {N}\). The same holds on any Stein manifold with the holomorphic density property X. We study a parametrized case: we consider m points on X parametrized by a Stein manifold W, and seek a family of automorphisms of X, parametrized by W, putting them into a standard form which does not depend on the parameter. This general transitivity is shown to enjoy an Oka principle, to the effect that the obstruction to a holomorphic solution is of a purely topological nature. In the presence of a volume form and of a corresponding density property, similar results for volume-preserving automorphisms are obtained.     

Λ-modules and holomorphic Lie algebroid connections

Let X be a complex smooth projective variety, and G a locally free sheaf on X. We show that there is a one-to-one correspondence between pairs (Λ, Ξ), where Λ is a sheaf of almost polynomial filtered algebras over X satisfying Simpson’s axioms and \( \equiv :Gr\Lambda \to Sym \bullet _{\mathcal{O}_X } \mathcal{G}\) is an isomorphism, and pairs (L, Σ), where L is a holomorphic Lie algebroid structure on \(\mathcal{G}\) and Σ is a class in F ¹ H ²(L, ?), the first Hodge filtration piece of the second cohomology of L.As an application, we construct moduli spaces of semistable flat L-connections for any holomorphic Lie algebroid L. Particular examples of these are given by generalized holomorphic bundles for any generalized complex structure associated to a holomorphic Poisson manifold.     

Algebraic Hamiltonian actions

In this paper we deal with a Hamiltonian action of a reductive algebraic group G on an irreducible normal affine Poisson variety X. We study the quotient morphism \({\mu_{G,X}//G : X//G \rightarrow \mathfrak{g} //G}\) of the moment map \({\mu_{G,X} : X\rightarrow \mathfrak{g}}\) . We prove that for a wide class of Hamiltonian actions (including, for example, actions on generically symplectic varieties) all fibers of the morphism μ _G,X //G have the same dimension. We also study the “Stein factorization” of μ _G,X //G. Namely, let C _G,X denote the spectrum of the integral closure of \({\mu_{G,X}^{*}(\mathbb{K}[\mathfrak{g}]^G)}\) in \({\mathbb{K}(X)^G}\) . We investigate the structure of the \({\mathfrak{g} //G}\) -scheme C _G,X . Our results partially generalize those obtained by F. Knop for the actions on cotangent bundles and symplectic vector spaces.     

Homological invariants of modules over contracting endomorphisms

It is proved that when R is a local ring of positive characteristic, \({\phi\colon R{\to} R}\) is its Frobenius endomorphism, and some non-zero finite R-module has finite flat dimension or finite injective dimension for the R-module structure induced through \({\phi}\) , then R is regular. This broad generalization of Kunz’s characterization of regularity in positive characteristic is deduced from a theorem concerning a local ring R with residue field of k of arbitrary characteristic: If \({\phi}\) is a contracting endomorphism of R, then the Betti numbers and the Bass numbers over \({\phi}\) of any non-zero finitely generated R-module grow at the same rate, on an exponential scale, as the Betti numbers of k over R.     

Unbounded disjointness preserving linear functionals and operators

Let E and F be Banach lattices. We show first that the disjointness preserving linear functionals separate the points of any infinite dimensional Banach lattice E, which shows that in this case the unbounded disjointness preserving operators from \({E\to F}\) separate the points of E. Then we show that every disjointness preserving operator \({T:E\to F}\) is norm bounded on an order dense ideal. In case E has order continuous norm, this implies that every unbounded disjointness preserving map \({T:E\to F}\) has a unique decomposition T = R + S, where R is a bounded disjointness preserving operator and S is an unbounded disjointness preserving operator, which is zero on a norm dense ideal. For the case that E = C(X), with X a compact Hausdorff space, we show that every disjointness preserving operator \({T:C(X)\to F}\) is norm bounded on a norm dense sublattice algebra of C(X), which leads then to a decomposition of T into a bounded disjointness preserving operator and a finite sum of unbounded disjointness preserving operators, which are zero on order dense ideals.     

Normal coverings of solvable groups

For a finite non cyclic group G, let γ(G) be the smallest integer k such that G contains k proper subgroups H ₁, . . . , H _k with the property that every element of G is contained in \({H_i^g}\) for some \({i \in \{1,\dots,k\}}\) and \({g \in G.}\) We prove that for every n ≥ 2, there exists a finite solvable group G with γ(G) = n.     

Cardinality of the Ellis semigroup on compact metric countable spaces

Let E(X, f) be the Ellis semigroup of a dynamical system (X, f) where X is a compact metric space. We analyze the cardinality of E(X, f) for a compact countable metric space X. A characterization when E(X, f) and \(E(X,f)^* = E(X,f) \setminus \{ f^n : n \in \mathbb {N}\}\) are both finite is given. We show that if the collection of all periods of the periodic points of (X, f) is infinite, then E(X, f) has size \(2^{\aleph _0}\). It is also proved that if (X, f) has a point with a dense orbit and all elements of E(X, f) are continuous, then \(|E(X,f)| \le |X|\). For dynamical systems of the form \((\omega ^2 +1,f)\), we show that if there is a point with a dense orbit, then all elements of \(E(\omega ^2+1,f)\) are continuous functions. We present several examples of dynamical systems which have a point with a dense orbit. Such systems provide examples where \(E(\omega ^2+1,f)\) and \(\omega ^2+1\) are homeomorphic but not algebraically homeomorphic, where \(\omega ^2+1\) is taken with the usual ordinal addition as semigroup operation.     

Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

Consider the restriction of an irreducible unitary representation π of a Lie group G to its subgroup H. Kirillov’s revolutionary idea on the orbit method suggests that the multiplicity of an irreducible H-module ν occurring in the restriction π|_H could be read from the coadjoint action of H on \(\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H})\), provided π and ν are ‘geometric quantizations’ of a G-coadjoint orbit \(\mathcal {O}^{G}\) and an H-coadjoint orbit \(\mathcal {O}^{H}\), respectively, where \(\text {pr} \colon \sqrt {-1}\mathfrak {g}^{\ast } \to \sqrt {-1}\mathfrak {h}^{\ast }\) is the projection dual to the inclusion \(\mathfrak {h} \subset \mathfrak {g}\) of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits \(\mathcal {O}^{G}\) of a semisimple Lie group G corresponding to highest weight modules of scalar type. We prove that the Corwin–Greenleaf number \(\sharp (\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H}))/H\) is either zero or one for any H-coadjoint orbit \(\mathcal {O}^{H}\), whenever (G,H) is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits \(\mathcal {O}^{H}\) with nonzero Corwin–Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as ‘classical limits’ of the multiplicity-free branching laws of holomorphic discrete series representations (Kobayashi [Progr. Math. 2007]).     

On <Emphasis Type="Italic">G</Emphasis>-covering subgroup systems of finite groups

Let \(\mathcal{F}\) be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for  \(\mathcal{F}\) if \(G\in \mathcal{F}\) whenever \(\Sigma \subseteq \mathcal{F}\). Let p be any prime dividing |G| and P a Sylow p-subgroup of G. Then we write Σ _p to denote the set of subgroups of G which contains at least one supplement to G of each maximal subgroup of P. We prove that the sets Σ _p and Σ _p ∪Σ _q , where q≠p, are G-covering subgroup systems for many classes of finite groups.     

Equivariant Euler–Poincaré characteristic in sheaf cohomology

Let X be a Hausdorff space equipped with a continuous action of a finite group G and a G-stable family of supports \({\Phi}\). Fix a number field F with ring of integers R. We study the class \({\chi = \sum_j (-1)^j [H^j_\Phi (X, \mathcal{E}) \otimes_R F]}\) in the character group of G over F for any flat G-sheaf \({\mathcal{E}}\) of R-modules over X. Under natural cohomological finiteness conditions we give a formula for \({\chi}\) with respect to the basis given by the irreducible characters of G. We discuss applications of our result concerning the cohomology of arithmetic groups.     

Fourier transforms and bent functions on faithful actions of finite abelian groups

Let G be a finite abelian group acting faithfully on a finite set X. The G-bentness and G-perfect nonlinearity of functions on X are studied by Poinsot and co-authors (Discret Appl Math 157:1848–1857, 2009; GESTS Int Trans Comput Sci Eng 12:1–14, 2005) via Fourier transforms of functions on G. In this paper we introduce the so-called \(G\)-dual set \(\widehat{X}\) of X, which plays the role similar to the dual group \(\widehat{G}\) of G, and develop a Fourier analysis on X, a generalization of the Fourier analysis on the group G. Then we characterize the bentness and perfect nonlinearity of functions on X by their own Fourier transforms on \(\widehat{X}\). Furthermore, we prove that the bentness of a function on X can be determined by its distance from the set of G-linear functions. As direct consequences, many known results in Logachev et al. (Discret Math Appl 7:547–564, 1997), Carlet and Ding (J Complex 20:205–244, 2004), Poinsot (2009), Poinsot et al. (2005) and some new results about bent functions on G are obtained. In order to explain the theory developed in this paper clearly, examples are also presented.     

Liouville Property,Wiener’s Test and Unavoidable Sets for Hunt Processes

Let \((X,\mathcal {W})\) be a balayage space, \(1\in \mathcal {W}\), or – equivalently – let \(\mathcal W\) be the set of excessive functions of a Hunt process on a locally compact space X with countable base such that \(\mathcal {W}\) separates points, every function in \(\mathcal {W}\) is the supremum of its continuous minorants and there exist strictly positive continuous \(u,v\in \mathcal {W}\) such that u/v → 0 at infinity. We suppose that there is a Green function G > 0 for X, a metric ρ for X and a decreasing function \(g\colon [0,\infty )\to (0,\infty ]\) having the doubling property such that G ≈ g ° ρ. Assuming that the constant function 1 is harmonic and balls of (X, ρ) are relatively compact, it is shown that every positive harmonic function on X is constant (Liouville property) and that Wiener’s test at infinity shows, if a given set A in X is unavoidable, that is, if the process hits A with probability one, wherever it starts. An application yields that locally finite unions of pairwise disjoint balls B(z, r_z), z ∈ Z, which have a certain separation property with respect to a suitable measure λ on X are unavoidable if and only if, for some/any point x₀ ∈ X, the series \({\sum }_{z\in Z} g(\rho (x_{0},z))/g(r_{z}) \) diverges. The results generalize and, exploiting a zero-one law for hitting probabilities, simplify recent work by S. Gardiner and M. Ghergu, A. Mimica and Z. Vondra?ek, and the author.     

Rational approximation to surfaces defined by polynomials in one variable

For a Tychonoff space X, we denote by C _p (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence.

In this paper we prove that:

• If every finite power of X is Lindelöf then C _p (X) is strongly sequentially separable iff X is \({\gamma}\)-set.
• \({B_{\alpha}(X)}\) (= functions of Baire class \({\alpha}\) (\({1 < \alpha \leq \omega_1}\)) on a Tychonoff space X with the pointwise topology) is sequentially separable iff there exists a Baire isomorphism class \({\alpha}\) from a space X onto a \({\sigma}\)-set.
• \({B_{\alpha}(X)}\) is strongly sequentially separable iff \({iw(X)=\aleph_0}\) and X is a \({Z^{\alpha}}\)-cover \({\gamma}\)-set for \({0 < \alpha \leq \omega_1}\).
• There is a consistent example of a set of reals X such that C _p (X) is strongly sequentially separable but B₁(X) is not strongly sequentially separable.
• B(X) is sequentially separable but is not strongly sequentially separable for a \({\mathfrak{b}}\)-Sierpiński set X.