Vector bundles and regulous maps

Let $X$ be a compact nonsingular affine real algebraic variety. We prove that every pre-algebraic vector bundle on $X$ becomes algebraic after finitely many blowing ups. Using this theorem, we then prove that the Stiefel-Whitney classes of any pre-algebraic $\mathbb{R }$ -vector bundle on $X$ are algebraic. We also derive that the Chern classes of any pre-algebraic $\mathbb{C }$ -vector bundles and the Pontryagin classes of any pre-algebraic $\mathbb{R }$ -vector bundle are blow- $\mathbb{C }$ -algebraic. We also provide several results on line bundles on $X$ .     

Rank-two stable sheaves with odd determinant on Fano threefolds of genus nine

According to Mukai and Iliev, a smooth prime Fano threefold $X$ of genus $9$ is associated with a surface $\mathbb{P }(\mathcal{V })$ , ruled over a smooth plane quartic $\varGamma $ , and the derived category of $\varGamma $ embeds into that of $X$ by a theorem of Kuznetsov. We use this setup to study the moduli spaces of rank- $2$ stable sheaves on $X$ with odd determinant. For each $c_2 \ge 7$ , we prove that a component of their moduli space $\mathsf{M}_X(2,1,c_2)$ is birational to a Brill–Noether locus of vector bundles with fixed rank and degree on $\varGamma $ , having enough sections when twisted by $\mathcal{V }$ . For $c_2=7$ , we prove that $\mathsf{M}_X(2,1,7)$ is isomorphic to the blow-up of the Picard variety $\text{ Pic}^{2}({\varGamma })$ along the curve parametrizing lines contained in $X$ .     

f-algebras with a \sigma -bounded approximate unit

Let $X$ be a lattice ordered algebra ( $\ell $ -algebra). A positive element $x\in $ $X$ is said to be totally bounded if $x^{2}\le x$ . The $\ell $ -algebra $X$ is said to have a $\sigma $ -bounded approximate unit if for each positive linear functional $f$ on $X$ the set $\left\{ f(x)\text{: } x \text{ totally } \text{ bounded }\right\} $ is bounded in $\mathbb R $ . In this paper we study the class of $f$ -algebras with a $\sigma $ -bounded approximate unit which contains the class of all unital $f$ -algebras. In particular It is shown that an $f$ -algebra $X$ has a $\sigma $ -bounded approximate unit if and only if the order bidual $X^{\sim \sim }$ is a unital $f$ -algebra.     

Equivariant map superalgebras

Suppose a group $\Gamma $ acts on a scheme $X$ and a Lie superalgebra $\mathfrak {g}$ . The corresponding equivariant map superalgebra is the Lie superalgebra of equivariant regular maps from $X$ to $\mathfrak {g}$ . We classify the irreducible finite dimensional modules for these superalgebras under the assumptions that the coordinate ring of $X$ is finitely generated, $\Gamma $ is finite abelian and acts freely on the rational points of $X$ , and $\mathfrak {g}$ is a basic classical Lie superalgebra (or $\mathfrak {sl}\,(n,n)$ , $n \ge 1$ , if $\Gamma $ is trivial). We show that they are all (tensor products of) generalized evaluation modules and are parameterized by a certain set of equivariant finitely supported maps defined on $X$ . Furthermore, in the case that the even part of $\mathfrak {g}$ is semisimple, we show that all such modules are in fact (tensor products of) evaluation modules. On the other hand, if the even part of $\mathfrak {g}$ is not semisimple (more generally, if $\mathfrak {g}$ is of type I), we introduce a natural generalization of Kac modules and show that all irreducible finite dimensional modules are quotients of these. As a special case, our results give the first classification of the irreducible finite dimensional modules for twisted loop superalgebras.     

On the bicanonical map of primitive varieties with q(X) = \mathrm{dim }X: the degree and the Euler number

Let $X$ be a variety of maximal Albanese dimension and of general type. Assume that $q(X) = \mathrm{dim }X$ , the Albanese variety $\mathrm {Alb} (X)$ is a simple abelian variety, and the bicanonical map is not birational. We prove that the Euler number $\chi (X, \omega _X)$ is equal to 1, and $|2K_X|$ separates two distinct points over the same general point on $\mathrm {Alb} (X)$ via $\mathrm {alb}_X$ (Theorem 1.1).     

Constructible \nabla -modules on curves

Let $\mathcal{V }$ be a complete discrete valuation ring of mixed characteristic with perfect residue field. Let $X$ be a geometrically connected smooth proper curve over $\mathcal{V }$ . We introduce the notion of constructible convergent $\nabla $ -module on the analytification $X_{K}^{\mathrm{an}}$ of the generic fiber of $X$ . A constructible module is an $\mathcal{O }_{X_{K}^{\mathrm{an}}}$ -module which is not necessarily coherent, but becomes coherent on a stratification by locally closed subsets of the special fiber $X_{k}$ of $X$ . The notions of connection, of (over-) convergence and of Frobenius structure carry over to this situation. We describe a specialization functor from the category of constructible convergent $\nabla $ -modules to the category of $\mathcal{D }^\dagger _{\hat{X} \mathbf{Q }}$ -modules. We show that specialization induces an equivalence between constructible $F$ - $\nabla $ -modules and perverse holonomic $F$ - $\mathcal{D }^\dagger _{\hat{X} \mathbf{Q }}$ -modules.     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

Actions of arithmetic groups on homology spheres and acyclic homology manifolds

We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary $p$ -groups in the groups concerned. In some cases, including ${\mathrm{Sp}}(2n,\mathbb Z )$ , the bounds we obtain are sharp: if $X$ is a generalized $\mathbb Z /3$ -homology sphere of dimension less than $2n-1$ or a $\mathbb Z /3$ -acyclic $\mathbb Z /3$ -homology manifold of dimension less than $2n$ , and if $n\ge 3$ , then any action of ${\mathrm{Sp}}(2n,\mathbb Z )$ by homeomorphisms on $X$ is trivial; if $n=2$ , then every action of ${\mathrm{Sp}}(2n,\mathbb Z )$ on $X$ factors through the abelianization of ${\mathrm{Sp}}(4,\mathbb Z )$ , which is $\mathbb Z /2$ .     

2-Nilpotent real section conjecture

We show a $2$ -nilpotent section conjecture over $\mathbb{R }$ : for a geometrically connected curve $X$ over $\mathbb{R }$ such that each irreducible component of its normalization has $\mathbb{R }$ -points, $\pi _0(X(\mathbb{R }))$ is determined by the maximal $2$ -nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that for $X$ smooth and proper, $X(\mathbb{R })^{\pm }$ is determined by the maximal $2$ -nilpotent quotient of $\mathrm{Gal }(\mathbb{C }(X))$ with its $\mathrm{Gal }(\mathbb{R })$ action, where $X(\mathbb{R })^{\pm }$ denotes the set of real points equipped with a real tangent direction, showing a $2$ -nilpotent birational real section conjecture.     

On the existence of Evans potentials

It is shown that, for every noncompact parabolic Riemannian manifold $X$ and every nonpolar compact $K$ in  $X$ , there exists a positive harmonic function on $X\setminus K$ which tends to $\infty $ at infinity. (This is trivial for $\mathbb{R }$ , easy for  $\mathbb{R }^2$ , and known for parabolic Riemann surfaces.) In fact, the statement is proven, more generally, for any noncompact connected Brelot harmonic space  $X$ , where constants are the only positive superharmonic functions and, for every nonpolar compact set  $K$ , there is a symmetric (positive) Green function for $X\setminus K$ . This includes the case of parabolic Riemannian manifolds. Without symmetry, however, the statement may fail. This is shown by an example, where the underlying space is a graph (the union of the parallel half-lines $\left[0,\infty \right)\times \{0\}, \left[0,\infty \right)\times \{1\}$ , and the line segments $\{n\}\times [0,1], n=0,1,2,\dots $ ).     

Minimal generating and normally generating sets for the braid and mapping class groups of \mathbb{D }^{2}, \mathbb S ^{2} and \mathbb{R P^2}

We consider the (pure) braid groups $B_{n}(M)$ and $P_{n}(M)$ , where $M$ is the $2$ -sphere $\mathbb S ^{2}$ or the real projective plane $\mathbb R P^2$ . We determine the minimal cardinality of (normal) generating sets $X$ of these groups, first when there is no restriction on $X$ , and secondly when $X$ consists of elements of finite order. This improves on results of Berrick and Matthey in the case of $\mathbb S ^{2}$ , and extends them in the case of $\mathbb R P^2$ . We begin by recalling the situation for the Artin braid groups ( $M=\mathbb{D }^{2}$ ). As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show that for $M=\mathbb S ^{2}$ or $\mathbb R P^2$ , the induced action of $B_n(M)$ on $H_3(\widetilde{F_n(M)};\mathbb{Z })$ is trivial, $F_{n}(M)$ being the $n^\mathrm{th}$ configuration space of $M$ .     

Weak quadratic overgroups for a class of Lie groups

Let $G$ be a connected and simply connected Lie group with Lie algebra $\mathfrak g $ . We say that a subset $X$ in the set $\mathfrak g ^\star / G$ of coadjoint orbits is convex hull separable when the convex hulls differ for any pair of distinct coadjoint orbits in $X$ . In this paper, we define a class of solvable Lie groups, and we give an explicit construction of an overgroup $G^+$ and a quadratic map $\varphi $ sending each generic orbit in $\mathfrak g ^\star $ to a $G^+$ -orbit in $\mathfrak{g ^+}^\star $ , in such a manner that the set $\varphi (\mathfrak g ^\star _{gen}){/ G^+}$ is convex hull separable. We then call $G^+$ a weak quadratic overgroup for $G$ . Thanks to this construction, we prove that any nilpotent Lie group, with dimension at most 7 admits such a weak quadratic overgroup. Finally, we produce different examples of solvable Lie groups, having weak quadratic overgroups, but which are not in our class of Lie groups and for which usual constructions fail to hold.     

Approximate Hermitian–Einstein connections on principal bundles over a compact Riemann surface

Let $X$ be a compact connected Riemann surface and $G$ a connected reductive complex affine algebraic group. Given a holomorphic principal $G$ -bundle $E_G$ over $X$ , we construct a $C^\infty $ Hermitian structure on $E_G$ together with a $1$ -parameter family of $C^\infty $ automorphisms $\{F_t\}_{t\in \mathbb R }$ of the principal $G$ -bundle $E_G$ with the following property: Let $\nabla ^t$ be the connection on $E_G$ corresponding to the Hermitian structure and the new holomorphic structure on $E_G$ constructed using $F_t$ from the original holomorphic structure. As $t\rightarrow -\infty $ , the connection $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . In particular, as $t\rightarrow -\infty $ , the curvature of $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the curvature of the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . The family $\{F_t\}_{t\in \mathbb R }$ is constructed by generalizing the method of [6]. Given a holomorphic vector bundle $E$ on $X$ , in [6] a $1$ -parameter family of $C^\infty $ automorphisms of $E$ is constructed such that as $t\rightarrow -\infty $ , the curvature converges, in $C^0$ topology, to the curvature of the Hermitian–Einstein connection of the associated graded bundle.     

On involutions with middle-dimensional fixed-point locus and holomorphic-symplectic manifolds

Let $g$ be an involution of a compact closed manifold $X$ such that the fixed-point set $X^{g}$ is middle dimensional. Under the assumption that the normal bundle of the fixed-point set is either the tangent or co-tangent bundle conditions on the equivariant invariants of $X$ arise. In particular if $X$ is a holomorphic-symplectic manifold and $g$ an anti holomorphic-symplectic involution one arrives at a generalisation of Beauville’s result that for $X$ a hyper-Kähler manifold the $\hat{A}$ genus of $X^{g}$ is one.     

On smooth extensions of vector-valued functions defined on closed subsets of Banach spaces

Let $X$ and $ Z$ be Banach spaces, $A$ a closed subset of $X$ and a mapping $f:A\rightarrow Z$ . We give necessary and sufficient conditions to obtain a $C^1$ smooth mapping $F:X \rightarrow Z$ such that $F_{\mid _A}=f$ , when either (i) $X$ and $Z$ are Hilbert spaces and $X$ is separable, or (ii) $X^*$ is separable and $Z$ is an absolute Lipschitz retract, or (iii) $X=L_2$ and $Z=L_p$ with $1<p<2$ , or (iv) $X=L_p$ and $Z=L_2$ with $2<p<\infty $ , where $L_p$ is any separable Banach space $L_p(S,\Sigma ,\mu )$ with $(S,\Sigma ,\mu )$ a $\sigma $ -finite measure space.     

A Characterization of Compact Operators Via the Non-Connectedness of the Attractors of a Family of IFSs

In this paper we present a result which establishes a connection between the theory of compact operators and the theory of iterated function systems. For a Banach space $X$ , $S$ and $T$ bounded linear operators from $X$ to $X$ such that $\Vert S\Vert , \Vert T\Vert <1$ and $w\in X$ , let us consider the IFS $\mathcal S _{w}=(X,f_{1},f_{2})$ , where $f_{1},f_{2}:X\rightarrow X$ are given by $f_{1}(x)=S(x)$ and $f_{2}(x)=T(x)+w$ , for all $x\in X$ . On one hand we prove that if the operator $S$ is compact, then there exists a family $(K_{n})_{n\in \mathbb N }$ of compact subsets of $X$ such that $A_{\mathcal S _{w}}$ is not connected, for all $w\in X-\bigcup _{n\in \mathbb N } K_{n}$ . On the other hand we prove that if $H$ is an infinite dimensional Hilbert space, then a bounded linear operator $S:H\rightarrow H$ having the property that $\Vert S\Vert <1$ is compact provided that for every bounded linear operator $T:H\rightarrow H$ such that $\Vert T\Vert <1$ there exists a sequence $(K_{T,n})_{n}$ of compact subsets of $H$ such that $A_{\mathcal S _{w}}$ is not connected for all $w\in H-\bigcup _{n}K_{T,n}$ . Consequently, given an infinite dimensional Hilbert space $H$ , there exists a complete characterization of the compactness of an operator $S:H\rightarrow H$ by means of the non-connectedness of the attractors of a family of IFSs related to the given operator. Finally we present three examples illustrating our results.     

Algebraic Boundaries of \mathrm{SO}(2)-Orbitopes

Let $X\subset \mathbb{A }^{2r}$ X ? A 2 r be a real curve embedded into an even-dimensional affine space. We characterise when the $r$ r th secant variety to $X$ X is an irreducible component of the algebraic boundary of the convex hull of the real points $X(\mathbb{R })$ X ( R ) of $X$ X . This fact is then applied to $4$ 4 -dimensional $\mathrm{SO}(2)$ SO ( 2 ) -orbitopes and to the so called Barvinok–Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of $4$ 4 -dimensional $\mathrm{SO}(2)$ SO ( 2 ) -orbitopes, we find all irreducible components of their algebraic boundary.     

Asymptotic behaviour of first passage time distributions for Lévy processes

Let $X$ be a real valued Lévy process that is in the domain of attraction of a stable law without centering with norming function $c.$ As an analogue of the random walk results in Vatutin and Wachtel (Probab Theory Relat Fields 143(1–2):177–217, 2009) and Doney (Probab Theory Relat Fields 152(3–4):559–588, 2012), we study the local behaviour of the distribution of the lifetime $\zeta $ under the characteristic measure $\underline{n}$ of excursions away from $0$ of the process $X$ reflected in its past infimum, and of the first passage time of $X$ below $0,$ $T_{0}=\inf \{t>0:X_{t}<0\},$ under $\mathbb{P }_{x}(\cdot ),$ for $x>0,$ in two different regimes for $x,$ viz. $x=o(c(\cdot ))$ and $x>D c(\cdot ),$ for some $D>0.$ We sharpen our estimates by distinguishing between two types of path behaviour, viz. continuous passage at $T_{0}$ and discontinuous passage. In order to prove our main results we establish some sharp local estimates for the entrance law of the excursion process associated to $X$ reflected in its past infimum.