On the Topology of Kac–Moody groups

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over $\mathbb C $ , let $\text {K}$ denote the unitary form with maximal torus ${{\mathrm{T}}}$ having normalizer ${{\mathrm{N}}}({{\mathrm{T}}})$ . In this article we study the cohomology of the flag manifold $\text {K}/{{{\mathrm{T}}}}$ as a module over the Nil-Hecke algebra, as well as the (co)homology of $\text {K}$ as a Hopf algebra. In particular, if $\mathbb F $ has positive characteristic, we show that $\text {H}_*(\text {K},\mathbb F )$ is a finitely generated algebra, and that $\text {H}^*(\text {K},\mathbb F )$ is finitely generated only if $\text {K}$ is a compact Lie group . We also study the stable homotopy type of the classifying space $\text {BK}$ and show that it is a retract of the classifying space $\text {BN(T)}$ of ${{\mathrm{N}}}({{\mathrm{T}}})$ . We illustrate our results with the example of rank two Kac–Moody groups.     

On the Hall algebra of semigroup representations over \mathbb F _1

Let $\mathrm{A }$ be a finitely generated semigroup with 0. An $\mathrm{A }$ -module over $\mathbb F _1$ (also called an $\mathrm{A }$ -set), is a pointed set $(M,*)$ together with an action of $\mathrm{A }$ . We define and study the Hall algebra $\mathbb H _{\mathrm{A }}$ of the category $\mathcal C _{\mathrm{A }}$ of finite $\mathrm{A }$ -modules. $\mathbb H _{\mathrm{A }}$ is shown to be the universal enveloping algebra of a Lie algebra $\mathfrak n _{\mathrm{A }}$ , called the Hall Lie algebra of $\mathcal C _{\mathrm{A }}$ . In the case of $\langle t \rangle $ —the free monoid on one generator $\langle t \rangle $ , the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\langle t \rangle $ -modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when $\mathrm{A }$ is a quotient of $\langle t \rangle $ by a congruence, and the monoid $G \cup \{ 0\}$ for a finite group $G$ .     

Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

We consider biharmonic maps $\phi :(M,g)\rightarrow (N,h)$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $ p $ satisfies $ 2\le p <\infty $ . If for such a $ p $ , $\int _M|\tau (\phi )|^{ p }\,\mathrm{d}v_g<\infty $ and $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty ,$ where $\tau (\phi )$ is the tension field of $\phi $ , then we show that $\phi $ is harmonic. For a biharmonic submanifold, we obtain that the above assumption $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty $ is not necessary. These results give affirmative partial answers to the global version of generalized Chen’s conjecture.     

A zoo of diffeomorphism groups on \mathbb{R }^{n}

We consider the groups ${\mathrm{Diff }}_\mathcal{B }(\mathbb{R }^n)$ , ${\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)$ , and ${\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)$ of smooth diffeomorphisms on $\mathbb{R }^n$ which differ from the identity by a function which is in either $\mathcal{B }$ (bounded in all derivatives), $H^\infty = \bigcap _{k\ge 0}H^k$ , or $\mathcal{S }$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.     

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).     

Strengthening the cohomological crepant resolution conjecture for Hilbert–Chow morphisms

Given any smooth toric surface $S$ , we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov–Witten invariants of the $n$ -fold symmetric product stack $[\mathrm{Sym}^n(S)]$ of $S$ to the 3-point extremal Gromov–Witten invariants of the Hilbert scheme $\mathrm{Hilb}^n(S)$ of $n$ points on $S$ . As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan’s Cohomological Crepant Resolution Conjecture for the Hilbert–Chow morphism $\mathrm{Hilb}^n(S) \rightarrow \mathrm{Sym}^n(S)$ and yields a method of reconstructing the cup product for $\mathrm{Hilb}^n(S)$ from the orbifold invariants of $[\mathrm{Sym}^n(S)]$ .     

Half-flat structures on S^3\times S^3

We describe left-invariant half-flat $ \mathrm{SU }(3) $ -structures on $ S^3\times S^3$ using the representation theory of $ \mathrm SO (4) $ and matrix algebra. This leads to a systematic study of the associated cohomogeneity one Ricci-flat metrics with holonomy $ \mathrm G _2$ obtained on $ 7 $ -manifolds with equidistant $ S^3\times S^3$ hypersurfaces. The generic case is analysed numerically.     

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 $M$ .     

Wild quotient singularities of surfaces

Let $(B,\mathcal{M }_B)$ be a noetherian regular local ring of dimension $2$ with residue field $B/\mathcal{M }_B$ of characteristic $p>0$ . Assume that $B$ is endowed with an action of a finite cyclic group $H$ whose order is divisible by $p$ . Associated with a resolution of singularities of $\mathrm{Spec}B^H$ is a resolution graph $G$ and an intersection matrix $N$ . We prove in this article three structural properties of wild quotient singularities, which suggest that in general, one should expect when $H= \mathbb{Z }/p\mathbb{Z }$ that the graph $G$ is a tree, that the Smith group $\mathbb{Z }^n/\mathrm{Im}(N)$ is killed by $p$ , and that the fundamental cycle $Z$ has self-intersection $|Z^2|\le p$ . We undertake a combinatorial study of intersection matrices $N$ with a view towards the explicit determination of the invariants $\mathbb{Z }^n/\mathrm{Im}(N)$ and $Z$ . We also exhibit explicitly the resolution graphs of an infinite set of wild $\mathbb{Z }/2\mathbb{Z }$ -singularities, using some results on elliptic curves with potentially good ordinary reduction which could be of independent interest.     

Homogeneous almost quaternion-Hermitian manifolds

An almost quaternion-Hermitian structure on a Riemannian manifold $(M^{4n},g)$ is a reduction of the structure group of $M$ to $\mathrm{Sp}(n)\mathrm{Sp}(1)\subset \text{ SO }(4n)$ . In this paper we show that a compact simply connected homogeneous almost quaternion-Hermitian manifold of non-vanishing Euler characteristic is either a Wolf space, or $\mathbb{S }^2\times \mathbb{S }^2$ , or the complex quadric $\text{ SO }(7)/\mathrm{U}(3)$ .     

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.     

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.     

Sharpening the Norm Bound in the Subspace Perturbation Theory

Let $A$ be a (possibly unbounded) self-adjoint operator on a separable Hilbert space $\mathfrak H .$ Assume that $\sigma $ is an isolated component of the spectrum of $A$ , that is, $\mathrm{dist}(\sigma ,\Sigma )=d>0$ where $\Sigma =\mathrm spec (A)\setminus \sigma .$ Suppose that $V$ is a bounded self-adjoint operator on $\mathfrak H $ such that $\Vert V\Vert <d/2$ and let $L=A+V$ , $\mathrm{Dom }(L)=\mathrm{Dom }(A).$ Denote by $P$ the spectral projection of $A$ associated with the spectral set $\sigma $ and let $Q$ be the spectral projection of $L$ corresponding to the closed $\Vert V\Vert $ -neighborhood of $\sigma .$ Introducing the sequence $$\begin{aligned} \varkappa _n=\frac{1}{2}\left(1-\frac{(\pi ^2-4)^n}{(\pi ^2+4)^n}\right), \quad n\in \{0\}\cup {\mathbb N }, \end{aligned}$$ we prove that the following bound holds: $$\begin{aligned} \arcsin (\Vert P-Q\Vert )\le M_\star \left(\frac{\Vert V\Vert }{d}\right), \end{aligned}$$ where the estimating function $M_\star (x)$ , $x\in \bigl [0,\frac{1}{2}\bigr )$ , is given by $$\begin{aligned} M_\star (x)=\frac{1}{2}\,\,n_{_\#}(x)\,\arcsin \left(\frac{4\pi }{\pi ^2+4}\right) +\frac{1}{2}\,\arcsin \left(\frac{\pi ( x-\varkappa _{n_{_\#}(x)})}{1-2\varkappa _{n_{_\#}(x)})}\right), \end{aligned}$$ with $n_{_\#}(x)=\max \left\{ n\,\bigr |\,\,n\in \{0\}\cup {\mathbb N }\,, \varkappa _n\le x\right\} $ . The bound obtained is essentially stronger than the previously known estimates for $\Vert P-Q\Vert .$ Furthermore, this bound ensures that $\Vert P-Q\Vert <1$ and, thus, that the spectral subspaces $\mathrm{Ran }(P)$ and $\mathrm{Ran }(Q)$ are in the acute-angle case whenever $\Vert V\Vert <c_\star \,d$ , where $$\begin{aligned} c_\star =16\,\,\frac{\pi ^6-2\pi ^4+32\pi ^2-32}{(\pi ^2+4)^4}=0.454169\ldots . \end{aligned}$$ Our proof of the above results is based on using the triangle inequality for the maximal angle between subspaces and on employing the a priori generic $\sin 2\theta $ estimate for the variation of a spectral subspace. As an example, the boundedly perturbed quantum harmonic oscillator is discussed.     

Tilting Theory and Functor Categories II. Generalized Tilting

In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $\mathrm{Mod}(\mathcal{C})$ , from a skeletally small preadditive category $\mathcal{C}$ to the category of abelian groups, initiated in [15]. We introduce the notion of a generalized tilting category $\mathcal{T}$ , and we concentrate here on extending Happel’s theorem to $\mathrm{Mod}(\mathcal{C})$ ; more specifically, we prove that there is an equivalence of triangulated categories $\mathcal{D}^{b}( \mathrm{Mod}(\mathcal{C}))\cong \mathcal{D}^{b}(\mathrm{Mod}(\mathcal{T}))$ . We then add some restrictions on our category $\mathcal{C}$ , in order to obtain a version of Happel’s theorem for the categories of finitely presented functors. We end the paper proving that some of the theorems for artin algebras, relating tilting with contravariantly finite categories proved in Auslander and Reiten (Adv Math 12(3):306–366, 1974; Adv Math 86(1):111–151, 1991), can be extended to the category of finitely presented functors $\mathrm{mod}(\mathcal{C})$ , with $\mathcal{C}$ a dualizing variety.     

On the largest conjugacy class length of a finite group

Let $G$ be a finite group and $\mathrm{bcl}(G)$ the largest conjugacy class length of $G$ . In this note we slightly improve He and Shi’s lower bound for $\mathrm{bcl}(G)$ , showing that $|\mathrm{bcl}(G)|\ge p^{\frac{1}{p}}(|G:O_{p}(G)|_{p})^{\frac{p-1}{p}}$ .     

Macdonald polynomials and BGG reciprocity for current algebras

We study the category $\mathcal I _{\mathrm{gr }}$ of graded representations with finite-dimensional graded pieces for the current algebra $\mathfrak{g }\otimes \mathbf{C }[t]$ where $\mathfrak{g }$ is a simple Lie algebra. This category has many similarities with the category $\mathcal O $ of modules for $\mathfrak{g }$ , and in this paper, we prove an analog of the famous BGG duality in the case of $\mathfrak{sl }_{n+1}$ .     

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 $\mathbb Z /2$ .     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

Quasilinear elliptic and parabolic Robin problems on Lipschitz domains

We prove Hölder continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This includes the p-Laplace operator for all ${p \in (1,\infty)}$ , but also operators with unbounded coefficients. Based on the elliptic result we show that the corresponding parabolic problem is well-posed in the space ${\mathrm{C}(\overline{\Omega})}$ provided that the coefficients satisfy a mild monotonicity condition. More precisely, we show that the realization of the elliptic operator in ${\mathrm{C}(\overline{\Omega})}$ is m-accretive and densely defined. Thus it generates a non-linear strongly continuous contraction semigroup on ${\mathrm{C}(\overline{\Omega})}$ .