In this paper we solve the problem of finding integrals of equations determining the Killing tensors on an $n$-dimensional differentiable manifold $M$ endowed with an equiaffine $SL(n,)$-structure and discuss possible applications of obtained results in Riemannian geometry.

We prove that any first order ${ℱ}_2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operator transforming projectable general connections on an $(m_1,m_2,n_1,n_2)$-dimensional fibred-fibred manifold $p=(p,p):(p_Y:Y\to Y)\to (p_M:M\to M)$ into general connections on the vertical prolongation $VY\to M$ of $p:Y\to M$ is the restriction of the (rather well-known) vertical prolongation operator $𝒱$ lifting general connections $\overline{\Gamma }$ on a fibred manifold $Y\to M$ into $𝒱\overline{\Gamma }$ (the vertical prolongation of $\overline{\Gamma }$) on $VY\to M$.

Given any compact manifold $M$, we construct a non-empty open subset $𝒪$ of the space ${\mathrm{Diff}}^1\left(M\right)$ of $C^1$-diffeomorphisms and a dense subset $𝒟\subset 𝒪$ such that the centralizer of every diffeomorphism in $𝒟$ is uncountable, hence non-trivial.

The object of the present paper is to study $\eta$-Ricci solitons on $\eta$-Einstein ${\left(LCS\right)}_n$-manifolds. It is shown that if $\xi$ is a recurrent torse forming $\eta$-Ricci soliton on an $\eta$-Einstein ${\left(LCS\right)}_n$-manifold then $\xi$ is (i) concurrent and (ii) Killing vector field.

Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$. We consider the image ${𝒩}_1(D,X)$ of ${𝒩}_1\left(D\right)$ in ${𝒩}_1\left(X\right)$ under the natural push-forward of $1$-cycles. We show that ${\rho }_X-{\rho }_D\le codim{𝒩}_1(D,X)\le 8$. Moreover if $codim{𝒩}_1(D,X)\ge 3$, then either $X\cong S\times T$ where $S$ is a Del Pezzo surface, or $codim{𝒩}_1(D,X)=3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that ${\rho }_X-{\rho }_T=4$.

Let $ℳf_m$ be the category of $m$-dimensional manifolds and local diffeomorphisms and  let $T$ be the tangent functor on $ℳf_m$. Let $𝒱$ be the category of real vector spaces and linear maps and let ${𝒱}_m$ be the category of $m$-dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors $F:{𝒱}_m\to 𝒱$ admitting $ℳf_m$-natural operators $\tilde{J}$ transforming classical linear connections $\nabla$ on $m$-dimensional manifolds $M$ into almost complex structures $\tilde{J}\left(\nabla \right)$ on $F\left(T\right)M={\bigcup }_{x\in M}F\left(T_{x}M\right)$.

Let $M$ be an $m$-dimensional manifold and $A={𝔻}_k^r/I=ℝ\oplus N_A$ a Weil algebra of height $r$. We prove that any $A$-covelocity $T_x^{A}f\in T_x^{A*}M$, $x\in M$ is determined by its values over arbitrary $max\{\mathrm{width}A,m\}$ regular and under the first jet projection linearly independent elements of $T_x^{A}M$. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result $T^{A*}M\simeq T^{r*}M$ without coordinate computations, which improves and generalizes the partial...

Let (U) denote the algebra of holomorphic functions on an open subset U ⊂ ℂⁿ and Z ⊂ (U) its finite-dimensional vector subspace. By the theory of least spaces of de Boor and Ron, there exists a projection ${}_b$ from the local ring ${}_{n,b}$ onto the space $Z_b$ of germs of elements of Z at b. At a general point b ∈ U its kernel is an ideal and ${}_b$ induces the structure of an Artinian algebra on $Z_b$. In particular, this holds at points where the kth jets of elements of Z form a vector bundle for each k ∈...

We study  how a projectable general connection $\Gamma$ in a 2-fibred manifold $Y^2\to Y^1\to Y^0$  and a general vertical connection $\Theta$ in $Y^2\to Y^1\to Y^0$ induce a general connection $A(\Gamma ,\Theta )$ in $Y^2\to Y^1$.

Let $D$ be an open set of a Stein manifold $X$ of dimension $n$ such that $H^k(D,𝒪)=0$ for $2\le k\le n-1$. We prove that $D$ is Stein if and only if every topologically trivial holomorphic line bundle $L$ on $D$ is associated to some Cartier divisor $𝔡$ on $D$.

Let $X$ be a Stein manifold of complex dimension $n\ge 2$ and $\Omega ⋐X$ be a relatively compact domain with $C^2$ smooth boundary in $X$. Assume that $\Omega$ is a weakly $q$-pseudoconvex domain in $X$. The purpose of this paper is to establish sufficient conditions for the closed range of $\overline{\partial }$ on $\Omega$. Moreover, we study the $\overline{\partial }$-problem on $\Omega$. Specifically, we use the modified weight function method to study the weighted $\overline{\partial }$-problem with exact support in $\Omega$. Our method relies on the $L^2$-estimates by Hörmander (1965) and by Kohn (1973). ...

We study equivalences for category ${𝒪}_p$ of the rational Cherednik algebras ${𝐇}_p$ of type $G_{\ell }\left(n\right)={\left({\mu }_{\ell }\right)}^n⋊{𝔖}_n$: a highest weight equivalence between ${𝒪}_p$ and ${𝒪}_{\sigma \left(p\right)}$ for $\sigma \in {𝔖}_{\ell }$ and an action of ${𝔖}_{\ell }$ on an explicit non-empty Zariski open set of parameters $p$; a derived equivalence between ${𝒪}_p$ and ${𝒪}_{p^{\text{'}}}$ whenever $p$ and $p^{\text{'}}$ have integral difference; a highest weight equivalence between ${𝒪}_p$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$. As a consequence, we confirm special cases...

We describe all ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝Q{T}^{*}$ transforming projectable-projectable classical torsion-free linear connections $\nabla$ on fibred-fibred manifolds $Y$ into classical linear connections $D\left(\nabla \right)$ on cotangent bundles $T^{*}Y$ of $Y$. We show that this problem can be reduced to finding ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝(T^{*},{\otimes }^p{T}^{*}\otimes {\otimes }^{q}T)$ for $p=2$, $q=1$ and $p=3$, $q=0$.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

Let $X_1$ and $X_2$ be two compact strongly pseudoconvex CR manifolds of dimension $2n-1\ge 5$ which bound complex varieties $V_1$ and $V_2$ with only isolated normal singularities in ${ℂ}^{N1}$ and ${ℂ}^{N2}$ respectively. Let $S_1$ and $S_2$ be the singular sets of $V_1$ and $V_2$ respectively and $S_2$ is nonempty. If $2n-N_2-1\ge 1$ and the cardinality of $S_1$ is less than 2 times the cardinality of $S_2$, then we prove that any non-constant CR morphism from $X_1$ to $X_2$ is necessarily a CR biholomorphism. On the other hand, let $X$ be a compact strongly pseudoconvex CR...

Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let ${\pi }_V$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an ${}^m$-measurable subset of ℝⁿ with ${}^m\left(A\right)<\infty$. Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $(V,v)\left|V\in G\right(n,m),v\in V$ such that, for all P ∈ A, one has ${}^{m(n-m)}\left(V\in G(n,m)\left|\right(V,{\pi }_V\left(P\right))\in Z\right)>0$. One can replace “for all P ∈ A” by “for...

For $n=2m\ge 4$, let $\Omega \in {ℝ}^n$ be a bounded smooth domain and $𝒩\subset {ℝ}^L$ a compact smooth Riemannian manifold without boundary. Suppose that $\left\{u_k\right\}\in W^{m,2}(\Omega ,𝒩)$ is a sequence of weak solutions in the critical dimension to the perturbed $m$-polyharmonic maps $$\frac{\mathrm{d}}{\mathrm{d}t}{|}_{t=0}{E}_m\left(\Pi (u+t\xi )\right)=0$$ with ${\Phi }_k\to 0$ in ${\left(W^{m,2}(\Omega ,𝒩)\right)}^{*}$ and $u_k\rightharpoonup u$ weakly in $W^{m,2}(\Omega ,𝒩)$. Then $u$ is an $m$-polyharmonic map. In particular, the space of $m$-polyharmonic maps is sequentially compact for the weak-$W^{m,2}$ topology.

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\&gt;2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if $N^m$ is any smooth and closed m-dimensional manifold with m > n and $T:N^m\to N^m$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions $(M^m;\Phi )$ of the group $G=Z{₂}^k$ on closed smooth m-dimensional manifolds $M^m$ for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F....