Search

We consider the matrix for the Satake isomorphism with respect to natural bases. We give a simple proof in the case of Chevalley groups that the matrix coefficients which are not obviously zero are in fact positive numbers. We also relate the matrix coefficients to Kazhdan–Lusztig polynomials and to Bernstein functions.

In the previous paper [T] we gave a classification of hermitian forms over the real function fieldk=R(t) and its completionsk v with respect to valuationsv trivial onR. Unfortunately in the local case the arguments given for cases A and D, in general, were not correct. Therefore the resulting local and local-global classifications obtained were incorrect. I would like also to thank Dr. D. Hoffmann for pointing out these mistakes and the referee for useful comments. Here we would like to make necessary corrections to [T]. We keep the same notation used there, except that in the first paragraph,J is not the standard involution of a quaternion division algebraD (with basis {1,i,j,ij}). All hermitian forms will be hermitian forms with respect toJ, with values inD.

This paper deals with Schrodinger operators as they were treated by Kato - Ikebe [3]. It is shown that every element of the domain of definition of the adjoint of such an operator has locally square integrable distributional derivatives up to the order 2. For this the potential of the Schrodinger operator must fulfil a local Stummel condition; if the potential is only locally square integrable a somewhat weaker statement is possible for three dimensions (see remark 2 at the end of this paper).

It is shown that unbranched proper holomorphic maps between pseudoconvex domains with smooth C∞ boundaries, one of which satisfies subelliptic estimates for the\(\bar \partial\)-Neumann problem on (0,1)-forms, extend to unbranched C∞-coverings between the closures of the domains.

In this short paper I describe special situations where a local ring is normally flat along a divisorial prime ideal.

This paper deals with the quantitative Schmidt's subspace theorem and the general from of the second main theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second main theorem in Nevanlinna theory for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with a short proof. Our results improve and generalize the previous results in these directions., Comment: 21 pages. arXiv admin note: text overlap with arXiv:math/0408381 by other authors

This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio conditions. Moreover, graphical contributions are introduced which provide a novel and structured way of understanding multiplicities of floor decomposed curves in $\mathbb{R}^3$. Additionally, so-called condition flows on a tropical curve are used to reflect how conditions imposed on a tropical curve yield different types of edges. This concept is applicable in arbitrary dimension., 36 pages, 15 figures; fixed minor issues, added references

In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum., Comment: This paper comes from the last section of arXiv:1905.10765v1 as an independent paper. Comments are welcome! To appear in manuscripta mathematica

Special units are a sort of predecessor of Euler systems, and they are mainly used to obtain annihilators for class groups. So one is interested in finding as many special units as possible (actually we use a technical generalization called “semispecial”). In this paper we show that in any abelian field having a real genus field in the narrow sense all Washington units are semispecial, and that a slightly weaker statement holds true for all abelian fields. The group of Washington units is very often larger than Sinnott’s group of cyclotomic units. In a companion paper we will show that in concrete families of abelian fields the group of Washington units is much larger than that of Sinnott units, by giving lower bounds on the index. Combining this with the present paper gives strong annihilation results.

Let $${\widetilde{k}}$$ be a fixed cubic field, F a quadratic field and $$L=\widetilde{k}\cdot F$$. In this paper and its companion paper, we determine the density of more or less the ratio of the residues of the Dedekind zeta functions of L, F where F runs through quadratic fields.

We have, in this paper, constructed a new non-geometric embedding of some braid group into the mapping class group of a surface which is induced by the 3-fold branched covering over a disk with some branch points. There is a lift $$\tilde{\beta }_i$$ of the half-Dehn twist $$\beta _i$$ on the disk with some marked points to some surface via the 3-fold covering. We show how this lift $$\tilde{\beta }_i$$ acts on the fundamental group of the surface, and also show that $$\tilde{\beta }_i$$ equals the product of two (inverse) Dehn twists. Two adjacent lifts satisfy the braid relation, hence such lifts induce a homomorphism $$\phi : B_k \rightarrow \Gamma _{g,b}$$ . In this paper we give a concrete description of this homomorphism and show that it is injective by the Birman–Hilden theory. Furthermore, we show that the map on the level of classifying spaces of groups is compatible with the action of little 2-cube operad so that it induces a trivial homomorphism on the stable homology.

For any symplectic manifold $${(M,\omega )}$$ , the set of Hamiltonian diffeomorphisms $${{\text {Ham}}^c(M,\omega )}$$ forms a group and $${{\text {Ham}}^c(M,\omega )}$$ contains an important subset $${{\text {Aut}}(M,\omega )}$$ which consists of time one flows of autonomous(time-independent) Hamiltonian vector fields on M. One might expect that $${{\text {Aut}}(M,\omega )}$$ is a very small subset of $${{\text {Ham}}^c(M,\omega )}$$ . In this paper, we estimate the size of the subset $${{\text {Aut}}(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric which was introduced by Hofer. Polterovich and Shelukhin proved that the complement $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed symplectically aspherical manifold where Conley conjecture is established (Polterovich and Schelukhin in Sel Math 22(1):227–296, 2016). In this paper, we generalize above theorem to general closed symplectic manifolds and general conv! ex symplectic manifolds. So, we prove that the set of all non-autonomous Hamiltonian diffeomorphisms $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed or convex symplectic manifold without relying on the solution of Conley conjecture.

This paper investigates the transitivity of the relation defined by degeneration of finitely generated modules over an associative algebra. It is proved in this paper that if L degenerates to M and M degenerates to N, then $$L^{\oplus e}$$ degenerates to $$N^{\oplus e}$$ for some (but explicitly given) integer $$e>0$$ .

In this paper, we consider some CM fields which we call of dihedral type and compute the Artin L-functions associated to all CM types of these CM fields. As a consequence of this calculation, we see that the Colmez conjecture in this case is very closely related to understanding the log derivatives of certain Hecke characters of real quadratic fields. Recall that the ‘abelian case’ of the Colmez conjecture, proved by Colmez himself, amounts to understanding the log derivatives of Hecke characters of $$\mathbb {Q}$$ (cyclotomic characters). In this paper, we also prove that the Colmez conjecture holds for ‘unitary CM types of signature $$(n-1, 1)$$ ’ and holds on average for ‘unitary CM types of a fixed CM number field of signature $$(n-r, r)$$ ’.

In this paper we assign, under reasonable hypothesis, to each pseudomanifold with isolated singularities a rational Poincar\'e duality space. These spaces are constructed with the formalism of intersections spaces defined by Markus Banagl and are indeed related to them in the even dimensional case., Comment: New version of the paper formerly known as "Spatialization of self dual complexes for spaces with isolated singularities". Final version to appear in manuscripta mathematica

In our preceding paper a generating set of the derived Picard group of a selfinjective Nakayama algebra was constructed combining some previous results for Brauer tree algebras and the technique of orbit categories developed there. In this paper we finish the computation of the derived Picard group of a selfinjective Nakayama algebra.

In this paper, an intersective polynomial is a monic polynomial in one variable with rational integer coefficients, with no rational root and having a root modulo m for all positive integers m. Let G be a finite noncyclic group and let r(G) be the smallest number of irreducible factors of an intersective polynomial with Galois group G over \(\mathbb {Q}\). Let s(G) be smallest number of proper subgroups of G having the property that the union of their conjugates is G and the intersection of all their conjugates is trivial. It is known that \(s(G)\le r(G)\). It is also known that if G is realizable as a Galois group over the rationals, then it is also realizable as the Galois group of an intersective polynomial. However it is not known, in general, whether there exists such a polynomial which is a product of the smallest feasible number s(G) of irreducible factors. In this paper, we study the case \(G=S_n\), the symmetric group on n letters. We prove that for every n, either \(r(S_n)=s(S_n)\) or \(r(S_n)=s(S_n)+1\) and that the optimal value \(s(S_n)\) is indeed attained for all odd n and for some even n. Moreover, we compute \(r(S_n)\) when n is the product of at most two odd primes and we give general upper and lower bounds for \(r(S_n)\).

The aim of this paper is to extend results of Rorlich, Villegas and Yang about the non-vanishing of central L-values of canonical characters of imaginary quadratic fields over the rationals. One of the new ingredients in our paper is the local computations at the place “2”. Therefore, we extend their non-vanishing results to include imaginary quadratic fields of even discriminant. As a consequence, we show that the rank of the Mordell–Weil groups of certain canonical CM elliptic curves are zero.

Let $${\mathcal{O}}$$ be the local ring at a singularity of a geometrically integral algebraic curve defined over a finite field $${\mathbb{F}_q}$$ , and let m be the number of branches centered at the singularity. In a previous paper the second author extended the notion of partial local zeta-functions, by considering for each pair of $${\mathcal O}$$ -ideals $${\mathfrak{a}}$$ and $${\mathfrak{b}}$$ a Poincare series $${P(\mathfrak{a},\mathfrak{b},t_{1},\ldots ,t_{m})}$$ in m variables, which encodes cardinalities of certain finite sets of ideals. To study the behavior of these power series under blow-ups, we generalize the theory by allowing that $${\mathcal{O}}$$ is a semilocal ring of the curve. In this context we establish an Euler product identity, which provides the connection between the local and semilocal theory. We further present a procedure to compute the Poincare series, and illustrate the method by some examples of local rings. Another purpose of this paper is to study the reduction $${\mod q-1}$$ of $${P(\mathfrak{a},\mathfrak{b},t_{1},\ldots,t_{m}),}$$ which becomes a polynomial if m > 1.

International audience; This paper is a complement to the paper " On p-adic differential equations on semistable varieties " by Di Proietto. Given an open variety over a DVR with semistable reduction, the author constructed in that paper a fully faithful algebraization functor from the category of certain log overconvergent isocrystals on the special fiber to the category of modules with regular integrable connection on the generic fiber. In this paper, we prove that, with convenable hypothesis, this functor is a tensor functor whose essential image is closed under extensions and subquotients. As a consequence, we can find suitable Tannakian subcat-egories of log overconvergent isocrystals and of modules with regular integrable connection on which the algebraization functor is an equivalence of Tannakian categories.

We shall prove that the threefold studied in the paper “ Remarks on an Example of K. Ueno” by F. Campana is unirational. This gives an affirmative answer to a question posed in the paper above and also in the book by K. Ueno, “Classification theory of algebraic varieties and compact complex spaces”.

In this paper we prove a refined version of the canonical key formula for projective abelian schemes in the sense of Moret-Bailly, we also extend this discussion to the context of Arakelov geometry. Precisely, let $\pi: A\to S$ be a projective abelian scheme over a locally noetherian scheme $S$ with unit section $e: S\to A$ and let $L$ be a symmetric, rigidified, relatively ample line bundle on $A$. Denote by $\omega_A$ the determinant of the sheaf of differentials of $\pi$ and by $d$ the rank of the locally free sheaf $\pi_*L$. In this paper, we shall prove the following results: (i). there is an isomorphism {\rm det}(\pi_*L)^{\otimes 24}\cong (e^*\omega_A^\vee)^{\otimes 12d} which is canonical in the sense that it is compatible with arbitrary base-change; (ii). if the generic fibre of $S$ is separated and smooth, then there exist positive integer $m$, canonical metrics on $L$ and on $\omega_A$ such that there exists an isometry {\rm det}(\pi_*\bar{L})^{\otimes 2m}\cong (e^*\bar{\omega}_A^\vee)^{\otimes md} which is canonical in the sense of (i). Here the constant $m$ only depends on $g,d$ and is independent of $L$., Comment: 30 pages

In this paper we are concerned with solutions, in particular with the univalent solutions, of the Loewner differential equation associated with non-normalized subordination chains on the Euclidean unit ball B in \({\mathbb{C}^n}\). We also give applications to univalence conditions and quasiconformal extensions to \({\mathbb{C}^n}\) of holomorphic mappings on B. Finally we consider the asymptotical case of these results. The results in this paper are complete generalizations to higher dimensions of well known results due to Becker. They improve and extend previous sufficient conditions for univalence and quasiconformal extension to \({\mathbb{C}^n}\) of holomorphic mappings on B.

In this paper, we investigate the Monge–Ampere equation $$\text{ det }D^2u=f $$ in $${\mathbb {R}}^n_+$$ , where f is bounded, positive and $$f(x)=1+O(|x|^{-\beta })$$ for some $$\beta >2$$ at infinity. If u is a quadratic polynomial on $$\{x_n=0\}$$ and satisfies $$ \mu |x|^2\le u\le \mu ^{-1}|x|^2$$ for some $$00$$ is some constant depending only on $$\beta $$ and n. Meanwhile, the existence and uniqueness of viscosity solutions of the Dirichlet problem with prescribed asymptotic behavior at infinity will be concerned. The condition $$\beta >2$$ is sharp.

Northcott’s book Finite Free Resolutions (1976), as well as the paper (J. Reine Angew. Math. 262/263:205–219, 1973), present some key results of Buchsbaum and Eisenbud (J. Algebra 25:259–268, 1973; Adv. Math. 12: 84–139, 1974) both in a simplified way and without Noetherian hypotheses, using the notion of latent nonzero divisor introduced by Hochster. The goal of this paper is to simplify further the proofs of these results, which become now elementary in a logical sense (no use of prime ideals, or minimal prime ideals) and, we hope, more perspicuous. Some formulations are new and more general than in the references (J. Algebra 25:259–268, 1973; Adv. Math. 12: 84–139, 1974; Finite Free Resolutions 1976) (Theorem 7.2, Lemma 8.2 and Corollary 8.5).

Let F be a p-adic field with odd residual characteristic. This work is the continuation of a previous paper that contains some detailed computations of the doubling integral for irreducible constituents \({(\pi, \mathcal{V}_{\pi})}\) of the genuine unramified principal series of \({\widetilde{Sp}_2(F)}\) using various “good test data”. This paper aims to interpret those results in terms of the non-vanishing of local theta lifts. Assuming a technical condition on order of a particular pole for the family of doubling integrals for \({(\pi, \mathcal{V}_{\pi})}\) , we aim to determine the so-called “dichotomy sign” of \({(\pi, \mathcal{V}_{\pi})}\) .

In this paper we study examples of \({\mathbb P^r}\)-scrolls defined over primitively polarized K3 surfaces S of genus g, which arise from Brill–Noether theory of the general curve in the primitive linear system on S and from some results of Lazarsfeld. We show that such scrolls form an open dense subset of a component \({\mathcal H}\) of their Hilbert scheme; moreover, we study some properties of \({\mathcal H}\) (e.g. smoothness, dimensional computation, etc.) just in terms of \({\mathfrak F_g}\), the moduli space of such K3’s, and Mv(S), the moduli space of semistable torsion-free sheaves of a given rank on S. One of the motivation of this analysis is to try to introducing the use of projective geometry and degeneration techniques in order to studying possible limits of semistable vector-bundles of any rank on a very general K3 as well as Brill–Noether theory of vector-bundles on suitable degenerations of projective curves. We conclude the paper by discussing some applications to the Hilbert schemes of geometrically ruled surfaces introduced and studied in Calabri et al. (Rend Lincei Mat Appl 17(2):95–123, 2006) and Calabri et al. (Rend Circ Mat Palermo 57(1):1–32, 2008).

In this paper, we establish the pointwise boundary differentiability on Reifenberg domains for viscosity solutions of fully nonlinear elliptic equations which extends the result under the usual $$C^{1,\mathrm {Dini}}$$ condition and generalizes the result for linear equations in Huang et al (Manuscr. Math 162(3–4):305–313, 2020). Moreover, our proofs are relatively simple.

In the paper we begin to explore relation between the question of unitarizability of classical p-adic groups, and Arthur packets., Comment: 35 pages

The Kneser–Poulsen conjecture claims that if some balls of Euclidean space are rearranged in such a way that the distances between their centers do not increase, then neither does the volume of the union of the balls. A special case of the conjecture, when the balls move continuously in such a way that the distances between the centers (weakly) decrease during the motion, is known to hold not only in Euclidean, but also in spherical and hyperbolic spaces. In the present paper, we show that this theorem cannot be extended to elliptic space by constructing three smoothly moving congruent balls with centers getting closer to one another in such a way that the volume of the union of the balls strictly increase during the motion. In spite of this counterexample, it is true that n + 1 balls in n-dimensional elliptic space cover maximal volume if the distances between the centers are all equal to the diameter π/2 of the space. The second part of the paper is devoted to the proof of this fact.

In this paper we consider immersions with constant anisotropic mean curvature (CAMC) of a smooth oriented connected and compact surface $$\varSigma $$ , with non-empty boundary $$\partial \varSigma $$ , in a region $$\varOmega $$ such that the boundary $$\partial \varOmega $$ is a rotational surface. We prove that, under a suitable condition on the anisotropic function, the flat disks are the only free boundary CAMC immersions in $$\varOmega $$ . Moreover, we study which disks are stable. Finally, we consider an interesting result that allows us to build a wide variety of examples of Wulff Shape.

In this paper, we study the global F-splitting of varieties admitting an int-amplified endomoprhism. We prove that surfaces admitting an int-amplified endomorphism are of dense globally F-split type and, in particular, of Calabi–Yau type.

In the present paper we study multiplicity results for a Yamabe-type problem proposed by Escobar in 1992. We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, having null scalar curvature and constant mean curvature on the boundary. We use some standard results from the bifurcation theory to prove the existence of an infinite number of conformal classes with at least two non-homothetic Riemannian metrics of null scalar curvature and constant mean curvature on the boundary of the product manifold. In addition, we obtain a convergence result for bifurcating branches.

Let X be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products $$\mathrm {Sym}^{n}X$$ when the cohomology of X is given by exterior products of cohomology classes with odd degree. We obtain an expression for the equivariant mixed Hodge polynomials $$\mu _{X^{n}}^{S_{n}}\left( t,u,v\right) $$ , codifying the permutation action of $$S_{n}$$ as well as its subgroups. This allows us to deduce formulas for the mixed Hodge polynomials of its symmetric products $$\mu _{\mathrm {Sym}^{n}X}\left( t,u,v\right) $$ . These formulas are then applied to the case of linear algebraic groups.

The thesis consists of four papers which all regard the study of critical point theory and its applications to boundary value problems of semilinear elliptic equations. More specifically, let Ω be a domain, and consider a boundary value problem of the form -L u + u = f(x,u) in Ω, and with the boundary condition u=0. L denotes a linear differential operator of second order, and in the papers, it is either the classical Laplacian or the Heisenberg Laplacian defined on the Heisenberg group. The function f is subject to some regularity and growth conditions.Paper I contains an abstract result about nonlinear eigenvalue problems. We give an application to the given equation when L is the classical Laplacian, Ω is a bounded domain,and f is odd in the u variable.In paper II, we study a similar equation, but with Ω being an unbounded domain of N-dimensional Euclidean space. We give a condition on Ω for which the equation has infinitely many weak solutions.In papers III and IV we work on the Heisenberg group instead of Euclidean space, and with L being the Heisenberg Laplacian.In paper III, we study a similar problem as in paper II, and give a condition on a subset Ω of the Heisenberg group for which the given equation has infinitely many solutions. Although the condition on Ω is directly transferred from the Euclidean to the Heisenberg group setting, it turns out that the condition is easier to fulfil in the Heisenberg group than in Euclidean space.In paper IV, we are still on the Heisenberg group, Ω is the whole group, and we study the equation when f is periodic in the x variable. The main result is that also in this case, the equation has infinitely many solutions.

Let (X,L) be a quasi-polarized variety of dim X=n. In the previous paper we gave a new invariant (the i-th sectional geometric genus) of (X,L), which is a generalization of the degree and the sectional genus of (X,L). In this paper we study some properties of the sectional geometric genus, and we consider the i-th sectional geometric genus of some special varieties.

Let $k$ be a field with char $k\neq 2$ and $k$ be not algebraically closed. Let $a\in k\setminus k^2$ and $L=k(\sqrt{a})(x,y)$ be a field extension of $k$ where $x,y$ are algebraically independent over $k$. Assume that $\sigma$ is a $k$-automorphism on $L$ defined by \[ \sigma: \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c(x+\frac{b}{x})+d}{y} \] where $b,c,d \in k$, $b\neq 0$ and at least one of $c,d$ is non-zero. Let $L^{\langle\sigma\rangle}=\{u\in L:\sigma(u)=u\}$ be the fixed subfield of $L$. We show that $L^{\langle\sigma\rangle}$ is isomorphic to the function field of a certain surface in $P^4_k$ which is given as the intersection of two quadrics. We give criteria for the $k$-rationality of $L^{\langle\sigma\rangle}$ by using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Th\'el\`ene., Comment: To appear in Manuscripta Math. The main theorems (old Theorem 1.7 and Theorem 1.8) incorporated into (new) Theorem 1.8. Section 3 and Section 4 interchanged

The paper deals with the following singular fractional problem $$\begin{aligned} \left\{ \begin{array}{lll} M\left( \displaystyle \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right) (-\Delta )^{s} u-\mu \displaystyle \frac{u}{|x|^{2s}}= \lambda f(x)u^{-\gamma }+ g(x){u^{2^*_s-1}}&{}\;\; \text {in}\; \Omega ,\\ u>0&{} \;\; \text {in}\; \Omega ,\\ u=0&{}\;\;\text {in}\;{\mathbb {R}}^N\setminus \Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N$$ is an open bounded domain, with $$0\in \Omega $$ , dimension $$N>2s$$ with $$s\in (0,1)$$ , $$2^*_s=2N/(N-2s)$$ is the fractional critical Sobolev exponent, $$\lambda $$ and $$\mu $$ are positive parameters, exponent $$\gamma \in (0,1)$$ , M models a Kirchhoff coefficient, f is a positive weight while g is a sign-changing function. The main feature and novelty of our problem is the combination of the critical Hardy and Sobolev nonlinearities with the bi-nonlocal framework and a singular nondifferentiable term. By exploiting the Nehari manifold approach, we provide the existence of at least two positive solutions.

This paper is concerned with a borderline case of double phase problems with a finite Radon measure on the right-hand side. We obtain sharp fractional regularity estimates for such non-uniformly elliptic problems.

In this paper we adopt an alternative, analytical approach to Arnol'd problem \cite{A1} about the existence of closed and embedded $K$-magnetic geodesics in the round $2$-sphere $\mathbb S^2$, where $K: \mathbb S^2 \rightarrow \mathbb R$ is a smooth scalar function. In particular, we use Lyapunov-Schmidt finite-dimensional reduction coupled with a local variational formulation in order to get some existence and multiplicity results bypassing the use of symplectic geometric tools such as the celebrated Viterbo's theorem and Bottkoll results., 27 pages. To appear in Manuscripta Mathematica

In this paper we establish a stability result for the nonnegative local weak solutions to $$\begin{aligned} u_t= \text {div} \big (|Dw|^{p-2}Dw\big ) , \quad p>1 \end{aligned}$$ where $$w= \frac{u^\gamma -1}{\gamma }$$ and $$\gamma = \frac{m+p-2}{p-1}$$ , as $$|\gamma |\rightarrow 0$$ .

In this paper we provide a counterexample to a 22-year-old theorem about the structure of the Kauffman bracket skein module of the connected sum of two handlebodies. We achieve this by analysing handle slidings on compressing discs in a handlebody. We find more relations than previously predicted for the Kauffman bracket skein module of the connected sum of handlebodies, when one of them is not a solid torus. Additionally, we speculate on the structure of the Kauffman bracket skein module of the connected sum of two solid tori.

Let X be a set of smooth points in P^2, and I = \oplus_{t >= d} I_t the defining ideal of X. In this paper, we give a set of defining equations for the Rees algebra R(I_{d+1}) of the ideal generated by I_{d+1}. This study give information to completely answer questions on the defining ideals of projective embeddings of the blowup of P^2 along the points in X, which has been studied by many authors. In this paper, we also study the asymptotic behaviour of the Rees algebras R(I_t) of the ideal generated by I_t, as t gets large., 26 pages

In this paper we essentially classify all locally finite Lie algebras with an involution and a compatible root decomposition which permit a faithful unitary highest weight representation. It turns out that these Lie algebras have many interesting relations to geometric structures such as infinite-dimensional bounded symmetric domains and coadjoint orbits of Banach–Lie groups which are strong Kahler manifolds. In the present paper we concentrate on the algebraic structure of these Lie algebras, such as the Levi decomposition, the structure of the almost reductive and locally nilpotent part, and the structure of the representation of the almost reductive algebra on the locally nilpotent ideal.

We study a variety for graded maximal Cohen–Macaulay modules, which was introduced by Dao and Shipman. The main result of this paper asserts that there are only a finite number of isomorphism classes of graded maximal Cohen–Macaulay modules with fixed Hilbert series over Cohen–Macaulay algebras of graded countable representation type.

Given a homogeneous ideal I in a polynomial ring over a field, one may record, for each degree d and for each polynomial $$f\in I_d$$ , the set of monomials in f with nonzero coefficients. These data collectively form the tropicalization of I. Tropicalizing ideals induces a “matroid stratification” on any (multigraded) Hilbert scheme. Very little is known about the structure of these stratifications. In this paper, we explore many examples of matroid strata, including some with interesting combinatorial structure, and give a convenient way of visualizing them. We show that the matroid stratification in the Hilbert scheme of points $$(\mathbb {P}^1)^{[k]}$$ is generated by all Schur polynomials in k variables. We end with an application to the T-graph problem of $$(\mathbb {A}^2)^{[n]};$$ classifying this graph is a longstanding open problem, and we establish the existence of an infinite class of edges.

The aim of this paper is to continue the study started in [10] and to give a topological description of the Milnor fibre at infinity. As an application, we show that the links at infinity of some hypersurfaces diffeomorphic to affine spaces C k , given in [3], are knotted spheres. In the last Section of this paper we give examples which show that the property of being M-tame depends on the algebraic coordinate system of C n , when n≥ 4.

The involutions in this paper are algebra anti-automorphisms of period two. Involutions on endomorphism algebras of finite-dimensional vector spaces are adjoint to symmetric or skew-symmetric bilinear forms, or to hermitian forms. Analogues of the classical invariants of quadratic forms (discriminant, Clifford algebra, signature) have been defined for arbitrary central simple algebras with involution. In this paper it is shown that over certain fields these invariants are sufficient to classify involutions up to conjugation. For algebras of low degree a classification is obtained over an arbitrary field.

A general method for computing irreducible representations of Weyl groups and Iwahori–Hecke algebras was introduced by the first author in [10]. In that paper the representations of the algebras of types An, Bn, Dn and Gn were computed and it is the purpose of this paper to extend these computations to F4. The main goal here is to compute irreducible representations of the Iwahori–Hecke algebra of type F4 by only using information in the character table of the Weyl group.