In this paper, we completely classify 3-dimensional complete lambda-translators with constant squared norm S of the second fundamental form and constant f3 in the Euclidean space IIg4 and the Minkowski space IIg41, where hijare components of the second fundamental form, S = & sum;i,jh2ijand f3 = & sum;i,j,k hijhjkhki.
We give a basis for the vector space generated by rational Hodge classes of type (2, 2) on the Hilbert square of a projective K3 surface, which is a subspace of the singular cohomology ring with rational coefficients: we use Nakajima operators and an algebraic model developed by Lehn and Sorger as main tools. We then obtain a basis of the lattice generated by integral Hodge classes of type (2, 2) on the Hilbert square of a projective K3 surface exploiting lattice theory, a theorem by Qin and Wang and a result by Ellingsrud, Göttsche and Lehn.
Over an algebraically closed field of positive characteristic, we classify smooth Fano threefolds of Picard number one whose anti-canonical linear systems are not very ample. Furthermore, we also prove that an anti-canonically embedded Fano threefold of genus at least five is an intersection of quadrics.
We study $g$-vector cones in a cluster algebra defined from a weighted orbifold of rank $n$ introduced by Felikson, Shapiro and Tumarkin. We determine the closure of the union of the $g$-vector cones. It is equal to $\mathbb{R}^n$ except for a weighted orbifold with empty boundary and exactly one puncture, in which case it is equal to the half space of a certain explicit hyperplane in $\mathbb{R}^n$.
We study K\"ahler manifolds that are (weak) relatives, that is, K\"ahler manifolds which share a (locally isometric) submanifold. In particular, we prove that if two K\"ahler manifolds are weak relatives and one of them is projective, then they are relatives. Moreover, we introduce the notion of strict relatives K\"ahler manifolds and provide several nontrivial examples.
For given positive integers $d$ and $m$, consider the projective klt pairs $(X,B)$ of dimension $d$, of Cartier index $m$, and with semi-ample $K_X+B$ defining a contraction $\pi\colon X\to Z$. We prove that it is not possible in general to write $n(K_X+B)\sim\pi^*A_Z$ for some $n$ depending only on $d$ and $m$, and some Cartier divisor $A_Z$ on $Z$.
The first author and Oguni introduced a wide class of metric spaces, called coarsely convex spaces. It includes Gromov hyperbolic metric spaces, CAT(0) spaces, systolic complexes, proper injective metric spaces. We introduce the notion of free products of metric spaces and show that free products of symmetric geodesic coarsely convex spaces are also symmetric geodesic coarsely convex spaces. As an application, it follows that free products of symmetric geodesic coarsely convex spaces satisfy the coarse Baum-Connes conjecture.
The $t=0$ specialization of the Mimachi-Noumi Cauchy-type identity rewrites certain infinite product in terms of specialized nonsymmetric Macdonald polynomials of type $GL_n$. We interpret the infinite product as a character of the space of functions on a certain matrix space. We show that the space of functions admits a filtration such that the graded pieces are isomorphic to the tensor products of certain generalized global Weyl modules of the Iwahori algebra. We identify the characters of the graded pieces with the terms of the specialized Mimachi-Noumi formula. We conjecture the existence of an analogous filtration on the space of functions on the Iwahori group for all simple Lie algebras and prove the conjecture for $SL_n$. Our construction can be seen as a current algebra extension of the van der Kallen filtration on functions on a Borel subgroup.
In his 1989 paper, Floer established a connection between holomorphic strips with boundary on a Lagrangian $L$ and a small Hamiltonian push-off $L_{f}$, and gradient flow lines for the function $f$. The present paper studies the compactness theory for holomorphic curves $u_{n}$ whose boundary components lie on Hamiltonian perturbations $L_{n}^{1},\dots,L^{N}_{n}$ of a fixed Lagrangian $L$, where each sequence of nearby Lagrangians $L^{j}_{n}$ converges to $L$ as $n\to\infty$. Generalizing earlier work of Oh, Fukaya, Ekholm, and Zhu, we prove that the limit of a sequence of such holomorphic maps is a configuration consisting of holomorphic curves with boundary on $L$ joined by gradient flow lines connecting points on the boundary of holomorphic pieces. The key new result is an exponential estimate analyzing the interface between the holomorphic parts and the gradient flow line parts.
We establish H & ouml;rmander-type L2-estimates for the 8-operators that hold uniformly for all nontrivial flat holomorphic line bundles on compact K & auml;hler manifolds. Our result can be regarded as a 8-version of Ueda's lemma on the operator norm of tech coboundaries for flat line bundles and indeed recovers the original version of Ueda's lemma for compact K & auml;hler manifolds. A partial generalization for (p, 0)-forms on Ricci-flat manifolds is also given.
The present article studies holomorphic isometric embeddings of arbitrary complex Grassmannians into quadrics, generalising results in [13]. The moduli spaces of these embeddings up to gauge and image equivalence are discussed using a generalisation of do Carmo-Wallach theory.
In this article we introduce the space of configurations of commuting elements in a topological group and show that it satisfies rational homological stability for the sequences of unitary, special unitary and symplectic groups. We also prove that it satisfies cohomological rational representation stability with respect to the number of elements in the tuple for finite products of such groups, in particular cohomological rational stability for the space of unordered configurations of commuting elements. Finally we present some computations of cohomology in the unstable range.
This work is dedicated to the development of the theory of Fourier hyperfunctions in one variable with values in a complex non-necessarily metrisable locally convex Hausdorff space E. Moreover, necessary and sufficient conditions are described such that a reasonable theory of E-valued Fourier hyperfunctions exists. In particular, if E is an ultrabornological PLS-space, such a theory is possible if and only if E satisfies the so-called property (PA). Furthermore, many examples of such spaces having (PA) resp. not having (PA) are provided. We also prove that the vector-valued Fourier hyperfunctions can be realized as the sheaf generated by equivalence classes of certain compactly supported E-valued functionals and interpreted as boundary values of slowly increasing holomorphic functions.
Let (L, Q) be a nondegenerate quadratic module over the ring o of integers of a non-Archimedean local field. An ordered basis (psi 1, ... ,psi n) of L over o is called optimal if it attains the Gross-Keating invariant of L. In the first part of this paper, we give a simple characterization of an optimal basis. In the second part, we show that a pair (V', V'') of filtered quadratic spaces over the residue field k is associated to (L, Q). We show that the extended Gross-Keating datum of L is determined by (V', V'').
Let Sigma be a subset of the set of prime numbers that is either equal to the entire set of prime numbers or of cardinality one. In the present paper, we continue our study of the pro-Sigma fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields in which the primes of Sigma are invertible. The focus of the present paper is on applications of the theory developed in previous papers to the theory of tempered fundamental groups, in the style of Andre. These applications are motivated by the goal of surmounting two fundamental technical difficulties that appear in previous work of Andre-namely, (a) the fact that the characterization of the local Galois groups in the global Galois image associated to a hyperbolic curve that is given in earlier work of Andre is proved only for a quite limited class of hyperbolic curves-that is, a class that is far from generic; (b) the proof given in earlier work of Andre of a certain key injectivity result, which is of central importance in establishing the theory of a p-adic local analogue of the well-known global theory of the Grothendieck-Teichmuller group, contains a fundamental gap. In the present paper, we surmount these technical difficulties by introducing the notion of an M-admissible or metric-admissible outer automorphism of the profinite geometric fundamental group of a p-adic hyperbolic curve. Roughly speaking, M-admissible outer automorphisms are outer automorphisms that are compatible with the data constituted by the indices at the various nodes of the special fiber of the p-adic curve under consideration. By combining this notion with combinatorial anabelian results and techniques developed in earlier papers by the authors, together with the theory of cyclotomic synchronization (also developed in earlier papers by the authors), we obtain a generalization of Andre's characterization of the local Galois groups in the global Galois image associated to a hyperbolic curve to the case of arbitrary hyperbolic curves [cf. (a)]. Moreover, by applying the theory of local contractibility of p-adic analytic spaces developed by Berkovich, we show that the techniques developed in the present and earlier papers by the authors allow one to relate the groups of M-admissible outer automorphisms treated in the present paper to the groups of outer automorphisms of tempered fundamental groups of higher-dimensional configuration spaces (associated to the given p-adic hyperbolic curve). These considerations allow one to repair the gap in Andre's proof-albeit at the expense of working with M-admissible outer automorphisms-and hence, to realize the goal of obtaining a local analogue of the Grothendieck-Teichmuller group [cf. (b)].
In this paper, we classify two-dimensional simplicial complexes based on their symbolic powers of the Stanley-Reisner ideal such that they are approximately Cohen-Macaulay for all t >= 1 or for some t >= 9. Moreover, we identify the sharp value oft = 9 for which this condition holds.
We present two generalizations of Roth's approximation theorem on proper adelic curves, assuming some technical conditions on the behavior of the logarithmic absolute values. We illustrate how tightening such assumptions makes our inequalities stronger. As special cases, we recover Corvaja's results for fields admitting a product formula and Vojta's ones for arithmetic function fields.
In this paper, we introduce the complete coprime cyclic quotient singularities. This singularity has a good toric resolution corresponding to the fan obtained by subdivision using only elements of the Hilbert basis. There is a one-to-one correspondence between exceptional divisors of this resolution and the coefficients of the multidimensional continued fraction. In addition, we provide a series of cyclic quotient singularities that hold a weak version of the McKay correspondence on the Hilbert-basis resolution.
The spectrum of a Schrodinger operator with a Gaussian random potential is I. This theorem appeared in a book by Pastur and Figotin and is proven by the theory on small ball probabilities. In this paper, the detail of the proof is discussed, and similar facts are proven for Dirac and Schrodinger operators with a Gaussian random magnetic field.
