Manuscripta Mathematica最新文献

We define Hecke correspondences and Hecke operators on unitary RZ spaces and study their basic geometric properties, including a commutativity conjecture on Hecke operators. Then we formulate the arithmetic fundamental lemma conjecture for the spherical Hecke algebra. We also formulate a conjecture on the abundance of spherical Hecke functions with identically vanishing first derivative of orbital integrals. We prove these conjectures for the case (textrm{U} (1)times textrm{U} (2)).

Given a scheme over a complete discrete valuation ring of mixed characteristic with perfect residue field, the Greenberg transform produces a new scheme over the residue field thicker than the special fiber. In this paper, we will generalize this transform to the case of imperfect residue field. We will then construct a certain kind of cycle class map defined on this generalized Greenberg transform applied to the Néron model of a semi-abelian variety, which takes values in the relatively perfect nearby cycle functor defined by Kato and the second author.

Over the function field of a complex algebraic curve, strong approximation off a non-empty finite set of places holds for the complement of a codimension 2 closed subset in a homogeneous space under a semisimple algebraic group, and for the complement of a codimension 2 closed subset in an affine smooth complete intersection of low degree.

For an effective Cartier divisor D on a scheme X we may form an ({n}^{text {th}}) root stack. Its derived category is known to have a semiorthogonal decomposition with components given by D and X. We show that this decomposition is (2n)-periodic. For (n=2) this gives a purely triangulated proof of the existence of a known spherical functor, namely the pushforward along the embedding of D. For (n > 2) we find a higher spherical functor in the sense of recent work of Dyckerhoff et al. (N-spherical functors and categorification of Euler’s continuants. arXiv:2306.13350, 2023). We use a realization of the root stack construction as a variation of GIT, which may be of independent interest.

We show that the p-adic Siegel–Eisenstein series of general degree attached to two kind of number sequences are both linear combinations of genus theta series of level dividing p, by applying the theory of mod p-power singular forms. As special cases of this result, we derive the results of Nagaoka and Katsurada–Nagaoka.

We show an equation of Euler characteristics of tautological sheaves on Hilbert schemes of points on the fibers of a double point degeneration. This equation resembles a computation of such Euler characteristics via a combinatorial inclusion–exclusion principle. As a consequence, we show the existence of universal polynomials for the Euler characteristics of tautological sheaves on the Hilbert scheme of points on smooth proper algebraic spaces. We apply this result to a conjecture of Zhou on tautological sheaves on Hilbert schemes of points, and reduce the conjecture to the cases of products of projective spaces. Our main tools are good degenerations and algebraic cobordism.

We prove Kuznetsov’s conjecture on the fullness of exceptional collections in the line bundles case for the blow-up of ({mathbb {P}}^{5}) along the image of Segre threefold ({mathbb {P}}^{1}times {mathbb {P}}^{2}) and its hyperplane section.

Certain Iwahori double cosets in the loop group of a reductive group, known under the names of P-alcoves or ((J,w,delta ))-alcoves, play an important role in the study of affine Deligne–Lusztig varieties. For such an Iwahori double coset, its Newton stratification is related to the Newton stratification of an Iwahori double coset in a Levi subgroup. We study this relationship further, providing in particular a bijection between the occurring Newton strata. As an application, we prove a conjecture of Dong-Gyu Lim, giving a non-emptiness criterion for basic affine Deligne–Lusztig varieties.

Petrecca and Röser (Mathematische Zeitschrift 291:395–419, 2018) and Schueth (Ann Global Anal Geom 52:187–200, 2017) had shown that for a generic G-invariant metric g on certain compact homogeneous spaces (M=G/K) (including symmetric spaces of rank 1 and some Lie groups), the spectrum of the Laplace-Beltrami operator (Delta _g) was real G-simple. The same is not true for the complex version of (Delta _g) when there is a presence of representations of complex or quaternionic type. We show that these types of representations induces a (Q_8)-action that commutes with the Laplacian in such way that G-properties of the real version of the operator have to be understood as ((Q_8 times G))-properties on its corresponding complex version.

A space X is W-trivial if for every real vector bundle (alpha ) over X the total Stiefel-Whitney class (w(alpha )) is 1. It follows from a result of Milnor that if X is an orientable closed smooth manifold of dimension 1, 2, 4 or 8, then X is not W-trivial. In this note we completely characterize W-trivial orientable connected closed smooth manifolds in dimensions 3, 5 and 6. In dimension 7, we describe necessary conditions for an orientable connected closed smooth 7-manifold to be W-trivial.