Mathematische Annalen最新文献

We give properties of the real-split retraction of the mixed weak Mumford–Tate domain and prove the Ax–Schanuel property of period mappings arising from variations of mixed Hodge structures. An ingredient in the proof is the definability of the mixed period mapping obtained by Bakker–Brunebarbe–Klingler–Tsimerman. In comparison with preceding results, in the point counting step, we count rational points on definable quotients instead.

We introduce the specialization map in Scholze’s theory of diamonds. We consider v-sheaves that “behave like formal schemes" and call them kimberlites. We attach to them: a reduced special fiber, an analytic locus, a specialization map, a Zariski site, and an étale site. When the kimberlite comes from a formal scheme, our sites recover the classical ones. We prove that unramified p-adic Beilinson–Drinfeld Grassmannians are kimberlites with finiteness and normality properties.

A separable quantum state shared between parties A and B can be symmetrically extended to a quantum state shared between party A and parties (B_1,ldots ,B_k) for every (kin textbf{N}). Quantum states that are not separable, i.e., entangled, do not have this property. This phenomenon is known as “monogamy of entanglement”. We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones (textsf{C}_A) and (textsf{C}_B): The elements of the minimal tensor product (textsf{C}_Aotimes _{min } textsf{C}_B) are precisely the tensors that can be symmetrically extended to elements in the maximal tensor product (textsf{C}_Aotimes _{max } textsf{C}^{otimes _{max } k}_B) for every (kin textbf{N}). Equivalently, the minimal tensor product of two cones is the intersection of the nested sets of k-extendible tensors. It is a natural question when the minimal tensor product (textsf{C}_Aotimes _{min } textsf{C}_B) coincides with the set of k-extendible tensors for some finite k. We show that this is universally the case for every cone (textsf{C}_A) if and only if (textsf{C}_B) is a polyhedral cone with a base given by a product of simplices. Our proof makes use of a new characterization of products of simplices up to affine equivalence that we believe is of independent interest.

The double Dyck path algebra (mathbb {A}_{q,t}) and its polynomial representation first arose as a key figure in the proof of the celebrated Shuffle Theorem of Carlsson and Mellit. A geometric formulation for an equivalent algebra (mathbb {B}_{q,t}) was then given by the second author and Carlsson and Mellit using the K-theory of parabolic flag Hilbert schemes. In this article, we initiate the systematic study of the representation theory of the double Dyck path algebra (mathbb {B}_{q,t}). We define a natural extension of this algebra and study its calibrated representations. We show that the polynomial representation is calibrated, and place it into a large family of calibrated representations constructed from posets satisfying certain conditions. We also define tensor products and duals of these representations, thus proving (under suitable conditions) the category of calibrated representations is generically monoidal. As an application, we prove that tensor powers of the polynomial representation can be constructed from the equivariant K-theory of parabolic Gieseker moduli spaces.

We establish resolvent estimates in (L^q) spaces for the Stokes operator in a bounded (C^1) domain (Omega ) in (mathbb {R}^{d}). As a corollary, it follows that the Stokes operator generates a bounded analytic semigroup in (L^q(Omega ; mathbb {C}^d)) for any (1< q< infty ) and (dge 2). The case of an exterior (C^1) domain is also studied.

In this article, we will study the isoperimetric problem by introducing a mean curvature type flow in the Riemannian manifold endowed with a non-trivial conformal vector field. This flow preserves the volume of the bounded domain enclosed by a star-shaped hypersurface and decreases the area of hypersurface under certain conditions. We will prove the long time existence and convergence of the flow. As a result, the isoperimetric inequality for such a domain is established. Especially, we solve the isoperimetric problem for the star-shaped hypersurfaces in the Riemannian manifold endowed with a closed, non-trivial conformal vector field, a wide class of warped product spaces studied by Guan, Li and Wang is included.

For a prime number (p>2) and a finite extension (F/mathbb {Q}_p), we explain the construction of the difference divisors on the unitary Rapoport–Zink spaces of hyperspecial level over (mathcal {O}_{breve{F}}), and the GSpin Rapoport–Zink spaces of hyperspecial level over (breve{mathbb {Z}}_{p}) associated to a minuscule cocharacter (mu ) and a basic element b. We prove the regularity of the difference divisors, find the formally smooth locus of both the special cycles and the difference divisors, by a purely deformation-theoretic approach.

The purpose of this article is to study Newton polygons of certain abelian L-functions on curves. Let X be a smooth affine curve over a finite field (mathbb {F}_q) and let (rho :pi _1(X) rightarrow mathbb {C}_p^times ) be a finite character of order (p^n). By previous work of the first author, the Newton polygon ({{,mathrm{text {NP}},}}(rho )) lies above a ‘Hodge polygon’ ({{,mathrm{text {HP}},}}(rho )) defined using ramification invariants of (rho ). In this article we study the contact between these two polygons. We prove that ({{,mathrm{text {NP}},}}(rho )) and ({{,mathrm{text {HP}},}}(rho )) share a vertex if and only if a corresponding vertex is shared between the Newton and Hodge polygons of ‘local’ L-functions associated to each ramified point of (rho ). As a consequence, we determine a necessary and sufficient condition for the coincidence of ({{,mathrm{text {NP}},}}(rho )) and ({{,mathrm{text {HP}},}}(rho )).