Mathematische Zeitschrift最新文献

The (textrm{SL}_d)-skein algebra (mathcal {S}^q_{textrm{SL}_d}(S)) of a surface S is a certain deformation of the coordinate ring of the character variety consisting of flat (textrm{SL}_d)-local systems over the surface. As a quantum topological object, (mathcal {S}^q_{textrm{SL}_d}(S)) is also closely related to the HOMFLYPT polynomial invariant of knots and links in ({mathbb {R}}^3). We exhibit a rich family of central elements in (mathcal {S}^q_{textrm{SL}_d}(S)) that appear when the quantum parameter q is a root of unity. These central elements are obtained by threading along framed links certain polynomials arising in the elementary theory of symmetric functions, and related to taking powers in the Lie group (textrm{SL}_d).

We study the structure of mod 2 cohomology rings of oriented Grassmannians (widetilde{{text {Gr}}}_k(n)) of oriented k-planes in ({mathbb {R}}^n). Our main focus is on the structure of the cohomology ring (textrm{H}^*(widetilde{{text {Gr}}}_k(n);{mathbb {F}}_2)) as a module over the characteristic subring C, which is the subring generated by the Stiefel–Whitney classes (w_2,ldots ,w_k). We identify this module structure using Koszul complexes, which involves the syzygies between the relations defining C. We give an infinite family of such syzygies, which results in a new upper bound on the characteristic rank of (widetilde{{text {Gr}}}_k(2^t)), (k<2^t), and formulate a conjecture on the exact value of the characteristic rank of (widetilde{{text {Gr}}}_k(n)). For the case (k=3), we use the Koszul complex to compute a presentation of the cohomology ring (H=textrm{H}^*(widetilde{{text {Gr}}}_3(n);{mathbb {F}}_2)) for (2^{t-1}<nle 2^t-4) for (tge 4), complementing existing descriptions in the cases (n=2^t-i), (i=0,1,2,3) for (tge 3). More precisely, as a C-module, H splits as a direct sum of the characteristic subring C and the anomalous module H/C, and we compute a complete presentation of H/C as a C-module from the Koszul complex. We also discuss various issues that arise for the cases (k>3), supported by computer calculation.

We show that there exists a suitable (C_2)-fixed points functor from calculus with Reality to the orthogonal calculus of Weiss which recovers orthogonal calculus “up to a shift” in an analogous way with the recovery of real topological K-theory from Atiyah’s K-theory with Reality via appropriate (C_2)-fixed points.

Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal singularities and passing through a generic conjugation-invariant configuration of appropriately many points in the plane. We show that an invariant like this is unique: it enumerates real rational three-cuspidal quartics that pass through generically chosen four pairs of complex conjugate points. As a consequence, we show that through any generic configuration of four pairs of complex conjugate points, one can always trace a pair of real rational three-cuspidal quartics.

A conjecture of Bondal–Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We show that the braid group acts transitively on the set of maximal numerically exceptional collections on rational surfaces up to isometries of the Picard lattice and twists with line bundles. Considering the blow-up of the projective plane in up to 9 points in very general position, these results lift to the derived category. More precisely, we prove that, under these assumptions, a maximal numerically exceptional collection consisting of line bundles is a full exceptional collection and any two of them are related by a sequence of mutations and shifts. The former extends a result of Elagin–Lunts and the latter a result of Kuleshov–Orlov, both concerning del Pezzo surfaces. In contrast, we show in concomitant work (Krah in Invent Math 235(3):1009–1018, 2024) that the blow-up of the projective plane in 10 points in general position admits a non-full exceptional collection of maximal length consisting of line bundles.

We consider a Serrin’s type problem in convex cones in the Euclidean space and motivated by recent rigidity results we study the quantitative stability issue for this problem. In particular, we prove both sharp Lipschitz estimates for an (L^2)-pseudodistance and estimates in terms of the Hausdorff distance.

Let G be a compact group with two given subgroups H and K. Let (pi ) be an irreducible representation of G such that its space of H-invariant vectors as well as the space of K-invariant vectors are both one dimensional. Let (v_H) (resp. (v_K)) denote an H-invariant (resp. K-invariant) vector of unit norm in a given G-invariant inner product (langle ~,~ rangle _pi ) on (pi ). We are interested in calculating the correlation coefficient

In this paper, we compute the correlation coefficient of an irreducible representation of the multiplicative group of the p-adic quaternion algebra with respect to any two tori. In particular, if (pi ) is such an irreducible representation of odd minimal conductor with non-trivial invariant vectors for two tori H and K, then its root number (varepsilon (pi )) is (pm 1) and (c(pi text {;}, H, K)) is non-vanishing precisely when (varepsilon (pi ) = 1).

In this paper, we give infinitely many diffeomorphic families of different Weinstein manifolds. The diffeomorphic families consist of well-known Weinstein manifolds which are the Milnor fibers of ADE-type, and Weinstein manifolds constructed by taking the end connected sums of Milnor fibers of A-type. In order to distinguish them as Weinstein manifolds, we study how to measure the number of connected components of wrapped Fukaya categories.

We study the eigenforms of the action of A. Baker’s Hecke operators on the holomorphic elliptic homology of various topological spaces. We prove a multiplicity one theorem (i.e., one-dimensionality of the space of these “topological Hecke eigenforms” for any given eigencharacter) for some classes of topological spaces, and we give examples of finite CW-complexes for which multiplicity one fails. We also develop some abstract “derived eigentheory” whose motivating examples arise from the failure of classical Hecke operators to commute with multiplication by various Eisenstein series, or non-cuspidal holomorphic modular forms in general. Part of this “derived eigentheory” is an identification of certain derived Hecke eigenforms as the obstructions to extending topological Hecke eigenforms from the top cell of a CW-complex to the rest of the CW-complex. Using these obstruction classes together with our multiplicity one theorem, we calculate the topological Hecke eigenforms explicitly, in terms of pairs of classical modular forms, on all 2-cell CW complexes obtained by coning off an element in (pi _n(S^m)) which stably has Adams–Novikov filtration 1.

In this paper we present different ways to parametrize subsets of the space of valuations on K[x] extending a given valuation on K. We discuss the methods using pseudo-Cauchy sequences and approximation types. The method presented here is slightly different than the ones in the literature and we believe that our approach is more accurate.