Manuscripta Mathematica最新文献

Let X be a toric variety. We establish vanishing (and non-vanishing) results for the sheaves (R^if_*Omega ^p_{tilde{X}}(log E)), where (f: tilde{X} rightarrow X) is a strong log resolution of singularities with reduced exceptional divisor E. These extend the local vanishing theorem for toric varieties in Mustaţă et al. (J. Inst. Math. Jussieu 19(3):801-819, 2020). Our consideration of these sheaves is motivated by the notion of k-rational singularities introduced by Friedman and Laza (Higher Du Bois and higher rational singularities, 2001). In particular, our results lead to criteria for toric varieties to have k-rational singularities, as defined in Shen et al. (On k-Du Bois and k-rational singularities, 2023).

We give some conditions on a family of abelian covers of ({mathbb P}^1) of genus g curves, that ensure that the family yields a subvariety of ({mathsf A}_g) which is not totally geodesic, hence it is not Shimura. As a consequence, we show that for any abelian group G, there exists an integer M which only depends on G such that if (g >M), then the family yields a subvariety of ({mathsf A}_g) which is not totally geodesic. We prove then analogous results for families of abelian covers of ({tilde{C}}_t rightarrow {mathbb P}^1 = {tilde{C}}_t/{tilde{G}}) with an abelian Galois group ({tilde{G}}) of even order, proving that under some conditions, if (sigma in {tilde{G}}) is an involution, the family of Pryms associated with the covers ({tilde{C}}_t rightarrow C_t= {tilde{C}}_t/langle sigma rangle ) yields a subvariety of ({mathsf A}_{p}^{delta }) which is not totally geodesic. As a consequence, we show that if ({tilde{G}}=(mathbb Z/Nmathbb Z)^m) with N even, and (sigma ) is an involution in ({tilde{G}}), there exists an integer M(N) which only depends on N such that, if ({tilde{g}}= g({tilde{C}}_t) > M(N)), then the subvariety of the Prym locus in ({{mathsf A}}^{delta }_{p}) induced by any such family is not totally geodesic (hence it is not Shimura).

Conformal/anticonformal actions of the quasi-abelian group (QA_{n}) of order (2^n), for (nge 4), on closed Riemann surfaces, pseudo-real Riemann surfaces and closed Klein surfaces are considered. We obtain several consequences, such as the solution of the minimum genus problem for the (QA_n)-actions, and for each of these actions, we study the topological rigidity action problem. In the case of pseudo-real Riemann surfaces, attention was typically restricted to group actions that admit anticonformal elements. In this paper, we consider two cases: either (QA_n) has anticonformal elements or only contains conformal elements.

We study the postcritically finite non-polynomial map (f(x)=frac{1}{(x-1)^2}) over a number field k and prove various results about the geometric (G^{textrm{geom}}(f)) and arithmetic (G^{textrm{arith}}(f)) iterated monodromy groups of f. We show that the elements of (G^{textrm{geom}}(f)) are the ones in (G^{textrm{arith}}(f)) that fix certain roots of unity by assuming a conjecture on the size of (G^{textrm{geom}}_n(f)). Furthermore, we describe exactly for which (a in k) the Arboreal Galois group (G_a(f)) and (G^{textrm{arith}}(f)) are equal.

For n-dimensional weighted Riemannian manifolds, lower m-Bakry–Émery–Ricci curvature bounds with ({varepsilon })-range, introduced by Lu-Minguzzi-Ohta (Anal Geom Metr Spaces 10(1):1–30, 2022), integrate constant lower bounds and certain variable lower bounds in terms of weight functions. In this paper, we prove a Cheng type inequality and a local Sobolev inequality under lower m-Bakry–Émery–Ricci curvature bounds with ({varepsilon })-range. These generalize those inequalities under constant curvature bounds for (m in (n,infty )) to (min (-infty ,1]cup {infty }).

We classify isometric immersions (f:M^{n}rightarrow mathbb {R}^{n+p}), (n ge 5) and (2p le n), with constant Moebius curvature and flat normal bundle.

For an additive polynomial and a positive integer, we define an irreducible smooth representation of a Weil group of a non-archimedean local field. We study several invariants of this representation. We obtain a necessary and sufficient condition for it to be primitive.

We discuss how to resolve generic skew-symmetric and generic symmetric determinantal singularities. The key ingredients are (skew-) symmetry preserving matrix operations in order to deduce an inductive argument.

We study weak approximation for Châtelet surfaces over number fields when all singular fibers are defined over rational points. We consider Châtelet surfaces which satisfy weak approximation over every finite extension of the ground field. We prove many of these results by showing that the Brauer–Manin obstruction vanishes, then apply results of Colliot-Thélène, Sansuc, and Swinnerton-Dyer.

In this note, we derive a Yau type gradient estimate for positive exponentially harmonic functions on Riemannian manifolds with compact boundary. As its application, we obtain a Liouville type theorem.