Recently Yu. Bilu, P. Habegger and L. Kühne proved that no singular modulus can be a unit in the ring of algebraic integers. In this paper we study for which sets S of prime numbers there is no singular modulus that is an S-unit. Here we prove that if S is the set of all primes p congruent to 1 modulo 3, no singular modulus is an S-unit. We then give some remarks on the general case and we study the

We relate trimmed sums of twists in cylinders along a typical Teichmüller geodesic to the area Siegel–Veech constant.

Riemannian geodesic orbit spaces (G/H, g) are natural generalizations of symmetric spaces, defined by the property that their geodesics are orbits of one-parameter subgroups of G. We study the geodesic orbit spaces of the form (G/S, g), where G is a compact, connected, semisimple Lie group and S is abelian. We give a simple geometric characterization of those spaces, namely that they are naturally

We show that any p-form on the smooth locus of a normal complex space extends to a resolution of singularities, possibly with logarithmic poles, as long as \(p \le \mathrm {codim}_{X}({X}_{\mathrm {sg}}) - 2\). A stronger version of this result, allowing no poles at all, is originally due to Flenner. Our proof, however, is not only completely different, but also shorter and technically simpler. We

We consider the following double phase problem: $$\begin{aligned} -\mathrm {div}(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u)+V(x)|u|^{\alpha -2}u=f(x,u),\quad \mathrm {in}\ \mathbb {R}^{N}, \end{aligned}$$ where \(N\ge 2,\) and \(\frac{q}{p}<1+\frac{1}{N}, a:\mathbb {R}^{N}\rightarrow [0,+\infty )~\mathrm {is~Lipschitz~continuous.}\) The Ambrosetti–Rabinowitz type condition, that is so-called

We show that a stratified submersion between stratified semialgebraic sets is locally \({{\mathcal {C}}}^p\)-semialgebraically trivial outside the set of stratified generalized critical values.

In this paper we study the existence of solution for a class of elliptic problem in whole \(\mathbb {R}^N\) without the well known Ambrosetti–Rabinowitz condition. Here, we do not assume any monotonicity condition on f(s)/s for \(s>0\).

In this paper, we prove that two linearly full holomorphic curves in a hyperquadric \(Q_n\), \(n\ge 2\), are congruent if their first fundamental forms and all kth covariant derivatives of the second fundamental forms, \(k=0,1,\ldots ,[\frac{|n-3|}{2}]\), are all the same.

In this text we show that one can generalize results showing that \(\mathrm {CH}^2(X)\), for various Severi–Brauer varieties X, is sometimes torsion free. In particular we show that for any pair of odd integers (n, m), with m dividing n and sharing the same prime factors, one can find a central simple k-algebra A of index n and exponent m that moreover has \(\mathrm {CH}^2(X)\) torsion free for \(X=\mathrm

We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension \(\le 6\), every nice nilpotent Lie group of dimension \(\le 7\) and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals

We prove that any smooth irreducible supersingular representation with central character of \({\text {GL}}_2(F)\) is never of finite presentation when F is a finite field extension of \(\mathbb {Q}_p\) such that \(F\ne \mathbb {Q}_p\), extending a result of Schraen in (J Reine Angew Math (Crelle’s J) 2015(704):187–208, 2015) for quadratic extensions.

Let M be a projective fine moduli space of stable sheaves on a smooth projective variety X with a universal family \({\mathcal {E}}\). We prove that in four examples, \({\mathcal {E}}\) can be realized as a complete flat family of stable sheaves on M parametrized by X, which identifies X with a smooth connected component of some moduli space of stable sheaves on M.

We extend the Altmann–Mavlyutov construction of homogeneous deformations of affine toric varieties to the case of toric pairs \((X, \partial X)\), where X is an affine or projective toric variety and \(\partial X\) is its toric boundary. As an application, we generalise a result due to Ilten to the case of Fano toric pairs.

For odd \(N\ge 5\), we establish a short exact sequence about motivic double zeta values \(\zeta ^{\mathfrak {m}}(r,N-r)\) with \(r\ge 3\) odd, \(N-r\ge 2\). From this we classify all the relations among depth-graded motivic double zeta values \(\zeta ^{\mathfrak {m}}(r,N-r)\) with \(r\ge 3\) odd, \(N-r\ge 2\). As a corollary, we confirm a conjecture of Zagier on the rank of a matrix which concerns

We determine the occurrence and explicitly describe the theta lifts on all levels of all the irreducible generic representations for the dual pair of groups \((\text {Sp}_{2n}, \text {O}(V))\) defined over a local nonarchimedean field \({\mathbb {F}}\) of characteristic 0. As a direct application of our results, we are able to produce a series of non-generic unitarizable representations of these groups

We characterize a k-th accumulation point of pseudo-effective thresholds of n-dimensional varieties as certain invariant associates to a numerically trivial pair of an \((n-k)\)-dimensional variety. This characterization is applied towards Fujita’s log spectrum conjecture for large k.

Suppose \(F:{\mathcal {D}}(X)\rightarrow {\mathcal {T}}\) is an exact functor from the bounded derived category of coherent sheaves on a smooth projective variety X to a triangulated category \({\mathcal {T}}\). If F possesses left and right adjoints, then the Bondal–Orlov criterion gives a simple way of determining if F is fully faithful. We prove a natural extension of this theorem to the case when

For a prime \(p>2\), let G be a semi-simple, simply connected, split Chevalley group over \({\mathbb {Z}}_p\), G(1) be the first congruence kernel of G and \(\Omega _{G(1)}\) be the mod-p Iwasawa algebra defined over the finite field \({\mathbb {F}}_p\). Ardakov et al. (Adv Math 218: 865–901, 2008) have shown that if p is a “nice prime ” (\(p \ge 5\) and \(p \not \mid (n+1)\) if the Lie algebra of

Let G be \(A_5\), \(A_4\), or a finite group with cyclic Sylow 2 subgroup. We show that every closed smooth G manifold M has a strongly algebraic model. This means, there exist a nonsingular real algebraic G variety X which is equivariantly diffeomorphic to M and all G vector bundles over X are strongly algebraic. Making use of improved blow-up techniques and the literature on equivariant bordism theory

We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of the geometric Galois groups as subgroups of automorphism groups of regular trees, in terms of iterated wreath products. Using results on the reduction of dynamical Belyi maps modulo certain primes, we obtain

Inspired by the Gan–Gross–Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent cuspidal representations of orthogonal and symplectic groups over finite fields.

We study the Laplacian flow of a \(\mathrm {G}_2\)-structure where this latter structure is claimed to be locally conformal parallel. The first examples of long time solutions of this flow with the locally conformal parallel condition are given. All of the solutions are ancient and Laplacian soliton of shrinking type. These examples are one-parameter families of locally conformal parallel \(\mathrm

We study entry loci of varieties and their irreducibility from the perspective of X-ranks with respect to a projective variety X. These loci are the closures of the points that appear in an X-rank decomposition of a general point in the ambient space. We look at entry loci of low degree normal surfaces in \({\mathbb {P}}^4\) using Segre points of curves; the smooth case was classically studied by Franchetta

In this paper, we first prove a topological obstruction for a four-dimensional manifold carrying an Einstein metric. More precisely, assume (M, g) is a closed Einstein four-manifold with \(Ric=\rho g\). Denote by K the sectional curvature of M. If \(K\ge \delta \ge \frac{2\rho -\sqrt{5}\left|\rho \right|}{6}\) (or \(K\le \delta \le \frac{2\rho +\sqrt{5}}{6}\)) for some constant \(\delta \), then the

We study existence and Lorentz regularity of distributional solutions to elliptic equations with measurable coefficients and either a convection or a drift first order term. The presence of such a term makes the problem not coercive. The main tools are pointwise estimates of the rearrangements of both the solution and its gradient.

We compute the Picard group of the moduli stack of smooth curves of genus g for \(3\le g\le 5\), using methods of equivariant intersection theory. We base our proof on the computation of some relations in the integral Chow ring of certain moduli stacks of smooth complete intersections. As a byproduct, we compute the cycle classes of some divisors on \(\mathcal {M}_g\).

Soit \(\mathbb {P}^n\) l’espace projectif complexe de dimension n. Nous introduisons une nouvelle fonctionnelle sur l’espace de métriques admissibles sur \(\mathcal {O}(1)\) de \(\mathbb {P}^n\). Nous la comparons à certaines fonctionnelles classiques. Comme conséquence nous retrouvons quelques résultats classiques.

An elliptic curve defined over a number field possesses only a finite number of torsion points defined over the cyclotomic closure of its field of definition. In analogy to the relative version of the Manin–Mumford conjecture stated by Masser and Zannier, we propose a family version of the above statement and prove it under a suitable integrality condition.

By a recent result, it is known that compact homogeneous spaces with co-index of symmetry 4 are quotients of a semisimple Lie group of dimension at most 10. In this paper we determine exactly which ones of these spaces actually admit such a metric. For all the admissible spaces we construct explicit examples of these metrics.

Fix a positive integer g and rational prime p. We prove the existence of a genus g curve \(C/\mathbb {Q}\) such that the mod p representation of its Jacobian is tame by imposing conditions on the endomorphism ring. As an application, we consider the tame inverse Galois problem and are able to realise general symplectic groups as Galois groups of tame extensions of \(\mathbb {Q}\).

Let k be an arbitrary field and X be a reduced k-scheme of finite type. Let \(\gamma \) be an arc on X, not entirely contained in the non-smooth locus of X. We show that the nilpotents in the local ring at \(\gamma \) vanish in the completion. Along the way, we also obtain informations on the connection between torsion differential forms and nilpotent functions on arc schemes in positive characteristic

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli–Kohn–Nirenberg inequality with the same exponent \(n(n\ge 2)\), then it has exactly n-dimensional volume growth. As application, we obtain geometric and topological properties of Alexandrov spaces, Riemannian manifolds and Finsler manifolds which support a Caffarelli–Kohn–Nirenberg inequality

We introduce the analogue of Manin’s universal coacting (bialgebra) Hopf algebra for Poisson algebras. First, for two given Poisson algebras P and U, where U is finite dimensional, we construct a Poisson algebra \(\mathcal {B}(P,\, U)\) together with a Poisson algebra homomorphism \(\psi _{\mathcal {B}(P,\,U)} :P \rightarrow U \otimes \mathcal {B}(P,\, U)\) satisfying a suitable universal property

We define a universal Teichmüller space for locally quasiconformal mappings whose dilatation grows not faster than a certain rate. We prove results of existence and uniqueness for extremal mappings in the generalized Teichmüller class. Further, we analyze the circle maps that arise.

We prove that if S is a smooth reflexive surface in \(\mathbb {P}^3\) defined over a finite field \(\mathbb {F}_q\), then there exists an \(\mathbb {F}_q\)-line meeting S transversely provided that \(q\ge c\deg (S)\), where \(c=\frac{3+\sqrt{17}}{4}\approx 1.7808\). Without the reflexivity hypothesis, we prove the existence of a transverse \(\mathbb {F}_q\)-line for \(q\ge \deg (S)^2\).

We prove the existence of metrics with prescribed Q-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a prescription result for open submanifolds which allow us to conclude that any smooth function

We show that if \(f:X\rightarrow B\) is a Lagrangian fibration from a compact connected hyperkähler (and thus Kähler) manifold X onto a projective normal variety B, then f is locally projective. This answers a question raised by L. Kamenova and strengthens a former result (see Campana in Math Z 252:147–156, 2005, Proposition 2.1), according to which the smooth fibres of f are projective.

By mirror symmetry, the quantum connection of a weighted projective line is closely related to the localized Fourier–Laplace transform of some Gauß–Manin system. Following an article of D’Agnolo, Hien, Morando, and Sabbah, we compute the Stokes matrices for the latter at \(\infty \) for the cases \({\mathbb {P}}(1,3)\) and \({\mathbb {P}}(2,2)\) by purely topological methods. We compare them to the

Assume X is a variety over \({\mathbb {C}}\), \(A \subseteq {\mathbb {C}}\) is a finitely generated \({\mathbb {Z}}\)-algebra and \(X_A\) a model of X (i.e. \(X_A \times _A {\mathbb {C}} \cong X\)). Assuming the weak ordinarity conjecture we show that there is a dense set \(S \subseteq {{\,\mathrm{Spec}\,}}A\) such that for every closed point s of S the reduction of the maximal non-lc ideal filtration

In this paper, using variational methods, we establish the existence and multiplicity of multi-bump solutions for the following nonlinear magnetic Schrödinger equation $$\begin{aligned} -(\nabla +\mathrm{i} A(x))^2 u+(\lambda V(x)+Z(x))u=f(\vert u\vert ^{2})u\quad \text {in}\, \,{\mathbb {R}}^{2}, \end{aligned}$$ where \(\lambda >0\), f(t) is a continuous function with exponential critical growth,

In this article we use the variational method developed by Szulkin (Ann Inst H Poincaré Anal Non Linéire 3:77–109, 1986) to prove the existence of a positive solution for the following logarithmic Schrödinger equation $$\begin{aligned} \left\{ \begin{array}{ll} -\,{\epsilon }^2\Delta u+ V(x)u=u \log u^2, &{}\quad \text{ in } \,\, {\mathbb {R}}^{N}, \\ u \in H^1({\mathbb {R}}^{N}), &{} \\ \end{array}

We show that the second residue map for hermitian Witt groups of an Azumaya algebra A with involution \(\tau \) of first- or second kind over a semilocal Dedekind domain R is surjective. This proves a generalization to hermitian Witt groups of an exact sequence for Witt groups of quadratic forms due to Springer. If R is a complete discrete valuation ring and \(\tau \) is of the first kind we show that

In this paper, we give a precise definition of the analytic \(\gamma \)-factor of irreducible representations of quaternionic unitary groups, which extends a work of Lapid–Rallis.

We define a Gauss map \(\gamma :M\rightarrow \mathbb {S}^{6}\) of an oriented hypersurface M of the unit sphere \(\mathbb {S}^{7}\) and prove that \(\gamma \) is harmonic if and only if M has CMC. Results on the geometry and topology of CMC hypersurfaces of \(\mathbb {S}^{7}\), under hypothesis on the image of \(\gamma \), are then obtained. By a Hopf symmetrization process we define a Gauss map for

We study the periodic orbits problem on energy levels of Tonelli Lagrangian systems over configuration spaces of arbitrary dimension. We show that, when the fundamental group is finite and the Lagrangian has no stationary orbit at the Mañé critical energy level, there is a waist on every energy level just above the Mañé critical value. With a suitable perturbation with a potential, we show that there

Our main question is whether the isomorphism class as a scheme of a curve X over an algebraic closure of a finite field can be reconstructed by the etale fundamental group of X. Tamagawa answered this question affirmatively when the genus of X is 0. In this paper, we will discuss the genus 1 case, and prove a similar result when the genus of X is 1, the cardinality of cusps is 1, and the characteristic

Let \((G,{\mathfrak {X}})\) be a Shimura datum and K a neat open compact subgroup of \(G(\mathbb {A}_f)\). Under mild hypothesis on \((G,{\mathfrak {X}})\), the canonical construction associates a variation of Hodge structure on \(\text {Sh}_K(G,{\mathfrak {X}})(\mathbb {C})\) to a representation of G. It is conjectured that this should be of motivic origin. Specifically, there should be a lift of

Let M be a complex n-dimensional projective manifold in \({\mathbb {P}}^{n+r}\) endowed with the Fubini-Study metric of constant holomorphic sectional curvature 1, \(\sigma \) its second fundamental form, and \(\underline{|\sigma |}^2\) the mean value of the squared length of \(\sigma \) on M. We derive a formula for \(\underline{|\sigma |}^2\) and classify them when \(\underline{|\sigma |}^2\le 2n\)

We introduce the notion of cracked polytope, and – making use of joint work with Coates and Kasprzyk—construct the associated toric variety X as a subvariety of a smooth toric variety Y under certain conditions. Restricting to the case in which this subvariety is a complete intersection, we present a sufficient condition for a smoothing of X to exist inside Y. We exhibit a relative anti-canonical divisor

The classical notion of the Darboux transformation of isothermic surfaces can be generalised to a transformation for conformal immersions. Since a minimal surface is Willmore, we can use the associated \(\mathbb { C}_*\)-family of flat connections of the harmonic conformal Gauss map to construct such transforms, the so-called \(\mu \)-Darboux transforms. We show that a \(\mu \)-Darboux transform of

We introduce tropical complexes, as an enrichment of the dual complex of a degeneration with additional data from non-transverse intersection numbers. We define cycles, divisors, and linear equivalence on tropical complexes, analogous both to the corresponding theories on algebraic varieties and to previous work on graphs and abstract tropical curves. In addition, we establish conditions for the divisor-curve

In this note, we construct nine families of projective complex minimal surfaces of general type having the canonical map of degree 8 and irregularity 0 or 1. For six of these families the canonical system has a non trivial fixed part.

We show various properties of numerical data of an embedded resolution of singularities for plane curves, which are inspired by a conjecture of Nguyen and motivated by a conjecture of Igusa on exponential sums.

Let \(K=\mathbb {Q}(\sqrt{D})\) be a real quadratic field. We consider the additive semigroup \(\mathcal {O}_K^+(+)\) of totally positive integers in K and determine its generators (indecomposable integers) and relations; they can be nicely described in terms of the periodic continued fraction for \(\sqrt{D}\). We also characterize all uniquely decomposable integers in K and estimate their norms. Using

We study semi-algebraic domains associated with symplectic tori and conjecturally identified with spaces of stability conditions on the Fukaya categories of these tori. Our motivation is to test which results from the theory of flat surfaces could hold for more general spaces of stability conditions. The main results concern systolic bounds and volume of the moduli space.

In this note we prove a compactness theorem for the space of connected closed embedded f-minimal surfaces, of bounded f-index, in a simply connected smooth metric measure space \((M^3,g,e^{-f}d\mu )\). This result is similar to that proved by Li and Wei (J Geom Anal 25:421–435, 2015). Li and Wei assumed \(Ric_f\ge k>0\), where \(Ric_f\) is the Bakry-Émery Ricci curvature, and that the embedded f-minimal

Let \(\pi \) be an irreducible generic representation of \(GL_m(F)\), where F is a non-Archimedean local field. In 1990, Jacquet and Shalika established an integral representation of exterior square L-functions of \(GL_m(F)\). Following the works of Cogdell and Piatetski-Shapiro, we characterize exceptional poles in terms of certain Shalika functionals on the derivatives of \(\pi \), where “derivatives”

In this article we introduce a generalization of locally conformally Kähler metrics from complex manifolds to complex analytic spaces with singularities and study which properties of locally conformally Kähler manifolds still hold in this new setting. We prove that if a complex analytic space has only quotient singularities, then it admits a locally conformally Kähler metric if and only if its universal

In this note, we provide a complete classification for the entire area maximizing hypersurfaces with an isolated singularity. We also construct an interesting illustrated example. For the area maximizing hypersurfaces over exterior domains, we obtain a partial result on their asymptotic behavior at infinity. We also establish the solvability of the exterior Dirichlet problems for the area maximizing

We obtain sharp estimate on p-spectral gaps, or equivalently optimal constant in p-Poincaré inequalities, for metric measure spaces satisfying measure contraction property. We also prove the rigidity for the sharp p-spectral gap.