The article title was originally published with typographical error.

We investigate the structure of geometrically ruled surfaces whose anti-canonical class is big. As an application we show that the Picard group of a normal projective surface whose anti-canonical class is nef and big is a free abelian group of finite rank.

We show that the triangulated category of bounded constructible complexes on an algebraic variety X over an algebraically closed field is equivalent to the bounded derived category of the abelian category of constructible sheaves on X, extending a theorem of Nori to the case of positive characteristic.

We study the positive Hermitian curvature flow of left-invariant metrics on complex 2-step nilpotent Lie groups. In this setting we completely characterize the long-time behaviour of the flow, showing that normalized solutions to the flow subconverge to a non-flat algebraic soliton, in Cheeger–Gromov topology. We also exhibit a uniqueness result for algebraic solitons on such Lie groups.

We study holomorphic \(\text {GL}_2({\mathbb {C}})\) and \(\text {SL}_2({\mathbb C})\) geometries on compact complex manifolds.

We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from \(\mathbf {PSL}_n(q)\) collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is \(\mathbf {PSp}_{2n}(q)\), \(\mathbf {P}{\varvec{\Omega

We add two new 1-parameter families to the short list of known embedded triply periodic minimal surfaces of genus 4 in \(\mathbb {R}^3\). Both surfaces can be tiled by minimal pentagons with two straight segments and three planar symmetry curves as boundary. In one case (which has the appearance of the CLP surface of Schwarz with an added handle) the two straight segments are parallel, while they are

We classify all surfaces with constant Gaussian curvature K in Euclidean 3-space that can be expressed by an implicit equation of type \(f(x)+g(y)+h(z)=0\), where f, g and h are real functions of one variable. If \(K=0\), we prove that the surface is a surface of revolution, a cylindrical surface or a conical surface, obtaining explicit parametrizations of such surfaces. If \(K\not =0\), we prove that

In this paper, we prove the boundary Lipschitz regularity and the Hopf Lemma by a unified method on Reifenberg domains for fully nonlinear elliptic equations. Precisely, if the domain \(\Omega \) satisfies the exterior Reifenberg \(C^{1,\mathrm {Dini}}\) condition at \(x_0\in \partial \Omega \) (see Definition 1.3), the solution is Lipschitz continuous at \(x_0\); if \(\Omega \) satisfies the interior

A Riemannian Einstein manifold is called an Einstein solvmanifold if there exists a transitive solvable group of isometries. In this short note, we show that every Einstein solvmanifold admits at least one pseudo-Riemannian Einstein metric.

In this work we obtain algebraicity results on special L-values attached to Siegel–Jacobi modular forms. Our method relies on a generalization of the doubling method to the Jacobi group obtained in our previous work, and on introducing a notion of near holomorphy for Siegel–Jacobi modular forms. Some of our results involve also holomorphic projection, which we obtain by using Siegel–Jacobi Poincaré

We show that a perfect obstruction theory for a \(\mathbb {G}_\text {m}\)-gerbe determines a semi-perfect obstruction theory for its base, which is perfect if the gerbe is quasi-compact and affine-pointed. These results streamline the construction of a semi-perfect obstruction theory for the base, and allow us to relate virtual classes of the gerbe and its base.

Special units are a sort of predecessor of Euler systems, and they are mainly used to obtain annihilators for class groups. So one is interested in finding as many special units as possible (actually we use a technical generalization called “semispecial”). In this paper we show that in any abelian field having a real genus field in the narrow sense all Washington units are semispecial, and that a slightly

Mustafin varieties are flat degenerations of projective spaces, induced by a set of lattices in a vector space over a non-archimedean field. They were introduced by Mustafin (Math USSR-Sbornik 34(2):187, 1978) in the 70s in order to generalise Mumford’s groundbreaking work on the unformisation of curves to higher dimension. These varieties have a rich combinatorial structure as can be seen in pioneering

Let X be an irreducible complex affine variety of dimension greater than one and let \(f:X \rightarrow \mathbb {C}^m\) be a polynomial mapping. Let \({|}*{|}\) be a semialgebraic norm on \(\mathbb {C}^m.\) Then for R large enough the sets \(f^{-1}(B_R), f^{-1}(S_R), X{\setminus } f^{-1}(B_R)\) are all connected, where \(B_R=\{ z\in \mathbb {C}^m : |z|\le R\}\) and \(S_R=\{ z\in \mathbb {C}^m : |z|

We study contact loci sets of arcs and the behavior of Hironaka’s order function defined in constructive Resolution of singularities. We show that this function can be read in terms of the irreducible components of the contact loci sets at a singular point of an algebraic variety.

By a theorem of Banagl–Chriestenson, intersection spaces of depth one pseudomanifolds exhibit generalized Poincaré duality of Betti numbers, provided that certain characteristic classes of the link bundles vanish. In this paper, we show that the middle-perversity intersection space of a depth one Witt space can be completed to a rational Poincaré duality space by means of a single cell attachment,

For a family of varieties, we prove that the alternating sum of the traces of “local” monodromy acting on the \(\ell \)-adic étale cohomology groups of the generic fiber is an integer that is independent of \(\ell \). In the course of the proof, we also establish a result on fixed points.

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}

The augmented moduli space \(\widehat{\mathcal M}_n\) is a compactification of moduli space \(\mathcal M_n\) obtained by adding stable hyperbolic surfaces. The different topological types of the added stable surfaces produce a stratification of \(\partial \widehat{\mathcal M}_n\). Let \(\mathcal {P}_n\subset {\mathcal M}_n\) be the pyramidal locus in moduli space, i.e., the set of hyperbolic surfaces