G-operators, a class of differential operators containing the differential operators of minimal order annihilating Siegel’s G-functions, satisfy a condition of moderate growth called Galochkin condition, encoded by a p-adic quantity, the size. Previous works of Chudnovsky, André and Dwork have provided inequalities between the size of a G-operator and certain computable constants depending among others

We study the partial ordering on isomorphism classes of central simple algebras over a given field F, defined by setting \(A_1 \le A_2\) if \(\deg A_1 = \deg A_2\) and every étale subalgebra of \(A_1\) is isomorphic to a subalgebra of \(A_2\), and generalizations of this notion to algebras with involution. In particular, we show that this partial ordering is invariant under passing to the completion

Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply this result in the context of Fano fibrations and prove a classification theorem for projective bundle and quadric fibration structures on ample subvarieties.

Let (X, L) be a polarized variety over a number field K. We suppose that L is an hermitian line bundle. Let M be a non compact Riemann Surface and \(U\subset M\) be a relatively compact open set. Let \(\varphi :M\rightarrow X(\mathbf{C})\) be a holomorphic map. For every positive real number T, let \(A_U(T)\) be the cardinality of the set of \(z\in U\) such that \(\varphi (z)\in X(K)\) and \(h_L(\varphi

Let X be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products \(\mathrm {Sym}^{n}X\) when the cohomology of X is given by exterior products of cohomology classes with odd degree. We obtain an expression for the equivariant mixed Hodge polynomials \(\mu _{X^{n}}^{S_{n}}\left( t,u,v\right) \), codifying the permutation action of \(S_{n}\)

We prove that for two germs of analytic mappings \(f,g:({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}}^p,0)\) with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family \(\{f_t\}\) of analytic maps with \(f_0=f, f_1=g\) which has a so-called uniform stable radius for

For a given field F of characteristic different from 2 and \(a,b,d\in F^*\) we construct an invariant \(\mathrm{inv}\) for an element \(D\in \,_2\mathrm{Br}(F(\sqrt{a},\sqrt{b},\sqrt{d})/F)\). This invariant takes value in the quotient group $$\begin{aligned} H^3(F,\mu _2)/D\cup {\mathrm{N}_{\mathrm{F}\left( \sqrt{\mathrm{d}}, \sqrt{\mathrm{ab}}\right) /\mathrm{F}}}F\left( \sqrt{d},\sqrt{ab}\right)

In this paper we derive a generating series for the number of cellular complexes known as pavings or three-dimensional maps, on n darts, thus solving an analogue of Tutte’s problem in dimension three. The generating series we derive also counts free subgroups of index n in \(\Delta ^+ = {\mathbb {Z}}_2*{\mathbb {Z}}_2*{\mathbb {Z}}_2\) via a simple bijection between pavings and finite index subgroups

In this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete

This is a follow-up paper of Goldner (Math Z 297(1–2):133–174, 2021), where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in Tyomkin (Adv Math 305:1356–1383, 2017) allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem

We investigate closed subgroups \(G \subseteq \mathrm {Sp}_{2g}(\mathbb {Z}_2)\) whose modulo-2 images coincide with the image \(\mathfrak {S}_{2g + 1} \subseteq \mathrm {Sp}_{2g}(\mathbb {F}_2)\) of \(S_{2g + 1}\) or the image \(\mathfrak {S}_{2g + 2} \subseteq \mathrm {Sp}_{2g}(\mathbb {F}_2)\) of \(S_{2g + 2}\) under the standard representation. We show that when \(g \ge 2\), the only closed subgroup

We study the Mumford–Tate conjecture for hyperkähler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O’Grady’s ten dimensional example, and all of their self-products. For an arbitrary hyperkähler variety whose second Betti number is not 3, we prove the Mumford–Tate conjecture in every codimension

We discuss the fibre bundle of co-adjoint orbits of compact Lie groups, and show how it admits a compatible Kähler structure. The case of the unitary group allows us to reformulate the geometric framework of quantum information theory. In particular, we show that the Fisher information tensor gives rise to a structure that is sufficiently close to a Kähler structure to generalise some classical result

Let k be a field with char \(k\ne 2\) and k be not algebraically closed. Let \(a\in k{\setminus } k^2\) and \(L=k(\sqrt{a})(x,y)\) be a field extension of k where x, y are algebraically independent over k. Assume that \(\sigma \) is a k-automorphism on L defined by $$\begin{aligned} \sigma : \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c\big (x+\frac{b}{x}\big )+d}{y} \end{aligned}$$

In this paper we consider the question of when all Seshadri constants on a product of two isogenous elliptic curves \(E_1\times E_2\) without complex multiplication are integers. By studying elliptic curves on \(E_1\times E_2\) we translate this question into a purely numerical problem expressed by quadratic forms. By solving that problem, we show that all Seshadri constants on \(E_1\times E_2\) are

We calculate Chow quotients of some families of symmetric T-varieties. In complexity two we obtain new examples of Kähler–Einstein metrics by bounding the symmetric alpha invariant of their orbifold quotients. As an additional application we determine the homeomorphism class of the orbit space of the compact torus action.

The paper deals with the following singular fractional problem $$\begin{aligned} \left\{ \begin{array}{lll} M\left( \displaystyle \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right) (-\Delta )^{s} u-\mu \displaystyle \frac{u}{|x|^{2s}}= \lambda f(x)u^{-\gamma }+ g(x){u^{2^*_s-1}}&{}\;\; \text {in}\; \Omega ,\\ u>0&{} \;\; \text {in}\; \Omega ,\\ u=0&{}\;\;\text {in}\;{\mathbb {R}}^N\setminus

We consider boundary blowup problems for k-curvature equations of the form \(H_k[u] = f(u)g(|Du|)\) in a bounded smooth domain \(\Omega \subset \mathbb {R}^n\), where f(s) behaves like \(s^p\) as \(s \rightarrow \infty \) and g(t) behaves like \(t^{-q}\) as \(t \rightarrow \infty \). We obtain the exact principal blowup rate of a solution u near the boundary \(\partial \Omega \) under some conditions

In this sequel to our article on Hopf hypersurfaces of the homogeneous NK (nearly Kähler) manifold \({\mathbf {S}}^3\times {\mathbf {S}}^3\) (Hu and Yao in Ann Mat Pura Appl, 199:1147–1170, 2020), we continue working on the problem of determining the Hopf hypersurfaces of this ambient space for which the holomorphic distributions are preserved by the almost product structure of \({\mathbf {S}}^3\times

Let F be a CM field with totally real subfield \(F^+\) and let \(\pi \) be a C-algebraic cuspidal automorphic representation of the unitary group \({{\,\mathrm{U}\,}}(a,b)(\mathbf {A}_{F^+})\), whose archimedean components are discrete series or non-degenerate limit of discrete series representations. We attach to \(\pi \) a Galois representation \(R_\pi :{{\,\mathrm{Gal}\,}}(\overline{F}/{F}^+)\rightarrow

This note is an erratum pointing out an error in Theorem 7.1 in the paper [4] which was published in Manuscripta Math. in the year 2000.

By the unfolding method, Rankin–Selberg L-functions for \({{\,\mathrm{GL}\,}}(n)\times {{\,\mathrm{GL}\,}}(n^{\prime })\) can be expressed in terms of period integrals. These period integrals actually define invariant forms on tensor products of the relevant automorphic representations. By the multiplicity-one theorems due to Sun–Zhu and Chen–Sun such invariant forms are unique up to scalar multiples

We investigate curved flats in Lie sphere geometry. We show that in this setting curved flats are in one-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces related by Darboux transformation.

Minimal algebraic surfaces of general type X such that \(K^2_X=2\chi ({\mathcal {O}}_X)-6\) or \(K^2_X=2\chi ({\mathcal {O}}_X)-5\) are called Horikawa surfaces. In this note we study \({\mathbb {Z}}^2_2\)-actions on Horikawa surfaces. The main result is that all the connected components of Gieseker’s moduli space of canonical models of surfaces of general type with invariants satisfying these relations

This paper is concerned with a borderline case of double phase problems with a finite Radon measure on the right-hand side. We obtain sharp fractional regularity estimates for such non-uniformly elliptic problems.

Over a global field any finite number of central simple algebras of exponent dividing m is split by a common cyclic field extension of degree m. We show that the same property holds for function fields of 2-dimensional excellent schemes over a henselian local domain of dimension one or two with algebraically closed residue field.

We study the depth properties of certain direct image sheaves on normal varieties. Let \(f: Y\rightarrow X\) be a proper morphism of relative dimension d from a smooth variety onto a normal variety such that the preimage E of the singular locus of X is a divisor. We show that for any integer \(m>0\), the higher direct image \(R^df_*\omega ^{\otimes m}_Y(aE)\) modulo the torsion subsheaf is \(S_2\)

In this paper we adopt an alternative, analytical approach to Arnol’d problem [4] about the existence of closed and embedded K-magnetic geodesics in the round 2-sphere \({\mathbb {S}}^2\), where \(K: {\mathbb {S}}^2 \rightarrow {\mathbb {R}}\) is a smooth scalar function. In particular, we use Lyapunov-Schmidt finite-dimensional reduction coupled with a local variational formulation in order to get

In this paper we establish a stability result for the nonnegative local weak solutions to $$\begin{aligned} u_t= \text {div} \big (|Dw|^{p-2}Dw\big ) , \quad p>1 \end{aligned}$$ where \(w= \frac{u^\gamma -1}{\gamma }\) and \(\gamma = \frac{m+p-2}{p-1}\), as \(|\gamma |\rightarrow 0\).

Differential systems of pure Gaussian type are examples of D-modules on the complex projective line with an irregular singularity at infinity, and as such are subject to the Stokes phenomenon. We employ the theory of enhanced ind-sheaves and the Riemann–Hilbert correspondence for holonomic D-modules of D’Agnolo and Kashiwara to describe the Stokes phenomenon topologically. Using this description, we

In this paper, we study variational solutions to parabolic equations of the type \(\partial _t u - \mathrm {div}_x (D_\xi f(Du)) + D_ug(x,u) = 0\), where u attains time-independent boundary values \(u_0\) on the parabolic boundary and f, g fulfill convexity assumptions. We establish a Haar-Rado type theorem: If the boundary values \(u_0\) admit a modulus of continuity \(\omega \) and the estimate \(|u(x

We study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for

For a bounded smooth domain \(\Omega \subset {\mathbb {R}}^N\) with \(N\ge 2\), we establish a weighted and an anisotropic version of Sobolev inequality related to the embedding \(W_{0}^{1,p}(\Omega )\hookrightarrow L^q(\Omega )\) for \(1

Let X be a CR manifold with transversal, proper CR action of a Lie group G. We show that the quotient X/G is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorizes uniquely over a holomorphic map on X/G. We then use this result and complex geometry to prove an embedding theorem for (non-compact) strongly

We prove that actions of complex reductive Lie groups on a complex compact manifold are locally extendable to its Kuranishi family. This can be seen as an analogue of Rim’s result (see Rim in Trans Am Math Soc 257(1):217–226, 1980) in the analytic setting.

For every even number n, and every n-dimensional smooth hypersurface of \({\mathbb {P}}^{n+1}\) of degree d, we compute the periods of all its \(\frac{n}{2}\)-dimensional complete intersection algebraic cycles. Furthermore, we determine the image of the given algebraic cycle under the cycle class map inside the De Rham cohomology group of the corresponding hypersurface in terms of its Griffiths basis

Let \(\lambda (t)\) be the first eigenvalue of \(-\Delta +aR\, (a>0)\) under the backward Ricci flow on locally homogeneous 3-manifolds, where R is the scalar curvature. In the Bianchi case, we get the upper and lower bounds of \(\lambda (t)\). In particular, we show that when the the backward Ricci flow converges to a sub-Riemannian geometry after a proper re-scaling, \(\lambda ^{+}(t)\) approaches

In this paper we provide a counterexample to a 22-year-old theorem about the structure of the Kauffman bracket skein module of the connected sum of two handlebodies. We achieve this by analysing handle slidings on compressing discs in a handlebody. We find more relations than previously predicted for the Kauffman bracket skein module of the connected sum of handlebodies, when one of them is not a solid

In this paper, we prove that for the oper stratification of the de Rham moduli space \(M_{\mathrm {dR}}(X,r)\), the closed oper stratum is the unique minimal stratum with minimal dimension \(r^2(g-1)+g+1\), and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum.

In this paper, we complete Jantzen’s algorithm to compute the highest derivatives of irreducible representations of p-adic odd special orthogonal groups or symplectic groups. As an application, we give some examples of the Langlands data of the Aubert duals of irreducible representations, which are in the integral reducibility case.

Let p be an odd prime and K an imaginary quadratic field where p splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a p-ordinary elliptic curve over the anticyclotomic \({\mathbb {Z}}_p\)-extension of K does not admit any proper \(\Lambda \)-submodule of finite index, where \(\Lambda \) is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer

We study the existence of the Green function for an elliptic system in divergence form \(-\nabla \cdot a\nabla \) in \({\mathbb {R}}^d\), with \(d>2\). The tensor field \(a=a(x)\) is only assumed to be bounded and \(\lambda \)-coercive. For almost every point \(y \in {\mathbb {R}}^d\), the existence of a Green’s function \(G(\cdot , y)\) centered in y has been proven in Conlon et al. (Calc Var PDEs

In this work, we verify all the conjectural formulas for the Mahler measure of the Laurent polynomial $$\begin{aligned} \left( X+\frac{1}{X}\right) ^{2}\left( Y+\frac{1}{Y}\right) ^{2}(1+Z)^{3}Z^{-2}-s \end{aligned}$$ parametrized by s posed by Samart using properties of spherical theta functions, and show that when s is induced by a CM point, these Mahler measures are all expressible in terms of special

We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms \(\bmod \, \ell \), partially answering a question of Bellaïche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto x which do not vanish \(\bmod \, \ell \). We also propose a precise conjecture as a possible answer to this question. Further, we prove several

We study a variety for graded maximal Cohen–Macaulay modules, which was introduced by Dao and Shipman. The main result of this paper asserts that there are only a finite number of isomorphism classes of graded maximal Cohen–Macaulay modules with fixed Hilbert series over Cohen–Macaulay algebras of graded countable representation type.

A nonlocal curvature flow is investigated to evolve compact locally convex hypersurfaces in the Euclidean space \({\mathbb {E}}^{n+1}\). It is shown that the flow exists globally in all dimensions and deforms the evolving curve into an m-fold circle in the plane if \(n=1\) and drives the evolving hypersurface into a Euclidean sphere if \(n>1\).

In this paper, we develop the rudiments of a tropical homology theory. We use the language of “triples” and “systems” to simultaneously treat structures from various approaches to tropical mathematics, including semirings, hyperfields, and super tropical algebra. We enrich the algebraic structures with a negation map where it does not exist naturally. We obtain an analogue to Schanuel’s lemma which

We investigate the number of straight lines contained in a K3 quartic surface X defined over an algebraically closed field of characteristic 3. We prove that if X contains 112 lines, then X is projectively equivalent to the Fermat quartic surface; otherwise, X contains at most 67 lines. We improve this bound to 58 if X contains a star (ie four distinct lines intersecting at a smooth point of X). Explicit

Let \({\mathbb {K}}={\mathbb {R}},\,{\mathbb {C}}\), the field of reals or complex numbers and \({\mathbb {H}}\), the skew \({\mathbb {R}}\)-algebra of quaternions. We study the homotopy nilpotency of the loop spaces \(\Omega (G_{n,m}({\mathbb {K}}))\), \(\Omega (F_{n;n_1,\ldots ,n_k}({\mathbb {K}}))\), and \(\Omega (V_{n,m}({\mathbb {K}}))\) of Grassmann \(G_{n,m}({\mathbb {K}})\), flag \(F_{n;n_1

Given a homogeneous ideal I in a polynomial ring over a field, one may record, for each degree d and for each polynomial \(f\in I_d\), the set of monomials in f with nonzero coefficients. These data collectively form the tropicalization of I. Tropicalizing ideals induces a “matroid stratification” on any (multigraded) Hilbert scheme. Very little is known about the structure of these stratifications

We study the nonlocal nonlinear problem $$\begin{aligned} \left\{ \begin{array}[c]{lll} (-\Delta )^s u = \lambda f(u) &{} \text{ in } \Omega , \\ u=0&{}\text{ on } \mathbb {R}^N{\setminus }\Omega , \quad (P_{\lambda }) \end{array} \right. \end{aligned}$$ where \(\Omega \) is a bounded smooth domain in \(\mathbb {R}^N\), \(N>2s\), \(0

We give a new proof of the well-known classification of finite subgroups of \({\text {SL}}(2,\mathbb C)\), that generalizes to the three dimensional case of \({\text {SL}}(3,\mathbb C)\); recall the geometric proof, based on study of the motions of the Platonic solids, that does not seem generalizable to higher \({\text {SL}}(n)\) nor to other fields, but gives geometric intuition; use the classification

We describe a natural q-deformation of Fock and Goncharov’s canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type A. We then describe an extension of this construction involving a cluster variety called the symplectic double.

Let S be a regular surface endowed with a very ample line bundle \(\mathcal O_S(h_S)\). Taking inspiration from a very recent result by D. Faenzi on K3 surfaces, we prove that if \({\mathcal O}_S(h_S)\) satisfies a short list of technical conditions, then such a polarized surface supports special Ulrich bundles of rank 2. As applications, we deal with general embeddings of regular surfaces, pluricanonically

In the proof of Theorem 1.1 in Matsubara (2019), there are two mistakes that are explained in Remark 4, and therefore the vanishing of the first and the second cohomologies in Theorem 1.1 is still open.

We give necessary and sufficient conditions for log smoothness of a proper regular arithmetic surface with smooth geometrically connected generic fibre over a discrete valuation ring with perfect residue field. As an application, we recover known criteria for log smooth reduction of minimal normal crossings models of curves.

We describe the structure of the supersingular locus of a Shimura variety for a quaternionic unitary similitude group of degree 2 over a ramified odd prime p if the level at p is given by a special maximal compact open subgroup. More precisely, we show that such a locus is purely 2-dimensional, and every irreducible component is birational to the Fermat surface. Furthermore, we have an estimation of

We construct a $k[[Q]]$-linear predifferential graded Lie algebra $L^*_{X/S}$ associated to a log smooth and saturated morphism $f: X \rightarrow S$ and prove that it controls the log smooth deformation functor. This provides a geometric interpretation of a construction by Chan-Leung-Ma whereof $L^*_{X/S}$ is a purely algebraic version. Our proof crucially relies on studying deformations of the Gerstenhaber

In this paper, using computations done through the LiE software, we compare the tensor product of irreducible selfdual representations of the special linear group with those of classical groups to formulate some conjectures relating the two. In the process a few other phenomenon present themselves which we record as questions. More precisely, under the natural correspondence of irreducible finite dimensional

This article complements the Lorentzian Aubry–Mather Theory in Suhr (Geom Dedicata 160:91–117, 2012; J Fixed Point Theory Appl 21:71, 2019) by giving optimal multiplicity results for the number of maximal invariant measures. As an application the optimal Lipschitz continuity of the time separation on the Abelian cover is established.