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 \(k\left[ \!\left[ Q\right] \!\right] \)-linear predifferential graded Lie algebra \(L^{\bullet }_{X_0/S_0}\) associated to a log smooth and saturated morphism \(f_0: X_0 \rightarrow S_0\) and prove that it controls the log smooth deformation functor. This provides a geometric interpretation of a construction in Chan et al. (Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau

There is a natural bijective correspondence between irreducible (algebraic) selfdual representations of the special linear group with those of classical groups. In this paper, using computations done through the LiE software, we compare tensor product of irreducible selfdual representations of the special linear group with those of classical groups to formulate some conjectures relating the two. More

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.

We consider the following problem involving supercritical exponent and polyharmonic operator: $$\begin{aligned} (-\Delta )^mu=K(|y|)u^{m^*-1+\varepsilon }, \;u>0, \hbox { in } B_1(0), \; u \in {\mathcal {D}}_0^{m,2}(B_1(0)), \end{aligned}$$ where \(B_1(0)\) is the unit ball in \({\mathbb {R}}^{N}\), \(m^*=\frac{2N}{N-2m}\) is the critical exponent,\(\; N\ge 2m+2\), \( m \in {\mathbb {N}}_+\), \(\varepsilon

The cohomology jump loci of a space X are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in either the cohomology ring, or an algebraic model for X. We explore here the geometry of these varieties and the delicate interplay between them in the context of closed

We deal with minimal surfaces in spheres that are locally isometric to a pseudoholomorphic curve in a totally geodesic \({\mathbb {S}}^{5}\) in the nearly Kähler sphere \({\mathbb {S}}^6\). Being locally isometric to a pseudoholomorphic curve in \({\mathbb {S}}^5\) turns out to be equivalent to the Ricci-like condition \(\Delta \log (1-K)=6K,\) where K is the Gaussian curvature of the induced metric

In this paper we look for necessary and sufficient conditions for a genus-one fibration to have rational curves. We show that a projective variety with log terminal singularities that admits a relatively minimal genus-one fibration \(X\rightarrow B\) does contain vertical rational curves if and only if it not isomorphic to a finite étale quotient of a product \(\tilde{B}\times E\) over B. Many sufficient

In this paper we investigate the multiplicity and the log canonical threshold of abelian quotient complete intersection singularities in terms of the notion of special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection singularities.

We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental form of the bundle of linear frames of the manifold. We show that the group of affine

We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains \(\Omega \subset \mathbb {R}^n\). For a rotationally invariant Cheeger set C, the free boundary \(\partial C \cap \Omega \) consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show that if \(\Omega \) is convex, then the free boundary of C consists

In this note, we establish a version of the local Cauchy-Crofton formula for definable sets in Henselian discretely valued fields of characteristic zero. It allows to compute the motivic local density of a set from the densities of its projections integrated over the Grassmannian.

In this paper we study congruences between Siegel Eisenstein series and Siegel cusp forms for \(\mathrm{Sp}_4(\mathbb {Z})\).

We introduce Kummer surfaces \(X={\text {Km}}(C\times C)\) with the group scheme \(G=\mu _2\) acting on the self-product of the rational cuspidal curve in characteristic two. The resulting quotients are normal surfaces having a configuration of sixteen rational double points of type \(A_1\), together with a rational double point of type \(D_4\). We show that our Kummer surfaces are precisely the supersingular

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