Abstract: In this paper, we study the Albanese morphisms in positive characteristic. We prove that the Albanese morphism of a variety with nef anti-canonical divisor is an algebraic fiber space, under the assumption that the general fiber is F-pure. Furthermore, we consider a notion of F-splitting for morphisms, and investigate it in the case of Albanese morphisms. We show that an F-split variety has F-split Albanese morphism, and that the F-split Albanese morphism is an algebraic fiber space. As an application, we provide a new characterization of abelian varieties.

Abstract: We prove the derived equivalence of a pair of non-compact Calabi–Yau 7-folds, which are the total spaces of certain rank 2 bundles on \(G_2\)-Grassmannians. The proof follows that of the derived equivalence of Calabi–Yau 3-folds in \(G_2\)-Grassmannians by Kuznetsov (Derived equivalence of Ito–Miura–Okawa–Ueda Calabi–Yau 3-folds. arXiv:1611.08386) closely.

Abstract: Using the gauge theoretic approach for Lie applicable surfaces, we characterise certain subclasses of surfaces in terms of polynomial conserved quantities. These include isothermic and Guichard surfaces of conformal geometry and L-isothermic surfaces of Laguerre geometry. In this setting one can see that the well known transformations available for these surfaces are induced by the transformations of the underlying Lie applicable surfaces. We also consider linear Weingarten surfaces in this setting and develop a new Backlund-type transformation for these surfaces.

Abstract: The aim of this paper is to investigate the static perfect fluid spacetime $$M^{4}\times _{f}\mathbb {R}$$ such that $$(M^4, g)$$ is a half conformally flat Riemannian manifold. We prove that $$(M^4, g)$$ is, in fact, locally isometric to a warped product manifold $$I\times _{\phi }N^{3}$$ where $$I\subset \mathbb {R}$$ and $$N^{3}$$ is a space form. Consequently, we make an analysis of the Fischer-Marsden conjecture for a 4-dimensional Riemannian manifold.

Abstract: A method of Gabber (Enseign Math (2) 48(1–2):127–146, 2002) produces unramified cohomology classes in the products of certain varieties with an elliptic curve. The connection between third unramified cohomology and integral Hodge conjecture for codimension 2 cycles (Colliot-Thelene et Voisin in Duke Math J 161(5):735–801, 2012) then gives many examples of such a product for which this conjecture fails. The special case of the product with an Enriques surface was established by Benoist and Ottem (Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero, 2018. arXiv:1802.01845v1 ).

Abstract: We study actions of discrete groups on 2-categories. The motivating examples are actions on the 2-category of representations of finite tensor categories and their relation with the extension theory of tensor categories by groups. Associated to a group action on a 2-category, we construct the 2-category of equivariant objects. We also introduce the G-equivariant notions of pseudofunctor, pseudonatural transformation and modification. Our first main result is a coherence theorem for 2-categories with an action of a group. For a 2-category $${\mathcal B}$$ with an action of a group G, we construct a braided G-crossed monoidal category $$\mathcal {Z}_G({\mathcal B})$$ with trivial component the Drinfeld center of $${\mathcal B}$$ . We prove that, in the case of a G-action on the 2-category of representation of a tensor category $${\mathcal C}$$ , the 2-category of equivariant objects is biequivalent to the module categories over an associated G-extension of $${\mathcal C}$$ . Finally, we prove that the center of the equivariant 2-category is monoidally equivalent to the equivariantization of a relative center, generalizing results obtained in Gelaki et al. (Algebra Number Theory 3(8):959–990, 2009).

Abstract: We describe the Fukaya–Seidel category of a Landau–Ginzburg model $$\mathrm {LG}(2)$$ for the semisimple adjoint orbit of $$\mathfrak {sl}(2, {\mathbb {C}})$$ . We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to $$\mathrm {LG}(2)$$ , and that this remains so after compactification.

Abstract: In this paper, we study the K-finite matrix coefficients of integrable representations of the metaplectic cover of $$ {\mathrm {SL}}_{2}({\mathbb {R}}) $$ and give a result on the non-vanishing of their Poincare series. We do this by adapting the techniques developed for $$ {\mathrm {SL}}_{2}({\mathbb {R}}) $$ by Muic to the case of the metaplectic group.

Abstract: We develop a modification of a notion of distance of an element in a valued field extension introduced by F.-V. Kuhlmann. We show that the new notion preserves the main properties of the distance and at the same time gives more complete information about a valued field extension. We study valued field extensions of prime degree to show the relation between the distances of the elements and the corresponding extensions of value groups and residue fields. In connection with questions related to defect extensions of valued function fields of positive characteristic, we present constructions of defect extensions of rational function fields K(x, y)|K generated by elements of various distances from K(x, y). In particular, we construct dependent Artin–Schreier defect extensions of K(x, y) of various distances.

Abstract: In this paper, we study the mean curvature flow of graphs with Neumann boundary conditions. The main aim is to use the maximum principle to get the boundary gradient estimate for solutions. As an application, we obtain the corresponding long time existence for the mean curvature flow of graphs.

Abstract: We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation’s coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positive smooth solutions to many special cases of the nonlinear equation. In particular, we apply gradient estimates to discuss some Yamabe-type problems of complete Riemannian manifolds and smooth metric measure spaces.

Abstract: We introduce a formalism of descent of moduli spaces, and use it to produce limit linear series moduli spaces for families of curves in which the components of geometric fibers may have nontrivial monodromy. We then construct a universal stack of limit linear series over the stack of semistable curves of compact type, and produce new results on existence of real curves with few real linear series.

Abstract: On a two-dimensional compact Riemannian manifold with boundary, we prove that the first nonzero Steklov eigenvalue is nondecreasing along the unnormalized geodesic curvature flow if the initial metric has positive geodesic curvature and vanishing Gaussian curvature. Using the normalized geodesic curvature flow, we also obtain some estimate for the first nonzero Steklov eigenvalue. On the other hand, we prove that the compact soliton of the geodesic curvature flow must be the trivial one.

Abstract: We study the nonlocal equation $$\begin{aligned} \Delta ^{2}u-m\left( \displaystyle \int _{\Omega }| abla u|^{2} dx \right) \Delta u = \lambda a(x) |u|^{q-2}u+ b(x)|u|^{p-2}u, \, \text{ in } \Omega , \end{aligned}$$ subject to the boundary condition $$u=\Delta u=0$$ on $$\partial \Omega $$ . For m continuous and positive we obtain a nonnegative solution if $$10$$ small. If the affine case $$m(t)=\alpha +\beta t$$ , we obtain a second solution if $$4

Abstract: A co-Higgs sheaf on a smooth complex projective variety X is a pair of a torsion-free coherent sheaf $$\mathcal {E}$$ and a global section of $$\mathcal {E}nd(\mathcal {E})\otimes T_X$$ with $$T_X$$ the tangent bundle. We construct 2-nilpotent co-Higgs sheaves of rank two for some rational surfaces and of rank three for $$\mathbb {P}^3$$ , using the Hartshorne-Serre correspondence. Then we investigate the non-existence, especially over projective spaces.

Abstract: Gross, Hacking, and Keel have shown that, in the absence of frozen indices, a cluster $$\mathcal {A}$$ -variety with generic coefficients is the universal torsor of the corresponding cluster $$\mathcal {X}$$ -variety with corresponding coefficients. We extend this to allow for (partial) compactifications of the $$\mathcal {A}$$ - and $$\mathcal {X}$$ -spaces, using frozen vectors to capture the data of the boundary divisors. This works even without assuming that the exchange matrix is skew-symmetrizable. When certain assumptions are satisfied, we conclude that the theta bases on the $$\mathcal {A}$$ -space yield additive bases of global sections for every line bundle on the leaves of the partially compactified $$\mathcal {X}$$ -space.

Abstract: We give another proof, using tools from Geometric Invariant Theory, of a result due to Sam and Snowden in (J Algebraic Comb 43(1):1–10, 2016), concerning the stability of Kronecker coefficients. This result states that some sequences of Kronecker coefficients eventually stabilise, and our method gives a nice geometric bound from which the stabilisation occurs. We perform the explicit computation of such a bound on two examples, one being the classical case of Murnaghan’s stability. Moreover, we see that our techniques apply to other coefficients arising in representation theory: namely to some plethysm coefficients, as well as multiplicities for tensor products of representations of the hyperoctahedral group.

Abstract: Let X be a pointed CW-complex. The generalized conjecture on spherical classes states that, the Hurewicz homomorphism$$H: \pi _{*}(Q_0 X) \rightarrow H_{*}(Q_0 X)$$vanishes on classes of$$\pi _* (Q_0 X)$$of Adams filtration greater than 2. Let $$ \varphi _s: {\rm Ext} _{\mathcal {A}}^{s}(\widetilde{H}^*(X), \mathbb {F}_2) \rightarrow {(\mathbb {F}_2 \otimes _{{\mathcal {A}}}R_s\widetilde{H}^*(X))}^* $$ denote the sth Lannes–Zarati homomorphism for the unstable $${\mathcal {A}}$$-module $$\widetilde{H}^*(X)$$. This homomorphism corresponds to an associated graded of the Hurewicz map. An algebraic version of the conjecture states that the sth Lannes–Zarati homomorphism vanishes in any positive stem for $$s>2$$ and any CW-complex X. We construct a chain level representation for the Lannes–Zarati homomorphism by means of modular invariant theory. We show the commutativity of the Lannes–Zarati homomorphism and the squaring operation. The second Lannes–Zarati homomorphism for $$\mathbb {R}\mathbb {P}^{\infty }$$ vanishes in positive stems, while the first Lannes-Zatati homomorphism for any space is basically non-zero. We prove the algebraic conjecture for $$\mathbb {R}\mathbb {P}^{\infty }$$ and $$\mathbb {R}\mathbb {P}^{n}$$ with $$s=3$$, 4. We discuss the relation between the Lannes–Zarati homomorphisms for $$\mathbb {R}\mathbb {P}^{\infty }$$ and $$S^0$$. Consequently, the algebraic conjecture for $$X=S^0$$ is re-proved with $$s=3$$, 4, 5.

Abstract: In this paper we prove some gap theorem for complete immersed minimal submanifold of dimension no less than six or four, depending on the codimension, in a hyperbolic space $$\mathbb {H}^{n+m}(-1)$$ . That is, we show that a high dimensional complete immersed minimal submanifold M in $$ \mathbb {H}^{n+m}(-1)$$ , is totally geodesic if the $$L^d$$ norm of |A|, for some d, on geodesic balls centered at some point $$p \in M $$ has less than quadratic growth and if either $$\sup _{x \in M} |A|^2$$ is not too large or the $$L^n$$ norm of |A| on M is finite, were, A is the second fundamental form of M.

Abstract: Strata of k-differentials on smooth curves parameterize sections of the k-th power of the canonical bundle with prescribed orders of zeros and poles. Define the tautological ring of the projectivized strata using the $$\kappa $$ and $$\psi $$ classes of moduli spaces of pointed smooth curves along with the class $$\eta = \mathcal O(-1)$$ of the Hodge bundle. We show that if there is no pole of order k, then the tautological ring is generated by $$\eta $$ only, and otherwise it is generated by the $$\psi $$ classes corresponding to the poles of order k.

Abstract: We give an unconditional construction of K3 surfaces over finite fields with given L-function, up to finite extensions of the base fields, under some mild restrictions on the characteristic. Previously, such results were obtained by Taelman assuming semistable reduction. The main contribution of this paper is to make Taelman’s proof unconditional. We use some results of Nikulin and Bayer–Fluckiger to construct an appropriate complex projective K3 surface with CM which admits an elliptic fibration with a section, or an ample line bundle of low degree. Then using Saito’s construction of strictly semistable models and applying a slight refinement of Matsumoto’s good reduction criterion for K3 surfaces, we obtain a desired K3 surface over a finite field.

Abstract: This paper characterizes $${\mathrm {GL}}_n({\mathbb {F}}_q )$$ -distinguished cuspidal representations of $${\mathrm {GL}}_n({\mathbb {F}}_{q^2})$$ in terms of the special values of their twisted gamma factors.

Abstract: Taking symmetric powers of varieties can be seen as a functor from the category of varieties to the category of varieties with an action by the symmetric group. We study a corresponding map between ...

Abstract: We give an elementary proof of the nested Artin approximation theorem for linear equations with algebraic power series coefficients. Moreover, for any Noetherian local subring of the ring of formal power series, we clarify the relationship between this theorem and the problem of the commutation of two operations for ideals: the operation of replacing an ideal by its completion and the operation of replacing an ideal by one of its elimination ideals. In particular we prove that a Grothendieck conjecture about morphisms of analytic/formal algebras and Artin’s question about linear nested approximation problem are equivalent.

Abstract: In this paper, we establish $$L^{\infty }$$ and $$L^{p}$$ estimates for solutions of some polyharmonic elliptic equations via the Morse index. As far as we know, it seems to be the first time that such explicit estimates are obtained for polyharmonic problems.

Abstract: In this paper we prove the existence of a positive solution of the nonlinear and nonlocal elliptic equation in $$\mathbb {R}^n$$ $$\begin{aligned} (-\Delta )^s u =\varepsilon h u^q+u^{2_s^*-1} \end{aligned}$$ in the convex case $$1\le q<2_s^*-1$$ , where $$ 2_s^*={2n}/({n-2s}) $$ is the critical fractional Sobolev exponent, $$(-\Delta )^s$$ is the fractional Laplace operator, $$\varepsilon $$ is a small parameter and h is a given bounded, integrable function. The problem has a variational structure and we prove the existence of a solution by using the classical Mountain-Pass Theorem. We work here with the harmonic extension of the fractional Laplacian, which allows us to deal with a weighted (but possibly degenerate) local operator, rather than with a nonlocal energy. In order to overcome the loss of compactness induced by the critical power we use a Concentration-Compactness principle. Moreover, a finer analysis of the geometry of the energy functional is needed in this convex case with respect to the concave–convex case studied in Dipierro et al. (Fractional elliptic problems with critical growth in the whole of $$\mathbb {R}^n$$ . Lecture Notes Scuola Normale Superiore di Pisa, vol 15. Springer, Berlin, 2017).

Abstract: In this paper, we consider infinite Galois extensions of number fields and study the relation between their local degrees and the structure of their Galois groups. It is known that, if K is a number field and L / K is an infinite Galois extension of group G, then the local degrees of L are uniformly bounded at all rational primes if and only if G has finite exponent. In this note we show that the non uniform boundedness of the local degrees is not equivalent to any group theoretical property. More precisely, we exhibit several groups that admit two different realisations over a given number field, one with bounded local degrees at a given set of primes and one with infinite local degrees at the same primes.

Abstract: In this note we show the Holder regularity for bounded solutions to a class of anisotropic elliptic operators. This result is the dual of the one proved by Liskevich and Skrypnik (Nonlinear Anal 71:1699–1708, 2009).

Abstract: This paper investigates the transitivity of the relation defined by degeneration of finitely generated modules over an associative algebra. It is proved in this paper that if L degenerates to M and M degenerates to N, then $$L^{\oplus e}$$ degenerates to $$N^{\oplus e}$$ for some (but explicitly given) integer $$e>0$$ .