Abstract: For germs of holomorphic functions $$f: (\mathbf {C}^{m+1},0) \rightarrow (\mathbf {C},0)$$ , $$g: (\mathbf {C}^{n+1},0) \rightarrow (\mathbf {C},0)$$ having an isolated critical point at 0 with value 0, the classical Thom–Sebastiani theorem describes the vanishing cycles group $$\Phi ^{m+n+1}(f \oplus g)$$ (and its monodromy) as a tensor product $$\Phi ^m(f) \otimes \Phi ^n(g)$$ , where $$(f \oplus g)(x,y) = f(x) + g(y), x = (x_0,{\ldots },x_m), y = (y_0,{\ldots },y_n)$$ . We prove algebraic variants and generalizations of this result in etale cohomology over fields of any characteristic, where the tensor product is replaced by a certain local convolution product, as suggested by Deligne. They generalize Fu (Math Res Lett 21:101–119, 2014). The main ingredient is a Kunneth formula for $$R\Psi $$ in the framework of Deligne’s theory of nearby cycles over general bases. In the last section, we study the tame case, and the relations between tensor and convolution products, in both global and local situations.

Abstract: Zelevinsky’s classification theory of discrete series of p-adic general linear groups has been well known. Mœglin and Tadic gave the same kind of theory for p-adic classical groups, which is more complicated due to the occurrence of nontrivial structure of L-packets. Nonetheless, their work is independent of the endoscopic classification theory of Arthur (also Mok in the unitary case), which concerns the structure of L-packets in these cases. So our goal in this paper is to make more explicit the connection between these two very different types of theories. To do so, we reprove the results of Mœglin and Tadic in the case of quasisplit symplectic groups and orthogonal groups by using Arthur’s theory.

Abstract: We show partial regularity up to the boundary \(\partial \varOmega \) of a bounded open set \(\varOmega \subset \mathbb {R}^m\) for minimizers u for p(x)-growth functionals of the following type $$\begin{aligned} {\mathcal A}(u)=\int _\varOmega \left( A^{\alpha \beta }_{ij}(x,u) D_{\alpha }u^i(x) D_{\beta }u^j(x)\right) ^{p(x)/2}dx, \end{aligned}$$ assuming that \(A^{\alpha \beta }_{ij}(x,u)\) are bounded uniformly continuous functions satisfying Legendre condition and that p(x) is a Holder continuous function with \(p(x)>1\). When \(A^{\alpha \beta }_{ij}(x,u)\) are given as \(A^{\alpha \beta }_{ij}(x,u)=g^{\alpha \beta }(x)G_{ij}(x,u)\), we can also prove that minimizers have no singular points on the boundary.

Abstract: We give a defining equation of a complex smooth quartic surface containing 56 lines, and investigate its reductions to positive characteristics. This surface is isomorphic to the complex Fermat quartic surface, which contains only 48 lines. We give the isomorphism explicitly.

Abstract: Let A be a Koszul Artin–Schelter regular algebra and \(\sigma \) an algebra homomorphism from A to \(M_{2\times 2}(A)\). We compute the Nakayama automorphisms of a trimmed double Ore extension \(A_P[y_1, y_2; \sigma ]\) [introduced in Zhang and Zhang (J Pure Appl Algebra 212:2668–2690, 2008)]. Using a similar method, we also obtain the Nakayama automorphism of a skew polynomial extension \(A[t; \theta ]\), where \(\theta \) is a graded algebra automorphism of A. These lead to a characterization of the Calabi–Yau property of \(A_P[y_1, y_2; \sigma ]\), the skew Laurent extension \(A[t^{\pm 1}; \theta ]\) and \(A[y_1^{\pm 1}, y_2^{\pm 1}; \sigma ]\) with \(\sigma \) a diagonal type.

Abstract: We consider here operators which are sum of (possibly) fractional derivatives, with (possibly different) order. The main constructive assumption is that the operator is of order 2 in one variable. By constructing an explicit barrier, we prove a Lipschitz estimate which controls the oscillation of the solutions in such direction with respect to the oscillation of the nonlinearity in the same direction. As a consequence, we obtain a rigidity result that, roughly speaking, states that if the nonlinearity is independent of a coordinate direction, then so is any global solution (provided that the solution does not grow too much at infinity). A Liouville type result then follows as a byproduct.

Abstract: In the setting of a variety X admitting a tilting bundle T we consider the problem of constructing X as a quiver GIT quotient of the algebra \(A:=\mathrm{End}_{X}(T)^{\mathrm{op}}\) We prove that if the tilting equivalence restricts to a bijection between the skyscraper sheaves of X and the closed points of a quiver representation moduli functor for \(A=\mathrm{End}_{X}(T)^{\mathrm{op}}\) then X is indeed a fine moduli space for this moduli functor, and we prove this result without any assumptions on the singularities of X As an application we consider varieties which are projective over an affine base such that the fibres are of dimension 1, and the derived pushforward of the structure sheaf on X is the structure sheaf on the base In this situation there is a particular tilting bundle on X constructed by Van den Bergh, and our result allows us to reconstruct X as a quiver GIT quotient for an easy to describe stability condition and dimension vector This result applies to flips and flops in the minimal model program, and in the situation of flops shows that both a variety and its flop appear as moduli spaces for algebras produced from different tilting bundles on the variety We also give an application to rational surface singularities, showing that their minimal resolutions can always be constructed as quiver GIT quotients for specific dimension vectors and stability conditions This gives a construction of minimal resolutions as moduli spaces for all rational surface singularities, generalising the G-Hilbert scheme moduli space construction which exists only for quotient singularities

Abstract: We classify non-smooth del Pezzo surfaces with \(\frac{1}{3}(1,1)\) points in 29 qG-deformation families grouped into six unprojection cascades (this overlaps with work of Fujita and Yasutake in Classification of log del Pezzo surfaces of index three, arXiv:1401.1283 [math.AG]), we tabulate their biregular invariants, we give good model constructions for surfaces in all families as degeneracy loci in rep quotient varieties, and we prove that precisely 26 families admit qG-degenerations to toric surfaces. This work is part of a program to study mirror symmetry for orbifold del Pezzo surfaces (Akhtar et al. in Proc Am Math Soc 144(2):513–527, 2016).

Abstract: Variational methods are used in order to establish the existence and the multiplicity of nontrivial periodic solutions of a second order dynamical system. The main results are obtained when the potential satisfies different superquadratic conditions at infinity. The particular case of equations with a concave-convex nonlinear term is covered.

Abstract: We prove that if a linear equation, whose coefficients are continuous rational functions on a nonsingular real algebraic surface, has a continuous solution, then it also has a continuous rational solution. This is known to fail in higher dimensions.

Abstract: It is proved that Green’s functions associated to the linearized Monge–Ampere operator satisfy certain Sobolev-type estimates within the natural first-order calculus. Our main result extends the classical Sobolev estimates for Green’s functions due to Gruter and Widman (Manuscr Math 37(3):303–342, 1982) in the uniformly elliptic case and it addresses a question posed by Le (Manuscr Math 149:45–62, 2016) in the degenerate and/or singular Monge–Ampere setting.

Abstract: In our preceding paper a generating set of the derived Picard group of a selfinjective Nakayama algebra was constructed combining some previous results for Brauer tree algebras and the technique of orbit categories developed there. In this paper we finish the computation of the derived Picard group of a selfinjective Nakayama algebra.

Abstract: The purpose of this note is to prove the existence of global weak solutions to the flow associated to integro-differential harmonic maps into spheres and Riemannian homogeneous manifolds.

Abstract: In this note we study periodic homogenization of Dirichlet problem for divergence type elliptic systems when both the coefficients and the boundary data are oscillating. One of the key difficulties ...

Abstract: We associate to every central simple algebra with involution of orthogonal type in characteristic two a totally singular quadratic form which reflects certain anisotropy properties of the involution. It is shown that this quadratic form can be used to classify totally decomposable algebras with orthogonal involution. Also, using this form, a criterion is obtained for an orthogonal involution on a split algebra to be conjugated to the transpose involution.

Abstract: In this paper we assign, under reasonable hypothesis, to each pseudomanifold with isolated singularities a rational Poincare duality space. These spaces are constructed with the formalism of intersection spaces defined by Markus Banagl and are indeed related to them in the even dimensional case.

Abstract: We establish a localization theorem of Borel–Atiyah-Segal type for the equivariant operational K-theory of Anderson and Payne (Doc Math 20:357–399, 2015). Inspired by the work of Chang–Skjelbred and Goresky–Kottwitz–MacPherson, we establish a general form of GKM theory in this setting, applicable to singular schemes with torus action. Our results are deduced from those in the smooth case via Gillet–Kimura’s technique of cohomological descent for equivariant envelopes. As an application, we extend Uma’s description of the equivariant K-theory of smooth compactifications of reductive groups to the equivariant operational K-theory of all, possibly singular, projective group embeddings.

Abstract: Let $\Gamma$ be a finitely presented group and $G$ a linear algebraic group over $\mathbb{R}$. A representation $\rho:\Gamma\rightarrow G(\mathbb{R})$ can be seen as an $\mathbb{R}$-point of the representation variety $\mathfrak{R}(\Gamma, G)$. It is known from the work of Goldman and Millson that if $\Gamma$ is the fundamental group of a compact Kahler manifold and $\rho$ has image contained in a compact subgroup then $\rho$ is analytically defined by homogeneous quadratic equations in $\mathfrak{R}(\Gamma, G)$. When $X$ is a smooth complex algebraic variety, we study a certain criterion under which this same conclusion holds.

Abstract: Let X be a smooth projective variety of dimension 5 and L be an ample line bundle on X such that $$L^5>7^5$$ and $$L^d\cdot Z\ge 7^d$$ for any subvariety Z of dimension $$1\le d\le 4$$ . We show that $$\mathcal {O}_X(K_X+L)$$ is globally generated.

Abstract: We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results are new in characteristic 2, otherwise the symplectic result was shown in Becher (Invent Math 173(1):1–6, 2008) and the unitary result was partly shown in Black and Queguiner-Mathieu (Enseign Math 60(3–4):377–395, 2014).

Abstract: We construct p-adic L-functions of nearly p-ordinary cusp forms on \(\mathrm{GL}_2\) over number fields. As a consequence, we prove a result on the non-vanishing of the central value of L-function of \(\mathrm{GL_2}\) with cyclotomic twists.

Abstract: Let F be a p-adic field with residue field of cardinality q. To each irreducible representation of GL(n, F), we attach a local Euler factor $$L^{BF}(q^{-s},q^{-t},\pi )$$ via the Rankin–Selberg method, and show that it is equal to the expected factor $$L(s+t+1/2,\phi _\pi )L(2s,\Lambda ^2\circ \phi _\pi )$$ of the Langlands’ parameter $$\phi _\pi $$ of $$\pi $$ . The corresponding local integrals were introduced in Bump and Friedberg (The exterior square automorphic L-functions on $$\mathrm{GL}(n)$$ 47–65, 1990), and studied in Matringe (J Reine Angew Math doi: 10.1515/crelle-2013-0083 ). This work is in fact the continuation of Matringe (J Reine Angew Math doi: 10.1515/crelle-2013-0083 ). The result is a consequence of the fact that if $$\delta $$ is a discrete series representation of GL(2m, F), and $$\chi $$ is a character of Levi subgroup $$L=GL(m,F)\times GL(m,F)$$ which is trivial on GL(m, F) embedded diagonally, then $$\delta $$ is $$(L,\chi )$$ -distinguished if an only if it admits a Shalika model. This result was only established for $$\chi =\mathbf {1}$$ before.

Abstract: In this paper, we establish the interior gradient estimate of k-admissible solutions of prescribed Hessian quotient curvature equations \(\frac{\sigma _k (a_{ij})}{\sigma _l (a_{ij})} = f(x)\) with \(0 \le l < k \le n\). As an application, we get a Liouville type theorem.

Abstract: In our paper we develop a theory of harmonic mappings of Riemannian manifolds into non-negatively curved Riemannian manifolds and give the geometric applications of these results to the theory of contraction maps of Riemannian manifolds and of holomorphic maps of almost Kahlerian manifolds. In conclusion, we give the applications of our results to addressing the well known “prescribed Ricci curvature problem”.

Abstract: In this paper, we define a finite sum analogue of multiple polylogarithms inspired by the work of Kaneko and Zagier (in the article “Finite multiple zeta values” in preparation) and prove that they satisfy a certain analogue of the shuffle relation. Our result is obtained by using a certain partial fraction decomposition which is an idea due to Komori et al. (Math Z 268:993–1011, 2011). As a corollary, we give an algebraic interpretation of our shuffle product.

Abstract: Let K be a totally real number field of degree \(r\,=\,[K:\mathbb {Q}]\) and let p be an odd rational prime. Let \(K_{\infty }\) denote the cyclotomic \(\mathbb {Z}_{p}\)-extension of K and let \(L_{\infty }\) be a finite extension of \(K_{\infty }\), abelian over K. In this article, we extend results of Buyukboduk (Compos Math 145(5):1163–1195, 2009) relating characteristic ideal of the \(\chi \)-quotient of the projective limit of the ideal class groups to the \(\chi \)-quotient of the projective limit of the r-th exterior power of units modulo Rubin-Stark units, in the non semi-simple case, for some \(\overline{\mathbb {Q}_{p}}\)-irreducible characters \(\chi \) of \(\mathrm {Gal}(L_{\infty }/K_{\infty })\).

Abstract: We prove that the compact Kahler manifolds with \(c_{1} \ge 0\) that admit holomorphic parabolic geometries are the flat bundles of rational homogeneous varieties over complex tori. We also prove that the compact Kahler manifolds with \(c_{1} \ge 0\) that admit holomorphic cominiscule geometries are the locally Hermitian symmetric varieties.

Abstract: We consider second order uniformly elliptic operators of divergence form in \(\mathbb {R}^{d+1}\) whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators related with Poisson operators and Dirichlet–Neumann maps. Consequently, we obtain a solution formula for the inhomogeneous elliptic boundary value problem in the half space, which is useful to show the existence of solutions in a wider class of inhomogeneous data. We also establish \(L^2\) solvability of boundary value problems for a new class of the elliptic operators.

Abstract: We introduce Dolbeault cohomology valued characteristic classes of Higgs bundles over complex manifolds. Flat vector bundles have characteristic classes lying in odd degree de Rham cohomology and a theorem of Reznikov says that these must vanish in degrees three and higher over compact Kahler manifolds. We provide a simple and independent proof of Reznikov’s result and show that our characteristic classes of Higgs bundles and the characteristic classes of flat vector bundles are compatible via the nonabelian Hodge theorem.