We prove a Knörrer-periodicity-type equivalence between derived factorization categories of gauged Landau–Ginzburg models, which is an analogy of a theorem proved by Shipman and Isik independently. As an application, we obtain a gauged Landau–Ginzburg version of Orlov’s theorem describing a relationship between categories of graded matrix factorizations and derived categories of hypersurfaces in projective spaces, by combining the above Knörrer periodicity type equivalence and the theory of variations of geometric invariant theory quotients due to Ballard, Favero and Katzarkov.

This paper is a continuation of our previous work (D. Jiang and D. Soudry, On the genericity of cuspidal automorphic forms on${\rm SO}_{2n+1}$, J. reine angew. Math., to appear). We extend Moeglin's results (C. Moeglin, J. Lie Theory 7 (1997), 201–229, 231–238) from the even orthogonal groups to old orthogonal groups and complete our proof of the CAP conjecture for irreducible cuspidal automorphic representations of $\mathrm{SO}_{2n+1}(\mathbb{A})$ with special Bessel models. We also give a characterization of the vanishing of the central value of the standard $L$-function of $\mathrm{SO}_{2n+1}(\mathbb{A})$ in terms of theta correspondence. As a result, we obtain the weak Langlands functorial transfer from $\mathrm{SO}_{2n+1}(\mathbb{A})$ to $\mathrm{GL}_{2n}(\mathbb{A})$ for irreducible cuspidal automorphic representations of $\mathrm{SO}_{2n+1}(\mathbb{A})$ with special Bessel models.

We consider the problem of deforming simultaneously a pair of given structures. We show that such deformations are governed by an $L_{\infty }$-algebra, which we construct explicitly. Our machinery is based on Voronov’s derived bracket construction. In this paper we consider only geometric applications, including deformations of coisotropic submanifolds in Poisson manifolds, of twisted Poisson structures, and of complex structures within generalized complex geometry. These applications cannot be, to our knowledge, obtained by other methods such as operad theory.

We introduce a new class of autoequivalences that act on the derived categories of certain vector bundles over Grassmannians. These autoequivalences arise from Grassmannian flops: they generalize Seidel–Thomas spherical twists, which can be seen as arising from standard flops. We first give a simple algebraic construction, which is well suited to explicit computations. We then give a geometric construction using spherical functors which we prove is equivalent.

We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga–Gorenstein ring. We then apply this result to the Frobenius category of special Cohen–Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga–Gorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander–Solberg and Kong type results.

In this paper we generalize the work of Harris–Soudry–Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on ${\rm GL}_2$ over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semi-simplification.

let g be an algebraic group, $\gamma$ an arithmetic lattice of g and $x=\gamma\backslash g$. if h is an algebraic subgroup of g such that $h\cap \gamma$ is a lattice of h, then $\gamma\backslash \gamma h\subset x$ is endowed with a canonical h-invariant probability measure $\mu_h$. using ratner's theory, we give general examples where $\mu_{h_n}$ converges weakly to $\mu_g$ if hn is a strict sequence of algebraic subgroups of g. if $\gamma$ is a congruence subgroup of g, we define another probability measure $\mu_h^a$ on x by using the adelic description of the quotient. we conjecture that $\mu_{h_n}^a$ always converges weakly to $\mu_g$ if hn is a strict sequence. using automorphic forms and l-functions, we describe the case $g=\textit{sl}(2,f)$ for a number field f and a sequence of tori hn. the relation with similar problems on shimura varieties is explained.

We prove, assuming the generalized Riemann hypothesis for imaginary quadratic fields, the following special case of a conjecture of Oort, concerning Zarsiski closures of sets of CM points in Shimura varieties. Let X be an irreducible algebraic curve in C$^2$, containing infinitely many points of which both coordinates are j-invariants of CM elliptic curves. Suppose that both projections from X to C are not constant. Then there is an integer m [ges ] 1such that X is the image, under the usual map, of the modular curve Y$_0$(m). The proof uses some number theory and some topological arguments.

We prove the existence of complexified real arrangements with the same combinatorics but different embeddings in ${\mathbb P}^2$. Such a pair of arrangements has an additional property: they admit conjugated equations on the ring of polynomials over ${\mathbb Q}(\sqrt{5})$.

Let K be a discrete valuation field with ring of integers ${\mathcal O}$$_K$. Let f:X→Y be a finite morphism of curves over K. In this article, we study some possible relationships between the models over ${\mathcal O}$$_K$ of X and of Y. Three such relationships are listed below. Consider a Galois cover f:X→Y of degree prime to the characteristic of the residue field, with branch locus B. We show that if Y has semi-stable reduction over K, then X achieves semi-stable reduction over some explicit tame extension of K(B). When K is strictly henselian, we determine the minimal extension L/K with the property that X$_L$ has semi-stable reduction.

Let f:X→Y be a finite morphism, with g(Y)[ges ]2. We show that if X has a stable model ${\mathcal X}$ over ${\mathcal O}$$_K$, then Y has a stable model ${\mathcal Y}$ over ${\mathcal O}$$_K$, and the morphism f extends to a morphism ${\mathcal X}$→${\mathcal Y}$.

Finally, given any finite morphism f:X→Y, is it possible to choose suitable regular models ${\mathcal X}$ and ${\mathcal Y}$ of X and Y over ${\mathcal O}$$_K$ such that f extends to a finite morphism ${\mathcal X}$→${\mathcal Y}$? As was shown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situations, with f a cyclic cover of any order [ges ] 4. On the other hand, we prove, without any hypotheses on the residual characteristic, that this extension problem has a positive solution when f is cyclic of order 2 or 3.

We set up the basic theory of $\mathcal P$-adic modular forms over certain unitary PEL Shimura curves M′K′. For any PEL abelian scheme classified by M′K′, which is not ‘too supersingular’, we construct a canonical subgroup which is essentially a lifting of the kernel of Frobenius from characteristic p. Using this construction we define the U and Frob operators in this context. Following Coleman, we study the spectral theory of the action of U on families of overconvergent $\mathcal P$-adic modular forms and prove that the dimension of overconvergent eigenforms of U of a given slope is a locally constant function of the weight.

We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least five. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using an existence result of E. Legendre [Toric geometry of convex quadrilaterals, J. Symplectic Geom. 9 (2011), 343–385], we show that a co-oriented compact toric contact 5-manifold whose moment cone has four facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on S2×S3 admitting two non-isometric and non-transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

In 2014, Pila and Tsimerman gave a proof of the Ax–Schanuel conjecture for the$j$-function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax–Schanuel conjecture. In this article, we show that the hyperbolic Ax–Schanuel conjecture can be used to reduce the Zilber–Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila–Wilkie counting theorem and several arithmetic conjectures. Our methods are inspired by previous applications of the Pila–Zannier method and, in particular, the recent proof by Habegger and Pila of the Zilber–Pink conjecture for curves in abelian varieties.

Let $R$ be a discrete complete valuation ring, with algebraically closed residue field. Let $X$ be a semi-stable $R$-curve, with smooth generic fibre. In this paper we study tame coverings of $X$.

We calculate equivariant elliptic cohomology of the partial flag variety $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G/H$, where $H\subseteq G$ are compact connected Lie groups of equal rank. We identify the ${\rm RO}(G)$-graded coefficients ${\mathcal{E}} ll_G^*$ as powers of Looijenga’s line bundle and prove that transfer along the map

$$\begin{equation*} \pi \,{:}\,G/H\longrightarrow {\rm pt} \end{equation*}$$

$K$

We develop some basic homological theory of hopfological algebra as defined by Khovanov [Hopfological algebra and categorification at a root of unity: the first steps, Preprint (2006), arXiv:math/0509083v2]. Several properties in hopfological algebra analogous to those of usual homological theory of DG algebras are obtained.

We start developing a notion of reciprocity sheaves, generalizing Voevodsky’s homotopy invariant presheaves with transfers which were used in the construction of his triangulated categories of motives. We hope that reciprocity sheaves will eventually lead to the definition of larger triangulated categories of motivic nature, encompassing non-homotopy invariant phenomena.

We prove that the moduli spaces of n-pointed m-stable curves introduced in our previous paper have projective coarse moduli. We use the resulting spaces to run an analogue of Hassett’s log minimal model program for .

We give an algorithm for removing stackiness from smooth, tame Artin stacks with abelian stabilisers by repeatedly applying stacky blow-ups. The construction works over a general base and is functorial with respect to base change and compositions with gerbes and smooth, stabiliser-preserving maps. As applications, we indicate how the result can be used for destackifying general Deligne–Mumford stacks in characteristic $0$, and to obtain a weak factorisation theorem for such stacks. Over an arbitrary field, the method can be used to obtain a functorial algorithm for desingularising varieties with simplicial toric quotient singularities, without assuming the presence of a toroidal structure.

Let $\mathbb{L}\subset A\times I$ be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of $\mathfrak{sl}_{2}(\wedge )$, the exterior current algebra of $\mathfrak{sl}_{2}$. When $\mathbb{L}$ is an $m$-framed $n$-cable of a knot $K\subset S^{3}$, its sutured annular Khovanov homology carries a commuting action of the symmetric group $\mathfrak{S}_{n}$. One therefore obtains a ‘knotted’ Schur–Weyl representation that agrees with classical $\mathfrak{sl}_{2}$ Schur–Weyl duality when $K$ is the Seifert-framed unknot.