Showing papers in "Mathematische Zeitschrift in 2014"

Classifying \(\tau \)-tilting modules over preprojective algebras of Dynkin type

Abstract: We study support \(\tau \)-tilting modules over preprojective algebras of Dynkin type. We classify basic support \(\tau \)-tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and many other important objects in representation theory. We also study \(g\)-matrices of support \(\tau \)-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of \(g\)-matrices and prove that cones given by \(g\)-matrices coincide with chambers of the associated root systems.

Bridgeland’s stabilities on abelian surfaces

Abstract: In this paper, we shall study the structure of walls for Bridgeland’s stability conditions on abelian surfaces. In particular, we shall study the structure of walls for the moduli spaces of rank 1 complexes on an abelian surface with the Picard number 1.

Universal geometric cluster algebras

Abstract: We consider, for each exchange matrix $$B$$ , a category of geometric cluster algebras over $$B$$ and coefficient specializations between the cluster algebras. The category also depends on an underlying ring $$R$$ , usually $$\mathbb {Z},\,\mathbb {Q}$$ , or $$\mathbb {R}$$ . We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over $$B$$ with universal geometric coefficients, or the universal geometric cluster algebra over $$B$$ . Constructing universal geometric coefficients is equivalent to finding an $$R$$ -basis for $$B$$ (a “mutation-linear” analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan $${\mathcal {F}}_B$$ , which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between $${\mathcal {F}}_B$$ and $$\mathbf{g}$$ -vectors. We construct universal geometric coefficients in rank $$2$$ and in finite type and discuss the construction in affine type.

Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound

Abstract: We complete the picture of sharp eigenvalue estimates for the \(p\)-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator \(\Delta _p\) when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.

Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions

Abstract: We consider the distribution of the orbits of the number 1 under the -transformations as varies. Mainly, the size of the set of for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. The dimension of the following set is determined, where is a given point in and is a sequence of integers tending to infinity as . For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders (the set of with a common prefix in the expansion of 1) in the parameter space .

The inverse function theorem and the Jacobian conjecture for free analysis

Abstract: We establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices. We show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity, the function must be invertible. Thus, as a corollary, we establish the Jacobian conjecture in this context. Furthermore, our result holds for commutative polynomials evaluated on tuples of commuting matrices.

The supersingular locus of the Shimura variety for $$\hbox {GU}(1, n-1)$$ GU ( 1 , n - 1 ) over a ramified prime

Abstract: Consider the reduction modulo a prime ideal of an integral model of a Shimura variety. The present paper is a contribution to the general problem of giving a concrete description of its basic locus. This question has been addressed in the case of the Siegel moduli spaces by Katsura/Oort, Li/Oort, Kaiser, Richartz, and Kudla/Rapoport. We refer to the introduction of [28] for the precise references. In this case the basic locus coincides with the supersingular locus. In the case of Hilbert-Blumenthal moduli spaces, where again the basic locus coincides with the supersingular locus, there are results by Bachmat/Goren, by Goren, by Goren/Oort, by Stamm and by Yu. Again we refer to the introduction of [28] for the precise references. For the Shimura variety for GU(1, n− 1) at an inert prime, where again the basic locus coincides with the supersingular locus, there are the results of Vollaard [28] and by Vollaard/Wedhorn [29]. It is our purpose here to prove structure theorems, analogous to [29], for the Shimura variety for GU(1, n− 1) in the case of a ramified prime. By the uniformization theorem of [23], the general problem can be seen as a special case of the general problem of describing the underlying reduced scheme Nred of any RZ-space N . The experience of the work done so far on this problem seems to roughly indicate that the set of irreducible components of Nred should be describable in terms of the Bruhat-Tits simplicial complex associated to the corresponding algebraic group J over Qp. Furthermore, each irreducible component should be related to some Deligne-Lusztig variety, although it only rarely will be actually isomorphic to a Deligne-Lusztig variety. It should be pointed out that we can solve this kind of problem in only a limited number of cases. The analogue of this problem in the equi-characteristic case is a question about the structure of affine DeligneLusztig varieties (ADLV) [9]. U. Görtz and X. He have informed us that they produced an essentially complete list of (basic) ADLV for which the set of irreducible components are described by the Bruhat-Tits complex of J , and in which each irreducible component is isomorphic to a Deligne-Lusztig variety. It would be interesting to transpose their list to the unequal characteristic case. The case of the RZ-space associated to an unramified hermitian space of signature (2, 2) has been solved by B. Howard and G. Pappas [12].

Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension

Abstract: Let $$\mathrm{M }^n,\, n \in \{4,5,6\}$$ , be a compact, simply connected $$n$$ -manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on $$\mathrm{M }^n$$ by a torus $$\mathrm{T }^{n-2}$$ is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, non-negatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.

Extremal Eigenvalues of the Laplacian on Euclidean domains and closed surfaces

Abstract: We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part, we study sequences of extremal eigenvalues of the Laplace-Beltrami operator on closed surfaces of unit area.

Character varieties of abelian groups

Abstract: We prove that for every reductive group \(G\) with a maximal torus \({\mathbb {T}}\) and the Weyl group \(W,\, {\mathbb {T}}^N/W\) is the normalization of the irreducible component, \(X_G^0({\mathbb {Z}}^N)\), of the \(G\)-character variety \(X_G({\mathbb {Z}}^N)\) of \({\mathbb {Z}}^N\) containing the trivial representation. We also prove that \(X_G^0({\mathbb {Z}}^N)={\mathbb {T}}^N/W\) for all classical groups. Additionally, we prove that even though there are no irreducible representations in \(X_G^0({\mathbb {Z}}^N)\) for non-abelian \(G\), the tangent spaces to \(X_G^0({\mathbb {Z}}^N)\) coincide with \(H^1({\mathbb {Z}}^N, Ad\, \rho )\). Consequently, \(X_G^0({\mathbb {Z}}^2)\), has the “Goldman” symplectic form for which the combinatorial formulas for Goldman bracket hold.

On canonical metrics on Cartan–Hartogs domains

Abstract: The Cartan–Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan–Hartogs domain \(\Omega ^{B^{d_0}}(\mu )\) endowed with the canonical metric \(g(\mu ),\) we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space \(\mathcal {H}_{\alpha }\) of square integrable holomorphic functions on \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) with the weight \(\exp \{-\alpha \varphi \}\) (where \(\varphi \) is a globally defined Kahler potential for \(g(\mu )\)) for \(\alpha >0\), and, furthermore, we give an explicit expression of the Rawnsley’s \(\varepsilon \)-function expansion for \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) .\) Secondly, using the explicit expression of the Rawnsley’s \(\varepsilon \)-function expansion, we show that the coefficient \(a_2\) of the Rawnsley’s \(\varepsilon \)-function expansion for the Cartan–Hartogs domain \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is constant on \(\Omega ^{B^{d_0}}(\mu )\) if and only if \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.

Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to anosov

Abstract: We show that partially hyperbolic diffeomorphisms of $$d$$ -dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a global stability result, i.e. every partially hyperbolic diffeomorphism as above is leaf-conjugate to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.

Cohomology and base change for algebraic stacks

Abstract: We prove that cohomology and base change holds for algebraic stacks, generalizing work of Brochard in the tame case. We also show that Hom-spaces on algebraic stacks are represented by abelian cones, generalizing results of Grothendieck, Brochard, Olsson, Lieblich, and Roth–Starr. To accomplish all of this, we prove that a wide class of relative Ext-functors in algebraic geometry are coherent (in the sense of M. Auslander).

Reaching generalized critical values of a polynomial

Abstract: Let \(f: \mathbb K ^n \rightarrow \mathbb K \) be a polynomial, \(\mathbb K =\mathbb R , \,\mathbb C \). We give an algorithm to compute the set of generalized critical values. The algorithm uses a finite dimensional space of rational arcs along which we can reach all generalized critical values of \(f\).

Stable self-similar blowup in energy supercritical Yang-Mills theory

Abstract: We consider the Cauchy problem for an energy supercritical nonlinear wave equation that arises in -dimensional Yang-Mills theory A certain self-similar solution of this model is conjectured to act as an attractor for generic large data evolutions Assuming mode stability of , we prove a weak version of this conjecture, namely that the self-similar solution is (nonlinearly) stable Phrased differently, we prove that mode stability of implies its nonlinear stability The fact that this statement is not vacuous follows from careful numerical work by BizoA" and Chmaj that verifies the mode stability of beyond reasonable doubt

Volume growth and spectrum for general graph Laplacians

Abstract: We prove estimates relating exponential or sub-exponential volume growth of weighted graphs to the bottom of the essential spectrum for general graph Laplacians. The volume growth is computed with respect to a metric adapted to the Laplacian, and use of these metrics produces better results than those obtained from consideration of the graph metric only. Conditions for absence of the essential spectrum are also discussed. These estimates are shown to be optimal or near-optimal in certain settings and apply even if the Laplacian under consideration is an unbounded operator.

On a result of Moeglin and Waldspurger in residual characteristic 2

Abstract: Let \(F\) be a \(p\)-adic field, \(\mathbf G\) a connected reductive group over \(F\), and \(\pi \) an irreducible admissible representation of \(\mathbf G(F)\). A result of Moeglin and Waldspurger states that, if the residual characteristic of \(F\) is different from \(2\), then the ‘leading’ coefficients in the character expansion of \(\pi \) at the identity element of \(\mathbf G(F)\) give the dimensions of certain spaces of degenerate Whittaker forms. In this paper, we extend their result to residual characteristic 2. The outline of the proof is the same as in the original paper of Moeglin and Waldspurger, but certain constructions are modified to accommodate the case of even residual characteristic.

Explicit formulae for \(L\)-values in positive characteristic

Abstract: We focus on the generating series for the rational special values of Pellarin’s \(L\)-series in \(1 \le s \le 2(q-1)\) indeterminates, and using interpolation polynomials we prove a closed form formula relating this generating series to the Carlitz exponential, the Anderson–Thakur function, and the Anderson generating functions for the Carlitz module. We draw several corollaries, including explicit formulae and recursive relations for Pellarin’s \(L\)-series in the same range of \(s\), and divisibility results on the numerators of the Bernoulli–Carlitz numbers by monic irreducibles of degrees one and two.

Representations of quantum affine superalgebras

Abstract: We study the quantum affine superalgebra $$U_q(\mathcal {L}\mathfrak {sl}(M,N))$$ and its finite-dimensional representations. We prove a triangular decomposition and establish a system of Poincare-Birkhoff-Witt generators for this superalgebra, both in terms of Drinfel’d currents. We define the Weyl modules in the spirit of Chari–Pressley and prove that these Weyl modules are always finite-dimensional and non-zero. In consequence, we obtain a highest weight classification of finite-dimensional simple representations when $$M e N$$ . Some concrete simple representations are constructed via evaluation morphisms.

Spin-boson model through a Poisson-driven stochastic process

Abstract: We give a functional integral representation of the semigroup generated by the spin-boson Hamiltonian by making use of a Poisson point process and a Euclidean field. We present a method of constructing Gibbs path measures indexed by the full real line which can be applied also to more general stochastic processes with jump discontinuities. Using these tools we then show existence and uniqueness of the ground state of the spin-boson, and analyze ground state properties. In particular, we prove super-exponential decay of the number of bosons, Gaussian decay of the field operators, derive expressions for the positive integer, fractional and exponential moments of the field operator, and discuss the field fluctuations in the ground state.

From submodule categories to preprojective algebras

Abstract: Let \(S(n)\) be the category of invariant subspaces of nilpotent operators with nilpotency index at most \(n\). Such submodule categories have been studied already in 1934 by Birkhoff, they have attracted a lot of attention in recent years, for example in connection with some weighted projective lines (Kussin, Lenzing, Meltzer). On the other hand, we consider the preprojective algebra \(\Pi _n\) of type \(\mathbb {A}_n\); the preprojective algebras were introduced by Gelfand and Ponomarev, they are now of great interest, for example they form an important tool to study quantum groups (Lusztig) or cluster algebras (Geiss, Leclerc, Schroer). We are going to discuss the connection between the submodule category \(\mathcal {S}(n)\) and the module category \(\hbox {mod}\;\Pi _{n-1}\) of the preprojective algebra \(\Pi _{n-1}\). Dense functors \(\mathcal {S}(n) \rightarrow \hbox {mod}\;\Pi _{n-1}\) are known to exist: one has been constructed quite a long time ago by Auslander and Reiten, recently another one by Li and Zhang. We will show that these two functors are full, dense, objective functors with index \(2n\), thus \(\hbox {mod}\;\Pi _{n-1}\) is obtained from \(\mathcal {S}(n)\) by factoring out an ideal which is generated by \(2n\) indecomposable objects. As a byproduct we also obtain new examples of ideals in triangulated categories, namely ideals \(\mathcal {I}\) in a triangulated category \(\mathcal {T}\) which are generated by an idempotent such that the factor category \(\mathcal {T}/\mathcal {I}\) is an abelian category.

Robust vanishing of all Lyapunov exponents for iterated function systems

Abstract: Given any compact connected manifold $$M$$ , we describe $$C^2$$ -open sets of iterated functions systems (IFS’s) admitting fully-supported ergodic measures whose Lyapunov exponents along $$M$$ are all zero. Moreover, these measures are approximated by measures supported on periodic orbits. We also describe $$C^1$$ -open sets of IFS’s admitting ergodic measures of positive entropy whose Lyapunov exponents along $$M$$ are all zero. The proofs involve the construction of non-hyperbolic measures for the induced IFS’s on the flag manifold.

Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

Abstract: We consider maps preserving a foliation which is uniformly contracting and a one-dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for suitable Poincare maps of a large class of singular hyperbolic flows. From this we deduce a logarithm law for these flows.

The $$\mathfrak {sl}_{3}$$-web algebra

Abstract: In this paper we use Kuperberg’s \(\mathfrak {sl}_3\)-webs and Khovanov’s \(\mathfrak {sl}_3\)-foams to define a new algebra \(K^S\), which we call the \(\mathfrak {sl}_3\)-web algebra. It is the \(\mathfrak {sl}_3\) analogue of Khovanov’s arc algebra. We prove that \(K^S\) is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of \(q\)-skew Howe duality, which allows us to prove that \(K^S\) is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group \(K^{\oplus }_0(\mathcal {W}^S)_{\mathbb {Q}(q)}\), to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that \(K^S\) is a graded cellular algebra.

The Monge–Ampère equation on almost complex manifolds

Abstract: We study the Dirichlet problem for the Monge–Ampere equation on almost complex manifolds. We obtain the existence of the unique smooth solution in strictly pseudoconvex domains.

Fano 5-folds with nef tangent bundles and Picard numbers greater than one

Abstract: We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds.

Augmented base loci and restricted volumes on normal varieties

Abstract: We extend to normal projective varieties defined over an arbitrary algebraically closed field a result of Ein, Lazarsfeld, Mustaţa, Nakamaye and Popa characterizing the augmented base locus (aka non-ample locus) of a line bundle on a smooth projective complex variety as the union of subvarieties on which the restricted volume vanishes. We also give a proof of the folklore fact that the complement of the augmented base locus is the largest open subset on which the Kodaira map defined by large and divisible multiples of the line bundle is an isomorphism.

A characterization of annularity for area-preserving toral homeomorphisms

Abstract: We prove that if an area-preserving homeomorphism of the torus in the homotopy class of the identity has a rotation set which is a nondegenerate vertical segment containing the origin, then there exists an essential invariant annulus. In particular, some lift to the universal covering has uniformly bounded displacement in the horizontal direction.

Log canonical thresholds of certain Fano hypersurfaces

Abstract: We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an important application, we show that they have Kahler–Einstein metrics if they are general.

Equivariant map superalgebras

Abstract: Suppose a group $$\Gamma $$ acts on a scheme $$X$$ and a Lie superalgebra $$\mathfrak {g}$$ . The corresponding equivariant map superalgebra is the Lie superalgebra of equivariant regular maps from $$X$$ to $$\mathfrak {g}$$ . We classify the irreducible finite dimensional modules for these superalgebras under the assumptions that the coordinate ring of $$X$$ is finitely generated, $$\Gamma $$ is finite abelian and acts freely on the rational points of $$X$$ , and $$\mathfrak {g}$$ is a basic classical Lie superalgebra (or $$\mathfrak {sl}\,(n,n)$$ , $$n \ge 1$$ , if $$\Gamma $$ is trivial). We show that they are all (tensor products of) generalized evaluation modules and are parameterized by a certain set of equivariant finitely supported maps defined on $$X$$ . Furthermore, in the case that the even part of $$\mathfrak {g}$$ is semisimple, we show that all such modules are in fact (tensor products of) evaluation modules. On the other hand, if the even part of $$\mathfrak {g}$$ is not semisimple (more generally, if $$\mathfrak {g}$$ is of type I), we introduce a natural generalization of Kac modules and show that all irreducible finite dimensional modules are quotients of these. As a special case, our results give the first classification of the irreducible finite dimensional modules for twisted loop superalgebras.