Abstract: Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function \(f: ({\mathbb {C}}^n, 0)\rightarrow ({\mathbb {C}}, 0)\) . The Yau algebra L(V) is defined to be the Lie algebra of derivations of the moduli algebra \(A(V):= {\mathcal {O}}_n/(f, \frac{\partial f}{\partial x_1},\cdots , \frac{\partial f}{\partial x_n})\) , i.e., \(L(V)=\text {Der}(A(V), A(V))\) and plays an important role in singularity theory. It is known that L(V) is a finite dimensional Lie algebra and its dimension \(\lambda (V)\) is called Yau number. In this article, we generalize the Yau algebra and introduce a new series of k-th Yau algebras \(L^k(V)\) which are defined to be the Lie algebras of derivations of the moduli algebras \(A^k(V) = {\mathcal {O}}_n/(f, m^k J(f)), k\ge 0\) , i.e., \(L^k(V)=\text {Der}(A^k(V), A^k(V))\) and where m is the maximal ideal of \({\mathcal {O}}_n\) . In particular, it is Yau algebra when \(k=0\) . The dimension of \(L^k(V)\) is denoted by \(\lambda ^k(V)\) . These numbers i.e., k-th Yau numbers \(\lambda ^k(V)\) , are new numerical analytic invariants of an isolated singularity. In this paper we studied these new series of Lie algebras \(L^k(V)\) and also compute the Lie algebras \(L^1(V)\) for fewnomial isolated singularities. We also formulate a sharp upper estimate conjecture for the \(\lambda ^k(V)\) of weighted homogeneous isolated hypersurface singularities and we prove this conjecture in case of \(k=1\) for large class of singularities. PubDate: 2019-03-16
Abstract: A one-parameter family of coupled flows depending on a parameter \(\kappa >0\) is introduced which reduces when \(\kappa =1\) to the coupled flow of a metric \(\omega \) with a (1, 1)-form \(\alpha \) due recently to Y. Li, Y. Yuan, and Y. Zhang. It is shown in particular that, for \(\kappa \not =1\) , estimates for derivatives of all orders would follow from \(C^0\) estimates for \(\omega \) and \(\alpha \) . Together with the monotonicity of suitably adapted energy functionals, this can be applied to establish the convergence of the flow in some situations, including on Riemann surfaces. Very little is known as yet about the monotonicity and convergence of flows in presence of couplings, and conditions such as \(\kappa \not =1\) seem new and may be useful in the future. PubDate: 2019-03-13
Abstract: We consider hyperbolic Dirac-type operator with growing potential on a spatially non-compact globally hyperbolic manifold. We show that the Atiyah-Patodi-Singer boundary value problem for such operator is Fredholm and obtain a formula for this index in terms of the local integrals and the relative eta-invariant introduced by Braverman and Shi. This extends recent results of Bär and Strohmaier, who studied the index of a hyperbolic Dirac operator on a spatially compact globally hyperbolic manifold. PubDate: 2019-03-12
Abstract: This article is motivated by the original Casson invariant regarded as an integral lifting of the Rochlin invariant. We aim to defining an integral lifting of the Adams e-invariant of stably framed 3-manifolds, perhaps endowed with some additional structure. We succeed in doing so for manifolds which are links of normal complex Gorenstein smoothable singularities. These manifolds are naturally equipped with a canonical \(\mathrm{SU}(2)\) -frame. To start we notice that the set of homotopy classes of \(\mathrm{SU}(2)\) -frames on the stable tangent bundle of every closed oriented 3-manifold is canonically a \(\mathbb Z\) -torsor. Then we define the \(\widehat{E}\) -invariant for the manifolds in question, an integer that modulo 24 is the Adams e-invariant. The \(\widehat{E}\) -invariant for the canonical frame equals the Milnor number plus 1, so this brings a new viewpoint on the Milnor number of the smoothable Gorenstein surface singularities. PubDate: 2019-03-12
Abstract: Prime end boundaries \({\partial _\mathrm{P}}\Omega \) of domains \(\Omega \) are studied in the setting of complete doubling metric measure spaces supporting a p-Poincaré inequality. Notions of rectifiably (in)accessible- and (in)finitely far away prime ends are introduced and employed in classification of prime ends. We show that, for a given domain, the prime end capacity (defined by Estep and Shanmugalingam in Potential Anal 42:335–363, 2015) of the collection of all rectifiably inaccessible prime ends together will all non-singleton prime ends is zero. We show the resolutivity of continuous functions on \({\partial _\mathrm{P}}\Omega \) which are Lipschitz continuous with respect to the Mazurkiewicz metric when restricted to the collection \(\partial _\mathrm{SP}\Omega \) of all accessible prime ends. Furthermore, bounded perturbations of such functions in \({\partial _\mathrm{P}}\Omega {\setminus } \partial _\mathrm{SP}\Omega \) yield the same Perron solution. In the final part of the paper, we demonstrate the (resolutive) Kellogg property with respect to the prime end boundary of bounded domains in the metric space. Notions given in this paper are illustrated by a number of examples. PubDate: 2019-03-08
Abstract: Several constructive homological methods based on noncommutative Gröbner bases are known to compute free resolutions of associative algebras. In particular, these methods relate the Koszul property for an associative algebra to the existence of a quadratic Gröbner basis of its ideal of relations. In this article, using a higher-dimensional rewriting theory approach, we give several improvements of these methods. We define polygraphs for associative algebras as higher-dimensional linear rewriting systems that generalise the notion of noncommutative Gröbner bases, and allow more possibilities of termination orders than those associated to monomial orders. We introduce polygraphic resolutions of associative algebras, giving a categorical description of higher-dimensional syzygies for presentations of algebras. We show how to compute polygraphic resolutions starting from a convergent presentation, and how these resolutions can be linked with the Koszul property. PubDate: 2019-03-07
Abstract: We study the Fujita-type conjecture proposed by Popa and Schnell. We obtain an effective bound on the global generation of direct images of pluri-adjoint line bundles on the regular locus. We also obtain an effective bound on the generic global generation for a Kawamata log canonical \(\mathbb {Q}\) -pair. We use analytic methods such as \(L^2\) estimates, \(L^2\) extensions and injective theorems of cohomology groups. PubDate: 2019-02-27
Abstract: Let \((R, {\mathfrak {m}}, k)\) be a complete Cohen–Macaulay local ring. In this paper, we assign a numerical invariant, for any balanced big Cohen–Macaulay module, called \(\underline{\mathsf {h}}\) -length. Among other results, it is proved that, for a given balanced big Cohen–Macaulay R-module M with an \({\mathfrak {m}}\) -primary cohomological annihilator, if there is a bound on the \(\underline{\mathsf {h}}\) -length of all modules appearing in \(\mathsf {CM}\) -support of M, then it is fully decomposable, i.e. it is a direct sum of finitely generated modules. While the first Brauer–Thrall conjecture fails in general by a counterexample of Dieterich dealing with multiplicities to measure the size of maximal Cohen–Macaulay modules, our formalism establishes the validity of the conjecture for complete Cohen–Macaulay local rings. In addition, the pure-semisimplicity of a subcategory of balanced big Cohen–Macaulay modules is settled. Namely, it is shown that R is of finite \(\mathsf {CM}\) -type if and only if R is an isolated singularity and the category of all fully decomposable balanced big Cohen–Macaulay modules is closed under kernels of epimorphisms. Finally, we examine the mentioned results in the context of Cohen–Macaulay artin algebras admitting a dualizing bimodule \(\omega \) , as defined by Auslander and Reiten. It will turn out that, \(\omega \) -Gorenstein projective modules with bounded \(\mathsf {CM}\) -support are fully decomposable. In particular, a Cohen–Macaulay algebra \(\Lambda \) is of finite \(\mathsf {CM}\) -type if and only if every \(\omega \) -Gorenstein projective module is of finite \(\mathsf {CM}\) -type, which generalizes a result of Chen for Gorenstein algebras. Our main tool in the proof of results is Gabriel–Roiter (co)measure, an invariant assigned to modules of finite length, and defined by Gabriel and Ringel. This, in fact, provides an application of the Gabriel–Roiter (co)measure in the category of maximal Cohen–Macaulay modules. PubDate: 2019-02-15
Abstract: In the publication [1] there is an unfortunate computational error, which however does not affect the correctness of the main results. PubDate: 2019-02-14
Abstract: Let S be a compact Riemann surface and let H be a finite group. It is known that if H acts on S, then there is a H-equivariant isogeny decomposition of the Jacobian variety JS of S, called the group algebra decomposition of JS with respect to H. If \(S_1 \rightarrow S_2\) is a regular covering map, then it is also known that the group algebra decomposition of \(JS_1\) induces an isogeny decomposition of \(JS_2.\) In this article we deal with the converse situation. More precisely, we prove that the group algebra decomposition can be lifted under regular covering maps, under appropriate conditions. PubDate: 2019-02-13
Abstract: The main purpose of this work is to provide a non-local approach to study aspects of structural stability of 3D Filippov systems. We introduce a notion of semi-local structural stability which detects when a piecewise smooth vector field is robust around the entire switching manifold, as well as, provides a complete characterization of such systems. In particular, we present some methods in the qualitative theory of piecewise smooth vector fields, which make use of geometrical analysis of the foliations generated by their orbits. Such approach displays surprisingly rich dynamical behavior which is studied in detail in this work. It is worth mentioning that this subject has not been treated in dimensions higher than two from a non-local point of view, and we hope that the approach adopted herein contributes to the understanding of structural stability for piecewise-smooth vector fields in its most global sense. PubDate: 2019-02-13
Abstract: A very important question in geometric measure theory is how geometric features of a set translate into analytic information about it. Reifenberg (Bull Am Math Soc 66:312–313, 1960) proved that if a set is well approximated by planes at every point and at every scale, then the set is a bi-Hölder image of a plane. It is known today that Carleson-type conditions on these approximating planes guarantee a bi-Lipschitz parameterization of the set. In this paper, we consider an n-Ahlfors regular rectifiable set \(M \subset \mathbb {R}^{n+d}\) that satisfies a Poincaré-type inequality involving Lipschitz functions and their tangential derivatives. Then, we show that a Carleson-type condition on the oscillations of the tangent planes of M guarantees that M is contained in a bi-Lipschitz image of an n-plane. We also explore the Poincaré-type inequality considered here and show that it is in fact equivalent to other Poincaré-type inequalities considered on general metric measure spaces. PubDate: 2019-02-13
Abstract: We provide a description of certain invariance properties of completely bounded bimodule maps in terms of their symbols. If \(\mathbb {G}\) is a locally compact quantum group, we characterise the completely bounded \(L^{\infty }(\mathbb {G})'\) -bimodule maps that send \(C_0({\hat{\mathbb {G}}})\) into \(L^{\infty }({\hat{\mathbb {G}}})\) in terms of the properties of the corresponding elements of the normal Haagerup tensor product \(L^{\infty }(\mathbb {G}) \otimes _{\sigma \mathop {\mathrm{h}}} L^{\infty }(\mathbb {G})\) . As a consequence, we obtain an intrinsic characterisation of the normal completely bounded \(L^{\infty }(\mathbb {G})'\) -bimodule maps that leave \(L^{\infty }({\hat{\mathbb {G}}})\) invariant, extending and unifying results, formulated in the current literature separately for the commutative and the co-commutative cases. PubDate: 2019-02-12
Abstract: We study several problems related to the \(\ell ^p\) boundedness of Riesz transforms for graphs endowed with so-called bounded Laplacians. Introducing a proper notion of the gradient of a function, we prove for \(p\in (1,2]\) an \(\ell ^p\) estimate for the gradient of the continuous time heat semigroup, an \(\ell ^p\) interpolation inequality as well as the \(\ell ^p\) boundedness of the modified Littlewood–Paley–Stein function for a graph with bounded Laplacian. This yields an analogue to Dungey’s results in [21] while removing some additional assumptions. Coming back to the classical notion of the gradient, we give a counterexample to the interpolation inequality and hence to the boundedness of Riesz transforms for bounded Laplacians for \(1<p<2\) . Finally, we prove the boundedness of the Riesz transform for \(1< p<\infty \) under the assumption of positive spectral gap. PubDate: 2019-02-12
Abstract: In this paper, we compute the \(RO(C_{pq})\) -graded cohomology of \(C_{pq}\) -orbits. We deduce that in all the cases the Bredon cohomology groups are a function of the fixed point dimensions of the underlying virtual representations. Further, when thought of as a Mackey functor, the same independence result holds in almost all cases. This generalizes earlier computations of Stong and Lewis for the group \(C_p\) . The computations of cohomology of orbits are used to prove a freeness theorem. The analogous result for the group \(C_p\) was proved by Lewis. We demonstrate that certain complex projective spaces and complex Grassmannians satisfy the freeness theorem. PubDate: 2019-02-11
Abstract: We study the extension estimates for paraboloids in d-dimensional vector spaces over finite fields \(\mathbb F_q\) with q elements. We use the connection between \(L^2\) based restriction estimates and \(L^p\rightarrow L^r\) extension estimates for paraboloids. As a consequence, we improve the \(L^2\rightarrow L^r\) extension results obtained by Lewko and Lewko (Proc Am Math Soc 140:2013–2028, 2012) in even dimensions \(d\ge 6\) and odd dimensions \(d=4\ell +3\) for \(\ell \in \mathbb N.\) Our results extend the consequences for 3-D paraboloids due to Lewko (Adv Math 270(1):457–479, 2015) to higher dimensions. We also clarifies conjectures on finite field extension problems for paraboloids. PubDate: 2019-02-11
Abstract: The Landau–Ginzburg/Calabi–Yau correspondence claims that the Gromov–Witten invariant of the quintic Calabi–Yau 3-fold should be related to the Fan–Jarvis–Ruan–Witten invariant of the associated Landau–Ginzburg model via wall crossings. In this paper, we consider the stack of quasi-maps with a cosection and introduce sequences of stability conditions which enable us to interpolate between the moduli stack for Gromov–Witten invariants and the moduli stack for Fan–Jarvis–Ruan–Witten invariants. PubDate: 2019-02-11
Abstract: We prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices. This result generalizes the paraboloid restriction estimate in radial case from [Shao, Rev. Mat. Iberoam. 25(2009), 1127–1168], as well as the result from [Miao et al. Proc. AMS 140(2012), 2091–2102]. As an application, we show a local smoothing estimate for a solution of the linear Schrödinger equation under the assumption that the initial datum has additional angular regularity. PubDate: 2019-02-09
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