A compactification of the moduli variety of stable vector 2-bundles on a surface in the Hilbert scheme

A new compactification of the variety of moduli of stable vector 2-bundles with Chern classes c ₁ and c ₂ is constructed for the case in which the universal family of stable sheaves with given values of invariants is defined and there are no strictly semistable sheaves. The compactification is a subvariety in the Hilbert scheme of subschemes of a Grassmann manifold with fixed Hilbert polynomial; it is obtained from the variety of bundle moduli by adding points corresponding to locally free sheaves on surfaces which are modifications of the initial surface. Moreover, a morphism from the new compactification of the moduli space to its Gieseker-Maruyama compactification is constructed.     

Multiple cover formula of generalized DT invariants I: Parabolic stable pairs

In this paper, we introduce the notion of parabolic stable pairs on Calabi–Yau 3-folds and invariants counting them. By applying the wall-crossing formula developed by Joyce–Song, Kontsevich–Soibelman, we see that they are related to generalized Donaldson–Thomas invariants counting one dimensional semistable sheaves on Calabi–Yau 3-folds. Consequently, the conjectural multiple cover formula of generalized DT invariants is shown to be equivalent to a certain product expansion formula of the generating series of parabolic stable pair invariants. The application of this result to the multiple cover formula will be pursued in the subsequent paper.     

The Cohen–Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non-Cohen–Macaulay actually imply that no graded separating algebra is Cohen–Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen–Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen–Macaulay graded separating algebra implies the group is generated by bireections. Additionally, we give an example which shows that Cohen–Macaulay separating algebras can occur when the ring of invariants is not Cohen–Macaulay.     

Moduli stacks and invariants of semistable objects on K3 surfaces

For a K3 surface X and its bounded derived category of coherent sheaves D(X), we have the notion of stability conditions on D(X) in the sense of T. Bridgeland. In this paper, we show that the moduli stack of semistable objects in D(X) with a fixed numerical class and a phase is represented by an Artin stack of finite type over C. Then following D. Joyce's work, we introduce the invariants counting semistable objects in D(X), and show that the invariants are independent of a choice of a stability condition.     

Cobordism of algebraic knots

Summary The isolated singularities of complex hypersurfaces are studied by considering the topology of the highly connected submanifolds of spheres determined by the singularity. By introducing the notion of the link of a perturbation of the singularity and using techniques of surgery theory, we are able to describe which invariants associated to a singularity can be used to determine the cobordism type of the singularity.It is shown that the cobordism type is determined by the set of weakly distinguished bases. This result is used to draw a distinction between the classical case of two variables and the higher dimensional problem. That is, we show that the result of Le which states that the cobordism and topological classifications of singularities coincide in the classical dimension does not hold for singularities of functions of more than three variables. Examples of topologically distinct but cobordant singularities are obtained using results of Ebeling.     

On the invariants of base changes of pencils of curves,I

The main purpose of this paper is to prove the nonnegativity of the basic invariants of base changes of a surface fibration, which is conjectured by Xiao Gang. For this purpose we obtain some new inequalities between the invariants of the singularities ofz ^d =f(x, y). This work is supported by the National Natural Science Foundation of China and by the Science Foundation of the University Doctoral Program of CNEC. This paper is corrected while the author is visiting Max-Planck-Institut für Mathematik in Bonn.     

Modularity of certain potentially Barsotti-Tate Galois representations

We show that certain potentially semistable lifts of modular mod representations are themselves modular. As a result we show that any elliptic curve over the rational numbers with conductor not divisible by 27 is modular.

     

Local invariants attached to Weierstrass points

Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation d of the hyperelliptic discriminant of X/S, and the valuation δ of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable the invariants d and δ are known to satisfy certain inequalities. We prove an exact formula relating d and δ with intersection theoretic data determined by the distribution of Weierstrass points over the special fiber, in the semistable case. We also prove an exact formula for the stable Faltings height of an arbitrary curve over a number field, involving local contributions associated to its Weierstrass points.     

Rings of invariants for modular representations of elementary abelian <Emphasis Type="Italic">p</Emphasis>-groups

We initiate a study of the rings of invariants of modular representations of elementary abelian p-groups. With a few notable exceptions, the modular representation theory of an elementary abelian p-group is wild. However, for a given dimension, it is possible to parameterise the representations. We describe parameterisations for modular representations of dimension two and of dimension three. We compute the ring of invariants for all two-dimensional representations; these rings are generated by two algebraically independent elements. We compute the ring of invariants of the symmetric square of a two-dimensional representation; these rings are hypersurfaces. We compute the ring of invariants for all three-dimensional representations of rank at most three; these rings are complete intersections with embedding dimension at most five. We conjecture that the ring of invariants for any three-dimensional representation of an elementary abelian p-group is a complete intersection.     

Ramanujan's class invariants, Kronecker's limit formula, and modular equations

In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, but approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan's class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker's limit formula, the second employs modular equations, and the third uses class field theory to make Watson's ``empirical method'rigorous.

     

<Emphasis Type="Italic">SU</Emphasis>(2)/<Emphasis Type="Italic">SL</Emphasis>(2) knot invariants and Kontsevich–Soibelman monodromies

We review the Reshetikhin–Turaev approach for constructing noncompact knot invariants involving Rmatrices associated with infinite-dimensional representations, primarily those constructed from the Faddeev quantum dilogarithm. The corresponding formulas can be obtained from modular transformations of conformal blocks as their Kontsevich–Soibelman monodromies and are presented in the form of transcendental integrals, where the main issue is working with the integration contours. We discuss possibilities for extracting more explicit and convenient expressions that can be compared with the ordinary (compact) knot polynomials coming from finite-dimensional representations of simple Lie algebras, with their limits and properties. In particular, the quantum A-polynomials and difference equations for colored Jones polynomials are the same as in the compact case, but the equations in the noncompact case are homogeneous and have a nontrivial right-hand side for ordinary Jones polynomials.     

Similarity inner solutions for the Pulsar equation

Lie symmetries are applied to classify the source of the magnetic field for the Pulsar equation near to the surface of the neutron star. We find that there are six possible different admitted Lie algebras. We apply the corresponding Lie invariants to reduce the Pulsar equation close to the surface to an ordinary differential equation. This equation is solved either with the use of Lie symmetries or the application of the ARS algorithm for singularity analysis to write the analytic solution as a Laurent expansion. These solutions are called inner solutions.     

On pseudopoints of algebraic curves

Clifford indices for semistable vector bundles on a smooth projective curve of genus at least four were defined in a previous paper of the authors. The present paper studies bundles which compute these Clifford indices. We show that under certain conditions on the curve all such bundles and their Serre duals are generated.     

On the identification and stability of formal invariants for singular differential equations

Given a system of linear differential equations near an irregular singularity of pole type, formal invariants are quantities that remain unchanged with respect to linear transformations of the system. While certain “natural” formal invariants can easily be observed in formal fundamental solution matrices, the algorithms for constructing them do not readily show how the invariants can be universally described as properties of the coefficient matrix of the system, and in particular of the individual constant matrices in the power-series expansion. Other invariants have been abstractly defined by mapping properties of the differential operator, but they are not immediately related to either the natural invariants or the coefficients. In this paper we show how certain invariants in the formal solution may be described and calculated through matrix-theoretic properties of the coefficients and at the same time show how they are related to ones for the differential operator.     

Geometric bases and topological equivalence

There are many algebraic and topological invariants associated to a singular point of a complex analytic function. The intent here is to discuss some of these invariants and the topological classification of singularities. Specifically, we establish that the topological type is determined by the Lefschetz vanishing cycles obtained by unfolding the singularity and certain local monodromy operators defined by Gabrielov. In Brieskorn's terminology singularities with the same geometric bases are topologically indistinguishable. Thus the higher invariants in the hierarchy of Brieskorn are necessary to understand the geometry of higher singularities. As a corollary to our main theorem, we obtain the result of Lê-Ramanujam which states that the topological type is constant in a oneparameter family of singularities with constant Milnor number.     

Depth of modular invariant rings

It is well-known that the ring of invariants associated to a non-modular representation of a finite group is Cohen-Macaulay and hence has depth equal to the dimension of the representation. For modular representations the ring of invariants usually fails to be Cohen-Macaulay and computing the depth is often very difficult. In this paper¹ we obtain a simple formula for the depth of the ring of invariants for a family of modular representations. This family includes all modular representations of cyclic groups. In particular, we obtain an elementary proof of the celebrated theorem of Ellingsrud and Skjelbred [6].     

The Index of Vector Fields and Logarithmic Differential Forms

We introduce the notion of logarithmic index of a vector field on a hypersurface and prove that the homological index can be expressed via the logarithmic index. Then both invariants are described in terms of logarithmic differential forms for Saito free divisors, which are hypersurfaces with nonisolated singularities, and all contracting homology groups of the complex of regular holomorphic forms on such a hypersurface are computed. In conclusion, we consider the case of normal hypersurfaces, including the case of an isolated singularity, and describe the contracting homology of the complex of regular meromorphic forms with the help of the residue of logarithmic forms.     

KMS weights on higher rank buildings

We extend some of the results of Carey-Marcolli-Rennie on modular index invariants of Mumford curves to the case of higher rank buildings. We discuss notions of KMS weights on buildings, that generalize the construction of graph weights over graph C*-algebras.     

Three-dimensional extensions of the Alday-Gaiotto-Tachikawa relation

An extension of the two-dimensional (2d) Alday-Gaiotto-Tachikawa (AGT) relation to three dimensions starts from relating the theory on the domain wall between some two S-dual supersymmetric Yang-Mills (SYM) models to the 3d Chern-Simons (CS) theory. The simplest case of such a relation would presumably connect traces of the modular kernels in 2d conformal theory with knot invariants. Indeed, the two quantities are very similar, especially if represented as integrals of quantum dilogarithms. But there are also various differences, especially in the ??conservation laws?? for the integration variables holding for the monodromy traces but not for the knot invariants. We also consider another possibility: interpreting knot invariants as solutions of the Baxter equations for the relativistic Toda system. This implies another AGT-like relation: between the 3d CS theory and the Nekrasov-Shatashvili limit of the 5d SYM theory.