Volume 2011, Issue 650

All known examples of nontrivial homogeneous Ricci solitons are left-invariant metrics on simply connected solvable Lie groups whose Ricci operator is a multiple of the identity modulo derivations (called solsolitons , and nilsolitons in the nilpotent case). The tools from geometric invariant theory used to study Einstein solvmanifolds, turned out to be useful in the study of solsolitons as well. We prove that, up to isometry, any solsoliton can be obtained via a very simple construction from a nilsoliton N together with any abelian Lie algebra of symmetric derivations of its metric Lie algebra (𝔫, 〈·, ·〉). The following uniqueness result is also obtained: a given solvable Lie group can admit at most one solsoliton up to isometry and scaling. As an application, solsolitons of dimension ≦ 4 are classified.

Consider a height two ideal, I , which is minimally generated by m homogeneous forms of degree d in the polynomial ring R = k [ x, y ]. Suppose that one column in the homogeneous presenting matrix φ of I has entries of degree n and all of the other entries of φ are linear. We identify an explicit generating set for the ideal which defines the Rees algebra ℛ = R [ It ]; so for the polynomial ring S = R [ T 1 , . . . , T m ]. We resolve ℛ as an S -module and I s as an R -module, for all powers s . The proof uses the homogeneous coordinate ring, A = S / H , of a rational normal scroll, with . The ideal is isomorphic to the n th symbolic power of a height one prime ideal K of A . The ideal K ( n ) is generated by monomials. Whenever possible, we study A / K ( n ) in place of because the generators of K ( n ) are much less complicated then the generators of . We obtain a filtration of K ( n ) in which the factors are polynomial rings, hypersurface rings, or modules resolved by generalized Eagon–Northcott complexes. The generators of I parameterize an algebraic curve in projective m – 1 space. The defining equations of the special fiber ring ℛ/( x, y )ℛ yield a solution of the implicitization problem for .

We characterize all linear operators on finite or infinite-dimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and non-negative polynomials, respectively. This is done by means of Fischer–Fock dualities, Hankel forms, and convolutions with non-negative measures. We also establish higher-dimensional analogs of these results. In particular, our classification theorems solve the questions raised in [Borcea, Guterman, Shapiro, Preserving positive polynomials and beyond] originating from entire function theory and the literature pertaining to Hilbert's 17th problem.

Let A be a finite dimensional algebra over a field of characteristic zero graded by a finite abelian group G . Here we study a growth function related to the graded polynomial identities satisfied by A by computing the exponential rate of growth of the sequence of graded codimensions of A . We prove that the G -exponent of A exists and is an integer related in an explicit way to the dimension of a suitable semisimple subalgebra of A .

Let ( M, g ) be a compact conformally flat manifold of dimension n ≧ 4 with positive scalar curvature. According to a positive mass theorem by Schoen and Yau, the constant term in the development of the Green function of the conformal Laplacian is positive if ( M, g ) is not conformally equivalent to the sphere. On spin manifolds, there is an elementary proof of this fact by Ammann and Humbert, based on a proof of Witten. Using differential forms instead of spinors, we give an elementary proof on even dimensional manifolds, without any other topological assumption.

We establish the natural Calderón and Zygmund theory for solutions of elliptic and parabolic obstacle problems involving possibly degenerate operators in divergence form of p -Laplacian type, and proving that the (spatial) gradient of solutions is as integrable as that of the assigned obstacles. We also include an existence and regularity theorem where obstacles are not necessarily considered to be non-increasing in time.

Given a C *-algebra A with a KMS weight for a circle action, we construct and compute a secondary invariant on the equivariant K -theory of the mapping cone of , both in terms of equivariant KK -theory and in terms of a semifinite spectral flow. This in particular puts the previously considered examples of Cuntz algebras [Carey, Phillips, Rennie, Twisted cyclic theory and the modular index theory of Cuntz algebras] and SU q (2) [Carey, Rennie, Tong, J. Geom. Phys. 59: 1431–1452, 2009] in a general framework. As a new example we consider the Araki–Woods III λ representations of the Fermion algebra.

We study the motivic Serre invariant of a smoothly bounded algebraic or rigid variety X over a complete discretely valued field K with perfect residue field k . If K has characteristic zero, we extend the definition to arbitrary K -varieties using Bittner's presentation of the Grothendieck ring and a process of Néron smoothening of pairs of varieties. The motivic Serre invariant can be considered as a measure for the set of unramified points on X . Under certain tameness conditions, it admits a cohomological interpretation by means of a trace formula. In the curve case, we use T. Saito's geometric criterion for cohomological tameness to obtain more detailed results. We discuss some applications to Weil–Châtelet groups, Chow motives, and the structure of the Grothendieck ring of varieties.