Volume 2014, Issue 696

In this paper, we consider the moduli space 𝒮𝒰C(r,𝒪C)$\mathcal {S\hspace{-1.0pt}U}_C(r,\mathcal {O}_C)$ of rank r semistable vector bundles with trivial determinant on a smooth projective curve C of genus g . For r = 2, F. Kirwan constructed a smooth log resolution X¯→𝒮𝒰C(2,𝒪C)$\overline{X}\rightarrow \mathcal {S\hspace{-1.0pt}U}_C(2,\mathcal {O}_C)$. Based on earlier work of M. Kerr and J. Lewis, Lewis explains in the Appendix the notion of a relative Chow group (w.r.t. the normal crossing divisor), and a subsequent Abel–Jacobi map on the relative Chow group of null-homologous one-cycles (tensored with ℚ). This map takes values in the intermediate Jacobian of the compactly supported cohomology of the stable locus. We show that this is an isomorphism and since the intermediate Jacobian is identified with the Jacobian Jac(C)⊗ℚ$\operatorname{Jac}(C)\otimes \mathbb {Q}$, this can be thought of as a weak-representability result for open smooth varieties. A hard Lefschetz theorem is also proved for the odd degree bottom weight cohomology of the moduli space 𝒮𝒰Cs(2,𝒪C)$\mathcal {S\hspace{-1.0pt}U}_C^s(2,\mathcal {O}_C)$. When r ≥ 2, we compute the codimension two rational Chow groups of 𝒮𝒰C(r,𝒪C)$\mathcal {S\hspace{-1.0pt}U}_C(r,\mathcal {O}_C)$.

We investigate pairs of diagonal cubic equations with integral coefficients. For a class of such Diophantine systems with 11 or more variables, we are able to establish that the number of integral solutions in a large box is at least as large as the expected order of magnitude.

Suppose that X is a projective variety over an algebraically closed field of characteristic p > 0. Further suppose that L is an ample (or more generally in some sense positive) divisor. We study a natural linear system in |KX+L|$\vert K_X + L\vert $. We further generalize this to incorporate a boundary divisor Δ. We show that these subsystems behave like the global sections associated to multiplier ideals, H0(X,𝒥(X,Δ)⊗L)$H^0(X, \mathcal {J}(X, \Delta ) \otimes L)$ in characteristic zero. In particular, we show that these systems are in many cases base-point-free. While the original proof utilized Kawamata–Viehweg vanishing and variants of multiplier ideals, our proof uses test ideals.

We adapt the conjectural local Langlands parameterization to split metaplectic groups over local fields. When G ˜ is a central extension of a split connected reductive group over a local field (arising from the framework of Brylinski and Deligne), we construct a dual group 𝐆˜∨$\tilde{\mathbf {G}}^{\vee }$ and an L-group L𝐆˜∨${\,}^L \tilde{\mathbf {G}}^{\vee }$ as group schemes over ℤ. Such a construction leads to a definition of Weil–Deligne parameters (Langlands parameters) with values in this L-group, and to a conjectural parameterization of the irreducible genuine representations of G ˜. This conjectural parameterization is compatible with what is known about metaplectic tori, Iwahori–Hecke algebra isomorphisms between metaplectic and linear groups, and classical theta correspondences between 𝑀𝑝2n$\textit {Mp}_{2n}$ and special orthogonal groups.

We introduce a parabolic flow of almost Kähler structures, providing an approach to constructing canonical geometric structures on symplectic manifolds. We exhibit this flow as one of a family of parabolic flows of almost Hermitian structures, generalizing our previous work on parabolic flows of Hermitian metrics. We exhibit a long time existence obstruction for solutions to this flow by showing certain smoothing estimates for the curvature and torsion. We end with a discussion of the limiting objects as well as some open problems related to the symplectic curvature flow.

We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres (𝕊n,[g𝕊n])$(\mathbb {S}^n,[g_{\mathbb {S}^n}])$, it converges to the standard sphere up to a Möbius diffeomorphism. These arguments can be applied to obtain extinction profiles of solutions of some fractional porous medium equations. In the end, we use this fractional fast diffusion equation, together with its extinction profile and some estimates of its extinction time, to improve a Sobolev inequality via a quantitative estimate of the remainder term.

Grothendieck studied two filtrations – the γ and topological filtrations – on the ring K0(X)$K_0(X)$ for a projective variety X , together with the associated graded rings. He gave a necessary and sufficient condition for the graded ring ⊕iγi/γi+1$\oplus _i \gamma ^{i} / \gamma ^{i+1}$ to have non-zero torsion elements when X is the variety of Borel subgroups of a simply connected semisimple algebraic group G over an algebraically closed field. We sharpen his observation by calculating explicitly the torsion in γ2/γ3$\gamma ^{2} / \gamma ^3$, and we do this under weaker hypotheses on G . We apply this result to describing the torsion in the group CH3(X)$\operatorname{CH}^3(X)$ of codimension-3 cycles on X , and providing an extension of the Rost invariant.