Volume 2013, Issue 674

Building upon work of Clozel, Harris, Shepherd–Barron, and Taylor, this paper shows that certain Galois representations become automorphic after one makes a suitably large totally-real extension of the base field. The main innovation here is that the result applies to Galois representations to GL 2 n , where previous work dealt with representations to GSp n . The main technique is the consideration of the cohomology of the Dwork hypersurface, and in particular, of pieces of this cohomology other than the invariants under the natural group action.

We offer a solution to the long-standing problem of group completing within the context of rig categories (also known as bimonoidal categories). Given a rig category ℛ we construct a natural additive group completion ℛ̅ that retains the multiplicative structure, hence has become a ring category. If we start with a commutative rig category ℛ (also known as a symmetric bimonoidal category), the additive group completion ℛ̅ will be a commutative ring category. In an accompanying paper we show how to use this construction to prove the conjecture that the algebraic K -theory of the connective topological K -theory ring spectrum ku is equivalent to the algebraic K -theory of the rig category 𝒱 of complex vector spaces.

Soit K un corps infini et A une K -algèbre de dimension finie n . Soit 0 <  r  <  n un entier. Le résultat principal de cet article est le suivant. Sous certaines hypothèses sur A (satisfaites si A est étale), la grassmannienne 𝔾( r ,  A ) est K -birationnelle, de manière PGL 1 ( A )-équivariante, au produit de 𝔾( pgcd ( r ,  n ),  A ) par un espace projectif sur lequel PGL 1 ( A ) agit trivialement (Théorème 3.3). On en déduit par torsion plusieurs résultats nouveaux sur les variétés de Severi–Brauer généralisées, liés à la conjecture d'Amitsur. Les plus significatifs sont les suivants. Si A est une K -algèbre simple centrale de degré n , et si 0 <  r  <  n est un entier, la r -ième variété de Severi–Brauer généralisée SB ( r ,  A ) est K -birationnelle au produit de SB ( pgcd ( r ,  n ),  A ) par un K -espace projectif de dimension convenable. Si A et B sont deux K -algèbres simples centrales de degrés premiers entre eux, SB ( A  ⊗ K   B ) est birationnelle au produit de SB ( A ) × K   SB ( B ) par un K -espace projectif de dimension convenable.

The main result of this note is an effective uniform bound for the number of deformation types of certain nonisotrivial families of canonically polarized manifolds. It extends the author's earlier such bound for the classical Shafarevich Conjecture over function fields to the case of higher dimensional fibers, but without the disadvantageous iterated use of Chow or Hilbert varieties that was the core of the proofs in the earlier approaches.

We investigate the Dirichlet problem for multidimensional variational integrals with linear growth which is formulated in a generalized way in the space of functions of bounded variation. We prove uniqueness of minimizers up to additive constants and deduce additional assertions about these constants and the possible (non-)attainment of the boundary values. Moreover, we provide several related examples. In the case of the model integral our results extend classical results from the scalar case N  = 1—where the problem coincides with the non-parametric least area problem—to the general vectorial setting N  ∈ ℕ.

We define the pseudo-Calabi flow as , ∆ φ f ( φ ) =  S ( ϕ ) −  S ̲. Then we prove the well-posedness of this flow including the short time existence, the regularity of the solution and the continuous dependence on the initial data. Next, we point out that the L ∞ bound on Ricci curvature is an obstruction to the extension of the pseudo-Calabi flow. Finally, we show that if there is a constant scalar curvature Kähler metric ω in its Kähler class, then for any initial potential in a small C 2,  α neighborhood of this metric (defined in terms of the C 2,  α norm on the Kähler potential), the pseudo-Calabi flow must converge exponentially fast to a constant scalar curvature Kähler metric near ω within the same Kähler class.