Volume 2017, Issue 723

Dans cet article, nous nous intéressons aux dimensions de certaines variétés qui ont été introduites récemment par Kisin pour démontrer la modularité de certaines représentations galoisiennes. Nous étudions plus spécialement un cas particulier pour lequel nous donnons une estimation de la dimension en question, puis, en nous basant sur ce résultat, nous énonçons une conjecture dans le cas général. In this paper, we study dimensions of some varieties, that were introduced recently by Kisin in order to prove modularity of some Galois representations. In fact, we mainly consider a special case for which we obtain an estimation of the dimension we are interested in. Then, based on this result, we state a conjecture for the general case.

We prove a general criterion to establish existence and uniqueness of a short-time solution to an evolution equation involving “closed” sections of a vector bundle, generalizing a method used by Bryant and Xu [8] for studying the Laplacian flow in G2$G_{2}$-geometry. We apply this theorem in balanced geometry introducing a natural extension of the Calabi flow to the balanced case. We show that this flow has always a unique short-time solution belonging to the same Bott–Chern cohomology class of the initial balanced structure and that it preserves the Kähler condition. Finally, we study explicitly the flow on the Iwasawa manifold.

We develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne–Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over ℙ1${\mathbb{P}^{1}}$, into other Mori fibre spaces, using Cox rings and variation of geometric invariant theory. We show that the pliability of these Mori fibre spaces is at least three and they are not rational.

We show that the complex projective space ℙn${\mathbb{P}^{n}}$ has maximal degree (volume) among all n -dimensional Kähler–Einstein Fano manifolds admitting a non-trivial holomorphic ℂ*${\mathbb{C}^{*}}$-action with a finite number of fixed points. The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart’s volume conjecture for a large class of rational polytopes, including duals of lattice polytopes. The case of spherical varieties/multiplicity free symplectic manifolds is also discussed. The proof uses Moser–Trudinger type inequalities for Stein domains and also leads to criticality results for mean field type equations in ℂn${\mathbb{C}^{n}}$ of independent interest. The paper supersedes our previous preprint [5] concerning the case of toric Fano manifolds.

The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P)${(G,P)}$, we use Tanaka theory to derive a universal upper bound on the submaximal symmetry dimension. We use Kostant’s version of the Bott–Borel–Weil Theorem to show that this bound is in fact sharp in almost all complex and split-real cases by exhibiting (abstract) models. We explicitly compute all submaximal symmetry dimensions when G is any complex or split-real simple Lie group.

In the present paper, we show that for an irreducible cubic f∈ℤ⁢[x]${f\in\mathbb{Z}[x]}$ and a full norm form 𝐍⁢(x1,…,xk)${\mathbf{N}(x_{1},\ldots,x_{k})}$ for a number field K/ℚ${K/\mathbb{Q}}$ satisfying certain hypotheses the variety f⁢(t)=𝐍⁢(x1,…,xk)≠0$f(t)=\mathbf{N}(x_{1},\ldots,x_{k})\neq 0$ satisfies the Hasse principle. Our proof uses sieve methods.