Pascal Cherrier

Einstein-Kähler metrics on a class of bundles involving integral weights

Abstract We prove that some compact complex bundles, defined over P d 1 −1 ×⋯× P d n −1 (where P d is the complex projective space of complex dimension d ), and depending on n integral weights a 1 ,…, a n , have positive first Chern class if 1⩽ a h ⩽ d h −1 for all h , and carry Einstein–Kahler metrics when a 1 =⋯= a n and d 1 =⋯= d n .

Einstein–Kähler metrics on certain complex bundles

Abstract We study a class of compact complex manifolds, with positive first Chern class, fibered over products of projective spaces. We prove that these bundles carry Einstein–Kahler metrics when the projective spaces of the basis have the same dimension. When this dimensional condition is not satisfied, we estimate their Ricci tensor in another paper.

Equations de type monge-ampère sur certaines variétés de fano

Resume On met en evidence un intervalle de temps pour lequel des equations de Monge-Ampere sur certaines varietes de Fano admettent des solutions.

Semi-strong Solutions, m = 2, k = 1

In this chapter we prove Theorem 1.4.3 on the existence and uniqueness of semi-strong solutions of problem (VKH) when m = 2 (recall that, by Definition 1.4.1, if m = 2 there is only one kind of semi-strong solution, corresponding to k = 1). Accordingly, we assume that

The Hardy Space \(\mathcal{H}^{1}\) and the Case m = 1

\displaystyle{ u_{0} \in H^{3}\,,\qquad u_{ 1} \in H^{1}\,,\qquad \varphi \in S_{ 2,1}(T) = C([0,T];H^{5}) }

The Parabolic Case

(4.1) [recall ( 1.137)], and look for solutions of problem (VKH) in the space \(\mathcal{X}_{2,1}(\tau )\), for some τ ∈ ]0, T].

Métriques d'Einstein-Kähler sur certaines variétés kählériennes à première classe de Chern positive

In this chapter we introduce the function spaces in which we build our solution theory for problems (VKH) and (VKP), and study the main properties of the operator N defined in (8) in these spaces.

Estimation of Ricci tensor on certain Fano manifolds

In this chapter we first review a number of results on the regularity of the functions N = N(u1, … , u m ) and f = f(u) in the framework of the Hardy space \(\mathcal{H}^{1}\), and then use these results to prove the well-posedness of the von Karman equations (3) and (4) in \(\mathbb{R}^{2}\).