A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry

Abstract

For φ a metric on the anticanonical bundle, −K X , of a Fano manifold X we consider the volume of X X e −φ .In earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on −K X . Our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on X, even with very low regularity assumptions on the geodesic. As a consequence we get a simplified proof of the Bando-Mabuchi uniqueness theorem for Kähler -Einstein metrics. A generalization of this theorem to 'twisted' Kähler-Einstein metrics and some classes of manifolds that satisfy weaker hypotheses than being Fano is also given. We moreover discuss a generalization of the main result to other bundles than −K X , and finally use the same method to give a new proof of the theorem of Tian and Zhu on uniqueness of Kähler-Ricci solitons.

Kähler–Einstein metrics along the smooth continuity method

“…We need the following result, generalizing the classical results of Bando-Mabuchi [7] and Matsushima [35], which are essentially contained in Berndtsson [11], BoucksomBerman-Eyssidieux-Guedj-Zeriahi [9], Berman-Witt-Nyström [10] and Chen-DonaldsonSun [20]. We will give an outline proof in Section 6.…”

“…A crucial point is the reductivity of the automorphism group of the limiting variety. This essentially follows from the work of Berndtsson [11] as used in [20], but since we did not find the exact statement that we need in the literature, we give a brief exposition in Section 6.…”

“…We say that a metric h on K V | Ui∩W0 , we will write (2) |σ i | 2 h r = e −rφi , for continuous functions φ i on U i . We will write the metric h simply as e −φ following the notation in Berndtsson [11]. In particular e −φ defines a volume form on W 0 , given in a local chart U i by…”

“…If W, φ, ψ are smooth, then the twisted Kähler-Ricci soliton equation is equivalent (up to adding a constant to φ) to (11) Ric…”

“…Even when W is normal and φ is only continuous, it is useful to have an equation for twisted Kähler-Ricci solitons in the form (11). For this the extra condition needed is that the measure e θv ω n φ defines a singular metric…”