Kähler-Einstein metrics of negative Ricci curvature on general quasi-projective manifolds

Abstract

Let M be a quasi-projective manifold which can be compactified by adding a divisor D with simple normal crossings, which means that D = ∑p i=1Di, where the irreducible components Di are smooth and intersect transversely. Under certain positivity conditions of the adjoint bundle over the compactification, the existence of a complete Kahler–Einstein metric on the quasi-projective manifold was first addressed by Yau (see, for example, [29, p. 166]), right after his resolution of the Calabi conjecture [28]. This program has been followed by many authors, for example, [1, 5, 8, 22,24], and [31]. In fact, notice that the second part of [28] is essentially devoted to construct Kahler–Einstein metrics on algebraic manifolds of general type. In general such a metric has singularities. In a sense, Yau’s motivation for this program may be viewed as to understand these singularities in many important cases from the differential-geometric point of view. This appears in his later papers joint with Cheng [8], and with Tian [22]. In the last paper, they proved the following result. Let