Existence of complete Kahler Ricci-flat metrics on crepant resolutions

A Kummer Construction for Gravitational Instantons

“…These manifolds are of special interest in quantum gravity and string theory, hence some motivation to understand examples. Previous constructions of gravitational instantons were either explicit [8,9], based on hyperkähler reduction [16,7], gauge theory [2,5,6], twistorial methods [13,4] or on the Monge-Ampère method of Tian and Yau [26,27], see also [14,15,12,25].…”

“…For X 1 = R × T 3 /±, our construction yields an ALH gravitational instanton. These ALG and ALH examples have been constructed recently in a general Tian-Yau framework on rational elliptic surfaces in [12] (see also [15,25]); in [12], they correspond to isotrivial elliptic fibrations. Two other quotients,…”

Complete Calabi-Yau metrics from P^2 # 9 \bar P^2

“…Thus one can use the flat orbifold (C × E)/Z 2 as a background. Settings of this sort have been treated in [5,67] and heuristically in [27,36], cf. Remark 1.6(ii).…”

“…Crepant resolution and other constructions have produced examples for each of these types [3,5,17,36,49,67,73], and also there is work of Biquard-Minerbe [5,55] [56] towards a possible classification of ALF spaces, the basic conjecture being that such metrics must live on resolutions or deformations of C 2 /Γ with Γ < SU(2) cyclic or dihedral, except for finitely many sporadic examples of a slightly different nature. However, one major issue with the ALF, G, H picture is that, unlike the ALE case, there seems to be no clear understanding as to what tameness conditions have to be imposed in order for a complete Ricci-flat 4-manifold of less than maximal volume growth to belong to one of these categories.…”

“…Example 4.1 for a careful discussion of isotrivial rational elliptic surfaces. Isotrivial ALG spaces with a type I * 0 , II, III, or IV fiber at infinity can also be obtained from crepant resolution constructions, as in [5,67], or heuristically in [27,36]. The cones on the base are then honest quotients of R 2 by cyclic group actions.…”