Singular Semi-Flat Calabi-Yau Metrics on <i>S</i>²

Gravitational instantons from rational elliptic surfaces

“…Semi-flat metrics play a role in a number of recent works [14,17,20,25,40,50]. It is interesting to take a brief look at the connections between some of these.…”

“…• Affine flat structures [14,40]. Given an elliptic fibration without singular fibers over some base S, equipped with a holomorphic symplectic form and a section, we get a flat connection on (T 1,0 S) * .…”

“…Given an elliptic fibration without singular fibers over some base S, equipped with a holomorphic symplectic form and a section, we get a flat connection on (T 1,0 S) * . By [40], the difference between this and the Levi-Civita connection of β := ω S,ε can be viewed as a holomorphic cubic differential U on S, and ρ(β) = |U | 2 β (Ţ iţeica's equation), where |...| 2 denotes norm squared with respect to β. An observation in [40] states that this equation formally implies that ρ(ρ(β)) = −2ρ(β) if U = 0.…”

“…By [40], the difference between this and the Levi-Civita connection of β := ω S,ε can be viewed as a holomorphic cubic differential U on S, and ρ(β) = |U | 2 β (Ţ iţeica's equation), where |...| 2 denotes norm squared with respect to β. An observation in [40] states that this equation formally implies that ρ(ρ(β)) = −2ρ(β) if U = 0. The cubic differential on P 1 associated with a generic K3 elliptic fibration has 24 poles, and these are cusps of the Weil-Petersson metric ρ(β).…”

Isolated Singularities of Affine Special Kähler Metrics in Two Dimensions

“…We remark in passing that starting from a different perspective, Loftin [Lof05] utilized an equation equivalent to (5) to study special Kähler metrics on CP 1 .…”