Collapsing Ricci-flat metrics on elliptic K3 surfaces

Torelli-type theorems for gravitational instantons with quadratic volume growth

“…Away from all singular fibers, we choose the hyperkähler structure as ω sf , given by the Greene-Shapere-Vafa-Yau ansatz; see [CVZ20, Subsection 2.2]. Near the I 1 fibers, we glue in Ooguri-Vafa metrics as in [GW00,CVZ20]. These regions contribute exponentially small error terms to the weighted estimates, so in the following we will take this as understood, and will not consider those regions in any detail.…”

“…then there is a diffeomorphism Ψ : X β → X β which induces the identity map on H 2 dR (X β ) such that Ψ * g ′ = g and Ψ * ω ′ = ω. This will be proved using a modification of the gluing construction in [CVZ20] (see Theorem 6.2 below), and then invoking the Torelli Theorem for K3 surfaces. We remark that the order condition is essential to control the error term in the gluing construction.…”

“…This was generalized to arbitrary elliptic K3 surfaces in [GTZ16]; see also [OO18]. Subsequently, the authors gave a new construction on arbitrary elliptic K3 surfaces in [CVZ20], which also allowed for a detailed description of the degeneration near the singular fibers, which we briefly describe next. Away from singular fibers, the degeneration is modeled by Greene-Shapere-Vafa-Yau's semi-flat metric; see [GSVY90].…”

“…In the case of I * ν fibers, the model used was a Z 2 -quotient of certain multi-Ooguri-Vafa metrics with 2ν monopole points, together with 4 Eguchi-Hanson metrics due to the 4 orbifold singularities of the resulting quotient. It was moreover shown in [CVZ20] that such degenerations exist for metrics which are Kähler with respect to the fixed elliptic complex structure.…”

“…In this case of an I * b fiber, the construction in [CVZ20] was done to preserve the elliptic complex structure. In this new gluing construction, the original elliptic complex structure is not preserved.…”