3-26【Zexuan Ouyang】Collapsing of K3 Surfaces and Special Kähler Structures

We consider the Gromov-Hausdorff compactification of hyperkähler metrics on K3 surfaces. I will show that the 2-dimensional collapsing limits are endowed with integral singular special Kähler structures (SKS) on P1, confirming a conjecture of Sun-Zhang. Moreover, each such metric space arises from a Jacobian elliptic K3 surface and is therefore a Gromov-Hausdorff limit of a hyperkähler K3. Additionally, there is a natural map F from the moduli space of Jacobian elliptic K3 surfaces to the space of 2-dimensional collapsing limits. We will show that F is almost injective and finite-to-one, except in the case of the Kummer surface.