Limits if canonical metrics are important for moduli compactification problems. Asymptotic behavious of Kähler-Einstein Fano manifolds of the same deformation class are somehow easier than that of Calabi-Yau metrics case. Indeed, the diameters are bounded then and Donaldson-Sun shows they even limit to singular KE (K-stable) Fano varieties. Moreover, we now have AG reconstruction using so-called delta invariant (stability threshold) for K-stability, or certain volume type invariant.

In "more classical" CY case, different rescale and base points give different limits and the theory of K-stability is not enough to understand these limits. We discuss how to lay "algebrio-geometric" foundation for these limits, and report some progresses.