2026 frontier: complete noncompact Ricci-flat Kähler manifolds

Sun and Zhang, arXiv:2610.29891v1 (21 October 2026), report the following results. A complete Ricci-flat Kähler $n$-fold means a geodesically complete Kähler manifold with zero Ricci tensor; Euclidean volume growth means ${\mathrm{lim\hspace{0.17em}inf}}_{R\to \mathrm{\infty }}{R}^{-2n}{\mathrm{Vol}}_g(B_g(p,R))>0$ for one, equivalently every, base point. Under this growth hypothesis, their Theorem 1.1 states that the manifold is quasi-projective (a Zariski-open subvariety of a projective variety), its ring of polynomial-growth holomorphic functions (functions bounded by $C(1+d_g(p,\cdot ){)}^m$ for some $C,m>0$) is finitely generated, and its evaluation map is a projective crepant resolution (a projective birational map from a smooth variety that preserves the canonical divisor) of a normal affine variety (the spectrum of an integrally closed finitely generated complex algebra). Its asymptotic cone at infinity (a pointed Gromov–Hausdorff limit of distance-rescaled spaces $(X,R^{-1}{d}_g,p)$) is unique up to holomorphic isometry (an isometry preserving the complex structures); the theorem also gives a two-step flat $\mathbb{Q}$-Gorenstein degeneration, a chain of flat families whose relative canonical divisor is $\mathbb{Q}$-Cartier. Separately, their Theorem 1.3 constructs a complete Ricci-flat Kähler complex threefold that is diffeomorphic to the total space of a real rank-four vector bundle over $S^2$ (and hence has finite topological type), has uniformly bounded sectional curvature, and admits no Kähler compactification, meaning no holomorphic open embedding into any compact Kähler manifold.

Sun–Zhang, arXiv:2610.29891v1, Theorems 1.1–1.3

The theorem statements and introduction were read in the full PDF. This remark records reported literature scope; it does not represent an independent proof audit.