https://neckpinch.com/authors/liyou-zhang/New arXiv papers by Liyou Zhang, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2606.01153/arxiv:2606.01153Sun, 31 May 2026 00:00:00 GMTBo-Yong Chen, Yuanpu Xiong, Liyou Zhang — Motivated by the works of Gromov and LeBrun in Riemannian geometry, we study the analogous phenomena in complex geometry. We first show that both ∫_M |S_C⁻(g)|ⁿdV_g and rm vol_g(M) (normalized by S_C(g)≥ −1) are bounded below by ((nπ)ⁿ)/(n!)CanVol(M) for any Hermitian metric g on a compact complex n-manifold M. Here S_C denotes the Chern scalar curvature, S_C⁻=max\−S_C,0\ and rm CanVol(M) is the canonical volume of M, i.e., the volume of the canonical line bundle K_M. Moreover, if rm vol_g(M)=((nπ)ⁿ)/(n!)CanVol(M) holds for some Kähler metric with S_C≥ −1, then it has to be the Kähler-Einstein metric of negative scalar curvature. The completely new phenomenon is that if M is a compact Kähler manifold such that K_M is nef, then rm MinVol_C(M)=𝓘_C(M)=𝓘_C⁻(M)=((nπ)ⁿ)/(n!)CanVol(M), where rm MinVol_C(M) is the infimum of rm vol_g(M) with S_C(g)≥ −1 and 𝓘_C⁻(M)=inf_g ∫_M |S_C⁻(g)|ⁿdV_g, 𝓘_C(M)=inf_g ∫_M |S_C(g)|ⁿdV_g. It remains unknown whether the nef condition is superfluous. The answer is positive when M is obtained by blowing up a finite number of points from a projective manifold with big and nef canonical line bundle. The arguments are based on the asymptotic behaviour of the Bergman kernel of mK_M as m→ ∞, the theory of Kähler-Ricci flow and singular Kähler-Einstein metric, as well as a very delicate gluing technique, using the Burns-Simanca metric.Ricci flowKähler–Ricci flowEinstein metricsKähler geometryScalar curvatureKähler–Einstein & K-stabilityAnalytic minimal model program