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Zehao Sha

Publications and source records attributed to Zehao Sha.

8 recordsLinked to original sources
arXiv2026

Uniruledness and the sign of total scalar curvature

For every integer $n\ge3$, we construct a smooth projective manifold $X$ of complex dimension $n$ whose canonical bundle is not pseudoeffective, or equivalently, which is uniruled, but every Kähler metric has negative total scalar curvature. In particular, $X$ admits no Kähler metric of positive scalar curvature, while it admits a Riemannian metric of positive scalar curvature. Thus the equivalence between uniruledness and the existence of a Kähler metric with positive scalar curvature holds in complex dimensions one and two, but fails in higher dimensions.

math.AG↗
arXiv2026

Scalar curvature on Kähler blow-ups and systolic inequalities

In this paper, we develop the weighted level set method for a Kähler manifold $(X^n,ω)$ admitting an almost holomorphic map to a possibly singular base $Z$, which is not uniruled. As a key intermediate result, we prove that any blowup $\operatorname{Bl}_SX$ of $X$ along smooth submanifolds $S$ of $ \operatorname{codim} S\ge2$ admits a sequence of Kähler metrics with scalar curvature globally and arbitrarily $C^0$-close to the scalar curvature of $ω$. As a consequence, we establish the sharp \(2\)-systole estimate for every positive scalar curvature Kähler manifold $(X,ω)$ and prove $\min_XS(ω) \cdot\operatorname{sys}_2(ω) \le 2πr(r+1)$, where \(r\) is the rational dimension of $X$, with equality if and only if the universal cover splits as $(\widetilde X,\widetilde ω) \cong (\mathbb P^r,ω_{\mathrm{FS}}) \times(Y^{n-r},ω_{\mathrm{RF}})$ up to normalization where $ω_{\mathrm{FS}}$ is the Fubini-Study metric and $ω_{\mathrm{RF}}$ is Ricci-flat. We also show a sharp even-systolic inequality in the same setting when the general fibre is the projective space.

math.DG↗
arXiv2026

A Variational Characterization of Positive Scalar Curvature Kähler metrics

We introduce the prescribed scalar curvature measure equation on a compact Kähler manifold. For a Kähler class of positive total scalar curvature, we prove that the following are equivalent: the existence of a positive scalar curvature Kähler metric, solvability of this equation for every admissible measure, $d_1$-coercivity of the associated functionals, and uniform geodesic stability along finite-energy $d_1$-geodesic rays. As a consequence, in each fixed Kähler class, the space of positive scalar curvature Kähler metrics is either empty or contractible. We further prove that every Kähler class on a positive-dimensional compact smooth toric Kähler manifold contains a torus-invariant metric of positive scalar curvature. Therefore, for every Kähler class, the prescribed scalar curvature measure equation admits a smooth solution for every admissible measure, unique modulo constants.

math.DG↗
arXiv2026

The Kähler-Ricci soliton on bounded pseudoconvex domains

In this paper, we study Kähler-Ricci solitons on bounded pseudoconvex domains in $\mathbb{C}^n$ with $C^2$ boundary. Under suitable assumptions, we prove that such solitons must be Kähler-Einstein. Building on Huang and Xiao's resolution of Cheng's conjecture, we further establish an analogous result for Bergman Kähler-Ricci solitons. Several model domains are presented to illustrate our results.

math.CV↗
arXiv2026

Rigidity of complete Kähler-Einstein metrics under cscK perturbations

In this paper, we study constant scalar curvature Kähler (cscK) metrics on complete non-compact Kähler--Einstein manifolds. We give sufficient conditions under which a cscK perturbation of a Kähler--Einstein metric must remain Kähler--Einstein. As a model case, we prove that the Bergman metric on a bounded strictly pseudoconvex domain is Kähler--Einstein whenever it has constant scalar curvature. In particular, combined with Huang--Xiao's resolution of Cheng's conjecture, this yields the ball characterization for smooth bounded strictly pseudoconvex domains.

math.DG↗
arXiv2026

The 2-systole on compact Kähler surfaces with positive scalar curvature

We study the 2-systole on compact Kähler surfaces of positive scalar curvature. For any such surface $(X,ω)$, we prove the sharp estimate $\min_X S(ω)\cdot\operatorname{sys}_2(ω)\le 12π$, with equality if and only if $X=\mathbb{P}^2$ and $ω$ is the Fubini-Study metric. Using the classification of positive scalar curvature Kähler surfaces, we determine the optimal constant in each case and describe the corresponding rigid models. When $X$ is a non-rational ruled surface, we also give an independent analytic proof, adapting Stern's level set method to the holomorphic fibration in Kähler setting.

math.DG↗