AI 中文总结
研究紧致凯勒流形上二黑塞方程，通过相关准则产生高杜雄类，复三维时与可解性等价，据此证明了村上和塞克利希迪的相关猜想，并建立了边界版本。
AI 中文摘要
我们证明与复二黑塞方程相关的一个中井 - 莫伊谢宗型准则产生一个高杜雄类。在复三维中，此数值准则等同于存在一个光滑的二可允许代表元，从而等同于二黑塞方程的可解性。作为这些结果的推论，我们证明了村上对于复黑塞方程以及塞克利希迪对于三维黑塞商方程的相应猜想。我们还建立了上述结果的一个边界版本。
英文摘要
We show that a Nakai--Moishezon-type criterion associated with the complex $2$-Hessian equation produces a Gauduchon class. In complex dimension three, this numerical criterion is equivalent to the existence of a smooth $2$-admissible representative and hence to the solvability of the $2$-Hessian equation. As consequences of these results, we prove the corresponding conjectures of Murakami for the complex Hessian equation and of Székelyhidi for the Hessian quotient equation in dimension three. We also establish a boundary version of the above results.
