紧Kähler-Einstein流形的夹逼定理与反向Yau不等式
Pinched theorem and the reverse Yau's inequalities for compact Kähler-Einstein manifolds
AI总结：
该研究利用不变量理论推导紧Kähler-Einstein流形的曲率积分表达式，得到反向Yau不等式，改进经典夹逼定理，证实相关曲率猜想，并提供射影流形负全纯截面曲率埃尔米特度量的存在性判断方法。
摘要：
对于维数$n\ge 2$的紧Kähler-Einstein流形$M$，我们利用不变量理论，将表达式$-c_1^n(M)+\frac{2(n+1)}{n}c_2(M)c_1^{n-2}(M)$表示为$M$中某固定点处全纯截面曲率及其平均值的某种积分形式。作为应用，我们得到了反向Yau不等式，并将紧Kähler-Einstein流形的经典$\frac{1}{4}$-夹逼定理与负$\frac{1}{4}$-夹逼定理改进为仅依赖于$M$的维数和第一陈类的更小夹逼常数。若$M$不具有正或负全纯截面曲率，则存在点$x\in M$使得$x$处的全纯截面曲率平均值为零。特别地，我们通过某种曲率条件刻画了二维复环面。此外，若全纯截面曲率的绝对值足够小，我们甚至在高维情形下证实了关于正全纯截面曲率的Yau猜想与关于负全纯截面曲率的Siu-Yang猜想。最后，利用反向Yau不等式，我们可以判断一个射影流形是否不携带任何具有负全纯截面曲率的埃尔米特度量。
英文摘要：
For a compact Kähler-Einstein manifold $M$ of dimension $n\ge 2$, we explicitly write the expression $-c_1^n(M)+\frac{2(n+1)}{n}c_2(M)c_1^{n-2}(M)$ in the form of certain integral on the holomorphic sectional curvature and its average at a fixed point in $M$ using the invariant theory. As applications, we get a reverse Yau's inequality and improve the classical $\frac{1}{4}$-pinched theorem and negative $\frac{1}{4}$-pinched theorem for compact Kähler-Einstein manifolds to smaller pinching constant depending only on the dimension and the first Chern class of $M$. If $M$ is not with positive or negative holomorphic sectional curvature, then there exists a point $x\in M$ such that the average of the holomorphic sectional curvature at $x$ vanishes. In particular, we characterise the $2$-dimensional complex torus by certain curvature condition. Moreover, we confirm Yau's conjecture for positive holomorphic sectional curvature and Siu-Yang's conjecture for negative holomorphic sectional curvature even for higher dimensions if the absolute value of the holomorphic sectional curvature is small enough. Finally, using the reverse Yau's inequality, we can judge if a projective manifold doesn't carry any hermitian metric with negative holomorphic sectional curvature.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
