Narrow Results

In this paper, we show that K-semistable Fano manifolds with the smallest alpha invariant are projective spaces. Singular cases are also investigated.

In this paper, we prove the existence of coupled Kähler–Einstein metrics on complex manifolds whose canonical bundle is ample. These metrics were introduced and their existence in the said case was proven by Hultgren and Nyström using calculus of variations. We prove the result using the method of continuity. In the process of proving estimates, akin to the usual Kähler–Einstein metrics, we reduce existence in the Fano case to a $C^0$ estimate.

Let $X$ be an $n$-dimensional complex Fano manifold $(n\ge 3)$. Assume that $X$ contains a divisor $A$, which is isomorphic to a rational homogeneous space with Picard number one, such that the conormal bundle ${\mathcal{𝒩}}_{A/X}^{\ast }$ is ample over $A$. Building on the works of Tsukioka, Watanabe and Casagrande–Druel, we give a complete classification of such pairs $(X,A)$.

In this paper, we make a correction to Theorem 1.2 of the aforementioned paper [J. Liu, Fano manifolds containing a negative divisor isomorphic to a rational homogeneous space of Picard number one, Int. J. Math. 31(9) (2020) 2050066].