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Suppose a compact Kähler manifold $(M,\omega)$ admits a smooth circle action $g_t,~t\in S^1$. So the pull back of the Kähler form $g_t^{\ast}(\omega)$ is a nondegenerate two-form. Since the circle act …

Let $(M,g)$ be a complex $n$-dim compact connected Hermitian manifold, and $\Delta_c(\cdot):=g^{i\bar{j}}\partial_i\partial_{\bar{j}}(\cdot)$ the complex Laplacian acting on smooth functions on $M$. …

Suppose $\omega$ is a real closed $(1,1)$ form on a compact Kähler manifold. If we have a real $d$-closed two form $\sigma$ such that $[\sigma]=[\omega] \in H^2(M)$, can we claim that this two form $\ …

A well-known result of Kobayashi and Ochiai says that an $n$-dimensional Fano maniofold $M$ is biholomorphic to $\mathbb{C}P^n$ or complex quadrics if its index is $n+1$ or $n$ respectively. In these …

We know that if an $n$-dimensional Fano manifold admits a Kahler-Einstein metric, it satisfies the following Chern number inequality $$nc_1^n\leq 2(n+1)c_2c_1^{n-2}.$$ My question is whether there ex …

To my limited knowledge, many compact Kähler manifolds have trivial odd Betti numbers. For instance, flag manifolds $G/P$，where $G$ is a semisimple complex Lie group and $P$ a parabolic subgroup, and …

Yesterday I asked the following question to which abx has given a positive answer. examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes But I suddenly realized that t …

It is well-known that the curvature forms of the (complexified) Levi-Civita connection can be used to provide explicit representatives for Chern classes of compact Kähler manifolds. This is not true f …

A result of Borel-Remmert in 1961/1962 published in Math. Ann. states that a compact homogeneous Kähler manifold must be the product of a complex torus and a projective-rational manifold. This implies …

Suppose $M$ is a complex $n$-dimensioanl compact Kähler manifold and $\omega$ a Kähler class. Suppose $\alpha\in H^{1,1}(M,\mathbb{R})$ is a nef class belonging to the boundary of the Kähler cone of $ …

Suppose we have a real-valued smooth function on a complex torus: $$f: \mathbb{C}^n/(\mathbb{Z}+\sqrt{-1}\mathbb{Z})^n\longrightarrow\mathbb{R},$$ i.e., this $f$ is a real-valued smooth function on $ …