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We invoke the classical fact that the algebra of bi-invariant forms on a compact connected Lie group G is naturally isomorphic to the de Rham cohomology H*dR(G) itself. Then, we show that when a flat connection A exists on a principal G-bundle P, we may construct a homomorphism EA: H*dR(G)→H*dR(P), which eventually shows that the bundle satisfies a condition for the Leray–Hirsch theorem. A similar argument is shown to apply to its adjoint bundle. As a corollary, we show that that both the flat principal bundle and its adjoint bundle have the real coefficient cohomology isomorphic to that of the trivial bundle.

Let $(X,L)$ be a polarized manifold over the complex number field with dim$X=n$. In this paper, we consider a conjecture of M. C. Beltrametti and A. J. Sommese and we obtain that this conjecture is true if $n=3$ and ${{h}^{0}}\,(L)\,\ge \,2$, or $\dim\,\text{Bs}|L|\le 0$ for any $n\ge 3$. Moreover we can generalize the result of Sommese.

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Cohomology of Flat Principal Bundles

On the Nonemptiness of the Adjoint Linear System of Polarized Manifolds

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Cohomology of Flat Principal Bundles

On the Nonemptiness of the Adjoint Linear System of Polarized Manifolds