A Weitzenbock Formula for Compact Complex Manifolds and Applications to the Hopf Conjecture in Real Dimension 6

Abstract The Hopf Conjecture is a well-known problem in differential geometry which relates the geometry of a manifold to its topology \cite{Hopf 1}. In this thesis, we investigate this problem on compact complex real 6-dimensional manifolds. First, we prove a Weitzenbock formula on a complex manifoldinvolving the Hodge Laplacian $\Delta _{H}$, the Bochner Laplacian of theLevi-Civita connection $\Delta _{R}$, and another Laplacian\ we construct that is related to the Lefschetz operator and $\partial $ operator on acompact complex manifold $\Delta _{K}$ such that for any $(p,q)-$form in [%\ref{WF}], $\Delta _{K}+\Delta _{H}-2\Delta _{R}=F(R)+$"quadratic terms"where the curvature operator $F(R):E^{p,q}\rightarrow E^{p,q}$ and the quadratic terms are given in [\ref{WFR}]. This formula generalizes a Weitzenbock formula of Wu for K\"{a}hler manifolds in \cite{Wu}. Then, under certain conditions, we show that the Weitzenbock formula provides vanishing theorems for the Dolbeault cohomology groups of complex differential $(p,q)-$%forms and obtain information about the Hodge numbers of the manifold. We use these vanishing theorems to obtain information about the geometric and arithmetic genus and irregularity of a compact complex manifold undercertain conditions. Earlier result of Alfred Gray shows that a hypothetical integrable almost complex structure on a 6-dimensional sphere, $S^{6},$ hasto satisfy $h^{0,1}>0$ \cite{Gray}. We apply our vanishing theorem for $%(0,1)-$forms to show that $h^{0,1}=0$ and thus, under certain additional conditions a 6-dimensional sphere can not have integrable almost complex structure. We use the Fr\"{o}licher spectral sequence to obtain the Hodge-deRham cohomology groups of any compact complex manifold of real dimension 6 and show that under certain conditions, the Euler characteristicof a compact complex manifold of real dimension 6 is positive to prove the Hopf conjecture.