We know Hodge decomposition splits any $k$-form into three $L^2$ components. And I see some proofs, none of them provide an explicit constructive method. Is there any general method to construct one? Or it is just a theorem of existence.

There are strong connections between the Hodge and the Tate conjectures, mainly at the level of similarities and analogies. To quote from an answer of Matthew Emerton on MathOverflow: “[…] we also have a natural abelian (in fact Tannakian) category in play: in the complex case, the category of pure Hodge structures, and [when the […]

As we know, for $V$ vectoral space and a orientation $\mathcal{O}$ on $V$ and $e_{1},…,e_{n}$, the hodge star operator $\ast:\wedge V^*\rightarrow\wedge V^*$ is defined for $\ast(e_{1}\wedge…\wedge e_{n})=\pm 1$(acoording to the orientation preserve/invert for the basis $\{e_{i}\}$) $\ast(1)=\pm e_{1}\wedge…\wedge e_{n}$(again $+$ if $[\{e_{i}\}]=\mathcal{O}$ and $-$ if $[\{e_{i}\}]=-\mathcal{O}$) $e_{1}\wedge…\wedge e_{k}=\pm e_{k+1}\wedge…\wedge e_{n}$ Already prove that $\ast\ast=(-1)^{n(n-k)}\cdot$ and […]

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