Sanford Meisner Acting Master Class4sharedtorrenttorrent 📛

Sanford Meisner Acting Master Class4sharedtorrenttorrent

Sanford Meisner Acting Master Class4sharedtorrenttorrentQ:

If $\phi$ is of weight $0$ relative to a specific Hodge structure, then it is of weight $0$ in general?

I am trying to prove the following statement:
Let $H$ be a complex analytic (possibly non-compact) manifold $H$ and $E$ a holomorphic vector bundle on $H$. Suppose $E$ is equipped with a flat connection $D$. Let $\mathcal{H}$ be the space of sections of $E$ and let $F$ be a Hodge structure of weight $0$ on $H$ (i.e. $F^k=\{0\}$ for $k \leq -1$). Suppose $\phi:E \to F$ is a map from $\mathcal{H}$ into $F$. If there exists a section of $F$ such that the map $s:=\phi-\phi_0$, where $\phi_0$ is the component of $\phi$ which is homogeneous of weight $0$ with respect to the weight filtration, is flat (i.e. $Ds=0$), then $s$ is of weight $0$ with respect to the Hodge structure induced on $\mathcal{H}$.
In general, this theorem is not true, but it is true when we have the restriction that $E$ has a Hodge structure.
To me, this sounds quite strange, but maybe I am missing something. Any help would be appreciated. Thanks!

Ok I found the answer: I am not quite sure if this is an answer to my question but this statement is definitely true when $E$ has a Hodge structure and $s:E \to F$ is a map of modules (i.e. it commutes with direct sums and $\otimes$).

Leo Varadkar, photo: Brian Lawless/PA Wire

Leo Varad

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