<p>A Hodge module on an algebraic variety may be viewed as a variation of Hodge structure with singularities. Given an irreducible variety $X$, for any polarized variation of Hodge structure $\bold{H}$ on a smooth open subvariety $U\subset X,$ there exists a unique Hodge module $\cM \in HM_{X}(X)$ that extends $\bH.$ Conversely, for any Hodge module $\cM \in HM_{X}(X)$ with strict support on $X,$ there exists a polarized variation of Hodge structure $\bH$ on a smooth open subset $U \subset X$ such that $\cM \vert _{V} \cong \bH.$ In this thesis, we first study the singularities of a Hodge module $\cM \in HM_{X}(X)$ by using Morihiko Saito's theory of $S$-sheaves and duality. Then using local cohomology and the theory of mixed Hodge modules, we study the Hodge structure of $H^{i}(X, DR(\cM))$ when $X$ is a projective variety. Finally, we consider a variation of Hodge structure $\bH$ on $U$ as a Hodge module $\cN \in HM(U)$ on $U,$ and study the local cohomology of the complex $Gr^{F}_{p}DR(j_{!}\cN) \in D^{b}_{coh}(\cO_{X}),$ where $j: U \hookrightarrow X$ is the natural map.</p>
| Identifer | oai:union.ndltd.org:purdue.edu/oai:figshare.com:article/22685053 |
| Date | 25 April 2023 |
| Creators | Scott M Hiatt (15347473) |
| Source Sets | Purdue University |
| Detected Language | English |
| Type | Text, Thesis |
| Rights | CC BY 4.0 |
| Relation | https://figshare.com/articles/thesis/Duality_and_Local_Cohomology_in_Hodge_Theory/22685053 |
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