Examples of the Local $L^2$-Cohomology of Algebraic Varieties

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$L^2$-cohomology is a cohomology theory on Riemannian manifolds. It agrees with de Rham cohomology in the compact case, but is often different in the non-compact case. In some informative examples of stratified spaces, $L^2$-cohomology of the regular part often agrees with the middle-perversity intersection homology of the stratified space, giving a de Rham-style theorem. Unfortunately, this doesn't always happen; we give a family of examples of real algebraic varieties for which the $L^2$-cohomology and middle-perversity intersection homology are not equal.\\

In both this family and in the case of complex singularities, it often happens that we can decompose the space into regions where the metric looks like that of a multiply-warped product, or like interpolations between such regions. An illuminating class of examples is that of normal complex surface singularities. In this case, the decomposition was begun by Hsiang and Pati and completed by Nagase, and this decomposition played a heavy role in the computation of $L^2$-cohomology.\\

Cheeger, Goresky, and MacPherson conjectured that the intersection cohomology of complex projective varieties and the $L^2$-cohomology of their regular part are isomorphic. One hope at the time of the conjecture would be that the proof would shed light on the local structure of complex algebraic singularities. If one instead looks at the local $L^2$-cohomology and ask that it is isomorphic to the local intersection homology, then the conjecture does imply restrictions on the cohomology which is only apparent after a closer look at the local geometry around the singularity.\\

In this thesis, we calculate the local $L^2$-cohomology for several examples of affine real and complex algebraic varieties with isolated singularities with the metric induced with the Euclidean metric. We give examples of real algebraic varieties where the local $L^2$-cohomology is not isomorphic to the middle intersection homology. We give another example where the local $L^2$-cohomology is not even a subspace of the cohomology of the link. We also calculate the local $L^2$-cohomology for a class of weighted homogeneous hypersurfaces; this class of examples includes the $A_k$-singularities in arbitrary dimension. \\







Cruz, Joshua (2020). Examples of the Local $L^2$-Cohomology of Algebraic Varieties. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/20973.


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