Let $X$ be a complex algebraic variety. With $\mathbb{ Q}$-coefficients, the compactly supported cohomology groups $H^{ i}_{ c}(X, \mathbb{ Q})$ and the compactly supported intersection cohomology groups $IH^{ i}_{ c}(X, \mathbb{ Q})$ have mixed Hodge structures. We compare these two mixed Hodge structures for varieties with pre-$k$-rational singularities. We then study various notions of differential forms on varieties with pre-$k$-rational singularities. In particular, we investigate the depth of the complex $\underline{ \Omega}^{ p}_{ X}$, where $\underline{ \Omega}^{ p}_{ X}$ is the $p^{ th}$-graded piece of the Du Bois complex.
Categories: math.AG