arXiv subjects
Publications and source records attributed to Mads Bach Villadsen.
We prove the decomposition theorem for Hodge modules with integral structure along proper Kähler morphisms, partially generalizing M. Saito's theorem for projective morphisms. Our proof relies on compactifications of period maps of generically defined variations of Hodge structure, as well as the theorems of Cattani-Kaplan-Schmid on \(L^2\)-cohomology of variations of Hodge structure for the hard Lefschetz theorem.
We give a new proof that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point, first proven by Popa and Schnell using generic vanishing theorems for Hodge modules. Our proof relies on Simpson's results on the relation between rank one Higgs bundles and local systems of one-dimensional complex vectors spaces, and the structure of the cohomology jump loci in their moduli spaces.
We give a new proof of a theorem by Pareschi, Popa and Schnell that the direct image of the canonical bundle of a smooth projective variety along a morphism to an abelian variety admits a Chen-Jiang decomposition, without using the theory of Hodge modules.