  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.
