5 research outputs found
On vanishing theorems for Higgs bundles
We introduce the notion of Hermitian Higgs bundle as a natural generalization
of the notion of Hermitian vector bundle and we study some vanishing theorems
concerning Hermitian Higgs bundles when the base manifold is a compact complex
manifold. We show that a first vanishing result, proved for these objects when
the base manifold was K\"ahler, also holds when the manifold is compact
complex. From this fact and some basic properties of Hermitian Higgs bundles,
we conclude several results. In particular we show that, in analogy to the
classical case, there are vanishing theorems for invariant sections of tensor
products of Higgs bundles. Then, we prove that a Higgs bundle admits no nonzero
invariant sections if there is a condition of negativity on the greatest
eigenvalue of the Hitchin-Simpson mean curvature. Finally, we prove that
invariant sections of certain tensor products of a weak Hermitian-Yang-Mills
Higgs bundle are all parallel in the classical sense.Comment: 10 Pages, some typos corrected and minor change