Nonabelian Hodge theory and vector valued modular forms
release_l2p4ryksnfghhhe4sexouakaci
by
Cameron Franc, Steven Rayan
2018
Abstract
We examine the relationship between nonabelian Hodge theory for Riemann
surfaces and the theory of vector valued modular forms. In particular, we
explain how one might use this relationship to prove a conjectural three-term
inequality on the weights of free bases of vector valued modular forms
associated to complex, finite dimensional, irreducible representations of the
modular group. This conjecture is known for irreducible unitary representations
and for all irreducible representations of dimension at most 12. We prove new
instances of the three-term inequality for certain nonunitary representations,
corresponding to a class of maximally-decomposed variations of Hodge structure,
by considering the same inequality with respect to a new type of modular form,
called a "Higgs form", that arises naturally on the Dolbeault side of
nonabelian Hodge theory. The paper concludes with a discussion of a strategy
for reducing the general case of nilpotent Higgs bundles to the case under
consideration in our main theorem.
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