Doctoral Defense Announcement

Z/2-harmonic 1-forms and applications to special holonomy by Dashen Yan

This thesis concerns multivalued sections arising from gauge theory and their applications to special holonomy, in particular to Calabi–Yau 3-folds. The first part of the thesis studies non-degenerate Z/2-harmonic functions and 1-forms. We use a variant of ellipsoidal coordinates on Rn to construct a family of non-degenerate Z/2-harmonic functions on Rn, n ≥ 3. The branching set of these examples is a codimension-2 ellipsoid, giving the first known family of non-degenerate Z/2-harmonic 1-forms on Rn with compact branching sets. We also develop a gluing construction for non-degenerate Z/2-harmonic 1-forms on compact manifolds, in which local non-degenerate Z/2-harmonic models on Rn are glued into the Morse zeros of an ordinary harmonic 1-form. As aconsequence, we show that every compact oriented manifold Mn, n ≥ 3, with first Betti number b1(M) > 0, admits a family of non-degenerate Z/2harmonic 1-forms. This resolves a folklore conjecture. We also discuss several possible applications to special holonomy, especially to G2-geometry. The second part of the thesis studies a dimensional reduction of Donaldson’s adiabatic limit program for G2-manifolds with coassociative fibrations. In this reduced setting, one considers a Calabi–Yau 3-fold equipped with a holomorphic fibration over a two-dimensional base. We construct Calabi–Yau metrics on certain hypersurfaces in C4 from branched harmonic k-fold covers of C in T∗C, which may be viewed as a linear analogue of the dimensionally reduced adiabatic data. The resulting Calabi–Yau manifolds are complete and non-compact, and admit the tangent cone C2/Zk × C at infinity.