Some geometric flow approaches for deformed Hermitian-Yang-Mills equation

Abstract: On SYZ mirror symmetry, a deformed Hermitian-Yang-Mills (dHYM) metric is a fiber metric on a holomorphic line bundle, which is the mirror object to a special Lagrangian section of the dual torus fibration. As a parabolic analogue, Jacob-Yau introduced the Line Bundle Mean Curvature Flow (LBMCF) as the mirror of the Lagrangian Mean Curvature Flow. In this talk, we explore some geometric flow approaches for dHYM metrics: (A) On K\”ahler surfaces, it is known that the existence of dHYM metrics is equivalent to a K\”ahler condition for a certain cohomology class. We relax this condition and study how the LBMCF blows up. (B) Recently, Collins-Yau discovered a new variational characterization for dHYM metrics. Motivated by this, we introduce a new geometric flow which is designed to deform a given metric to a dHYM one. Then we show that this new flow potentially has more global existence and convergence properties than the LBMCF.

algebraic geometryalgebraic topologycomplex variablesdifferential geometrygeometric topologysymplectic geometry

Audience: researchers in the topic

2021 Pacific Rim Complex & Symplectic Geometry Conference