Futaki invariants and Yau’s Conjecture on the Hull–Strominger system

Abstract We find a new obstruction to the existence of solutions of the Hull–Strominger system, which goes beyond the balanced property of the Calabi–Yau manifold <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (X,\Omega) and the Mumford–Takemoto slope stability of the bundle over it. The basic principle is the construction of a (possibly indefinite) Hermitian Einstein metric on the holomorphic string algebroid associated to a solution of the system, provided that the connection ∇ on the tangent bundle is Hermitian Yang–Mills. Using this, we define a family of Futaki invariants obstructing the existence of solutions in a given balanced class. Our results are motivated by a strong version of a conjecture by Yau on the existence problem for these equations.