Let $X$ be a compact Kähler manifold of dimension $n$. Each Kähler class $\omega$ on $X$ defines an adjoint Lefschetz operator $\Lambda$, and using this we can make $H^{1,1}(X,\mathbb C)$ into an algebra by setting $$ u \cdot v := \tfrac 12 \Lambda(u \cup v). $$ Varying the class $\omega$ varies the algebra structure.
These structures are not super nice; they are commutative, but lack an identity and are not associative. They pop up naturally when we give the Kähler cone of $X$ the structure of a Riemannian manifold by saying that the Riemannian metric at a point $\omega$ is the one defined on real $(1,1)$-classes by $\omega$. The curvature tensor of that metric is then $$ R(u,v,z,w) = -\tfrac14 \langle \Lambda(u\cup w), \Lambda(v \cup z) \rangle + \tfrac14 \langle \Lambda(u \cup z), \Lambda(v \cup w) \rangle, $$ so it kind of feels like one should be able to say things about the curvature of that metric by studying that algebraic structure.
Are there any attempts in the literature to study those structures, and maybe to constrain their properties based on some underlying topological or complex geometric features of the underlying manifold?
