Triholomorphic map - Knowlpedia

Definition

Let $(M,I_M,J_M,K_M)$ and $(N,I_N,J_N,K_N)$ be hypercomplex manifolds. A triholomorphic map is a smooth map $f:M\to N$ satisfying

\begin{aligned} df\circ I_M&=I_N\circ df,\\ df\circ J_M&=J_N\circ df,\\ df\circ K_M&=K_N\circ df. \end{aligned}

Thus $f$ is holomorphic for each member of the specified ordered triples. Because $K=IJ$ , any two of these equations imply the third, but writing all three makes the symmetry and convention explicit.

The equation also holds for every induced complex structure

L=aI+bJ+cK,\qquad a^2+b^2+c^2=1,

using the same coefficients on source and target. Identities and composites are triholomorphic, so hypercomplex manifolds with ordered triples and triholomorphic maps form a category.

Metric independence

Triholomorphicity does not require a metric. Between hyperkähler manifolds, a triholomorphic map need not preserve the hyperkähler metrics or two-forms. Adding the pullback equation $f^*g_N=g_M$ makes it a hyperkähler isometric immersion; if it is also a diffeomorphism, it is a hyperkähler isometry.

Convention warning

The ordered triples matter. A rotating hyperkähler isometry with nontrivial rotation is not triholomorphic under this strict definition.

References

Dominic D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000. DOI record. Relevant: Chapters 6–7, hypercomplex and hyperkähler structures.
Daniel Huybrechts, “Compact Hyperkähler Manifolds: Basic Results,” Inventiones Mathematicae 135 (1999), 63–113. DOI record. Relevant: §1, induced complex structures and hyperkähler data.