Subvarieties of hypercomplex manifolds with holonomy in SL(n,H)

TL;DR

This paper investigates the structure of subvarieties in hypercomplex manifolds with special holonomy, proving the absence of divisors and the trianalytic nature of certain subvarieties under specific conditions.

Contribution

It establishes new results about the non-existence of divisors and the hypercomplexity of codimension 2 subvarieties in hypercomplex manifolds with SL(n,H) holonomy.

Findings

No divisors in (M,L) under given conditions

All codimension 2 subvarieties are trianalytic

Results apply to manifolds with Obata holonomy in SL(n,H) and HKT metrics

Abstract

A hypercomplex manifold M is a manifold with a triple I,J,K of complex structure operators satisfying quaternionic relations. For each quaternion L=aI +bJ+cK, L^2=-1, L is also a complex structure operator on M, called an induced complex structure. We are studying compact complex subvarieties of (M,L), when L is a generic induced complex structure. Under additional assumptions (Obata holonomy contained in SL(n,H), existence of an HKT metric), we prove that (M,L) contains no divisors, and all complex subvarieties of codimension 2 are trianalytic (that is, also hypercomplex).