Many evolutionarily distant pathogenic organisms have evolved similar survival strategies to evade the immune responses of their hosts. These include antigenic variation, through which an infecting organism prevents clearance by periodically altering the identity of proteins that are visible to the immune system of the host. Antigenic variation requires large reservoirs of immunologically diverse antigen genes, which are often generated through homologous recombination, as well as mechanisms to ensure the expression of one or very few antigens at any given time. Both homologous recombination and gene expression are affected by three-dimensional genome architecture and local DNA accessibility. Factors that link three-dimensional genome architecture, local chromatin conformation and antigenic variation have, to our knowledge, not yet been identified in any organism. One of the major obstacles to studying the role of genome architecture in antigenic variation has been the highly repetitive nature and heterozygosity of antigen-gene arrays, which has precluded complete genome assembly in many pathogens. Here we report the de novo haplotype-specific assembly and scaffolding of the long antigen-gene arrays of the model protozoan parasite Trypanosoma brucei, using long-read sequencing technology and conserved features of chromosome folding. Genome-wide chromosome conformation capture (Hi-C) reveals a distinct partitioning of the genome, with antigen-encoding subtelomeric regions that are folded into distinct, highly compact compartments. In addition, we performed a range of analyses—Hi-C, fluorescence in situ hybridization, assays for transposase-accessible chromatin using sequencing and single-cell RNA sequencing—that showed that deletion of the histone variants H3.V and H4.V increases antigen-gene clustering, DNA accessibility across sites of antigen expression and switching of the expressed antigen isoform, via homologous recombination. Our analyses identify histone variants as a molecular link between global genome architecture, local chromatin conformation and antigenic variation.