2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/162063We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of non-commutative Hodge structures, investigate existence and variations, and propose explicit construction and classification techniques. We study the main examples of non-commutative Hodge structures coming from a symplectic or a complex geometry possibly twisted by a potential. We study the interactions of the new Hodge theoretic invariants with mirror symmetry and derive non-commutative analogues of some of the more interesting consequences of Hodge theory such as unobstructedness and the construction of canonical coordinates on moduli.124 pages, 8 figuresAlgebraic GeometryHigh Energy Physics - TheorySymplectic Geometry14C30, 14J32, 19D55, 34M40text