Hodge Theory and A∞ Structures on Cohomology

We use Hodge theory and a construction of Merkulov to construct A∞ structures on de Rham cohomology and Dolbeault cohomology. Hodge theory is a powerful tool in differential geometry. Classically, it can be used to identify the de Rham cohomology of a closed oriented Riemannian manifold with the space of harmonic forms on it as vector spaces. The wedge product on differential forms provides an algebra structure on the de Rham cohomology. By virtue of being isomorphic to the de Rham cohomology, the space of harmonic forms has an induced associative multiplication. However, the wedge product of two harmonic forms may not be harmonic. One needs to define a multiplication of two harmonic forms by taking the harmonic part of their wedge product, and then show that this multiplication is indeed associative and can be identified with the wedge product on the de Rham cohomology. In this paper, we show that one can actually take advantage of this awkward situation to construct higher multiplications and define a structure of A∞ algebra on the space of harmonic forms. Originally, A∞ structures were introduced by Stasheff [7, 8] in 1963 in the study of H spaces. Together with various cousins, they have appeared in the last couple of decades in many places in Mathematics and Mathematical Physics. In particular, infinite structures are very useful in the formulation of mirror symmetry. For example, Fukaya [2] constructed A∞ categories from symplectic manifolds, which is used in the formulation of homological mirror symmetry by Kontsevich [4]. For Calabi-Yau manifold, where the notion of mirror symmetry was originally conceived [9, 3], recent formulations of the mirror symmetry use the notion of Frobenius manifolds introduced by Dubrovin [1]. As pointed out in Manin [5], a formal Frobenius manifold structure on a vector space with a nondegenerate pairing is equivalent to a cyclic Comm∞-structure on it. The popular theory of quantum cohomology provides construction of formal Frobenius manifold structures on de Rham cohomology of symplectic manifolds. The appearance of infinite algebra structures in the theory of quantum cohomology and mirror symmetry indicates the importance of the study of infinite algebras in differential geometry. Our construction is based on a recent paper of Merkulov [6], where he gave a nice construction of A∞ algebra. Together with Hodge theory of closed Kähler manifold, he constructed an A∞ structure on a subcomplex of the de Rham complex which contains the harmonic forms. Similar constructions can be carried out for the deformation complex of Calabi-Yau manifolds. The simple observation of this paper is that Merkulov’s construction can be used to give a construction of A∞ struction on the space of harmonic forms of any oriented closed Riemannian manifold. Similarly, Hodge theory of the deformation complex of any closed complex M manifold can be used to construct an A∞ structure on H (M). Similar constructions can