UC Irvine

On a symplectic manifold, Tsai, Tseng, and Yau introduced a coholomogy of differentialforms that is analogous to the Dolbeault cohomology for symplectic manifolds. Such forms are called primitive forms. We develop a Morse theory for these primitive forms, including a Morse-type Cone complex of pairs of critical points that has isomorphic cohomology to the primitive cohomology. The differential of the complex consists of gradient flows and an integration of the symplectic form over spaces of gradient flow lines. We prove that the complex is independent of the choice of metric and Morse function. We also derive Morse style inequalities for the cohomology of the Cone complex and thus the primitive cohomologies. Also, we develop a Witten deformation of the Cone complex, which provides a Witten deformation of the differential operators associated to the cohomology.