In this paper we prove the existence and uniqueness of the form-type equation on Khler manifolds of nonnegative orthogonal bisectional curvature.

We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number (in the appropriate cases) and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, hyperk¨ahler and G2-manifolds respectively. For vector bundles over such manifolds, we introduce (special) A-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Similarly we introduce (special) 1/2 A-Lagrangian submanifolds as maximally real submanifolds. They include (special) Lagrangian, complex Lagrangian, Cayley and (co-)associative submanifolds. We also discuss geometric dualities from this viewpoint: Fourier transformations on A-geometry for flat tori and a conjectural SYZ mirror transformation from (special) A-geometry to (special) 1/2 A-Lagrangian geometry on mirror special A-manifolds.

This is a continuation of our first paper in [WY16]. There are two purposes of this paper: One is to give a proof of the main result in [WY16] without going through the argument depending on numerical effectiveness. The other one is to provide a proof of our conjecture, mentioned in [TY], where the assumption of negative holomorphic sectional curvature is dropped to quasi-negative. We should note that a solution to our conjecture is also provided by Diverio-Trapani [DT]. Both proofs depend on our argument in [WY16]. But our argument here makes use of the argument given by the second author and Cheng in [CY75].

Vector cross product structures on manifolds include symplectic, volume, G2- and Spin(7)-structures. We show that the knot spaces of such manifolds have natural symplectic structures, and relate instantons and branes in these manifolds to holomorphic disks and Lagrangian submanifolds in their knot spaces. For the complex case, the holomorphic volume form on a Calabi–Yau manifold defines a complex vector cross product structure. We show that its isotropic knot space admits a natural holomorphic symplectic structure.We also relate the Calabi–Yau geometry of the manifold to the holomorphic symplectic geometry of its isotropic knot space.

In this paper, we compute the adiabatic limit of the scalar curvature and prove several vanishing theorems by taking adiabatic limits. As an application, we give a Kastler-Kalau-Walze type theorem for foliations.