We obtain an effective lower bound on the distance of the sum of co-adjoint orbits from the origin. Even when the distance is zero (thus the symplectic quotient is well defined) our result gives a nontrivial constraint on these co-adjoint orbits. In the particular case of unitary groups, we obtain the quadratic inequality for eigenvalues of Hermitian matrices satisfying A + B = C. This quadratic inequality can be interpreted as the Chern number inequality for semi-stable reflexive toric sheaves.

We compute the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds continuing the study of [CO]. Related <i>A</i>-formulas hold for a transversal choice of chains. Two different computations are provided: a direct calculation using the classification of holomorphic discs by Oh and the author in [CO], and another method by using an <i>analogue of divisor equation</i> in Gromov-Witten invariants to the case of discs. Floer cohomology rings are shown to be isomorphic to Clifford algebras, whose quadratic forms are given by the Hessians of functions <i>W</i>, which turn out to be the superpotentials of Landau-Ginzburg mirrors. In the case of , this proves the prediction made by Hori, Kapustin and Li by B-model calculations via physical arguments. The latter method also provides correspondence between higher derivatives of the superpotential of LG mirror with

In this paper we study the birational geometry of HyperKaehler manifolds by combining the method of minimal model program and the traditional approach of symplectic geometry.