For a one-parameter family (V, \Omega_{i}) with I=1,...,p_g of general type hypersurfaces with bases of holomorphic n-forms, we construct open covers V = Spg i=1 Ui using tropical geometry. We show that after normalization, each i is approximately supported on a unique Ui and such a pair approximates a Calabi-Yau hypersurface together with its holomorphic n-form as the parameter becomes large. We also show that the Lagrangian fibers in the fibration constructed by Mikhalkin [9] are asymptotically special Lagrangian. As the holomorphic n-form plays an important role in mirror symmetry for Calabi-Yau manifolds, our results is a step toward understanding mirror symmetry for general type manifolds.

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.

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Almost toric manifolds form a class of singular Lagrangian fibered symplectic manifolds that include both toric manifolds and the K3 surface. We classify closed almost toric four-manifolds up to diffeomorphism and indicate precisely the structure of all almost toric fibrations of closed symplectic four-manifolds. A key step in the proof is a geometric classification of the singular integral affine structures that can occur on the base of an almost toric fibration of a closed four-manifold. As a byproduct we provide a geometric explanation for why a generic Lagrangian fibration over the two-sphere must have 24 singular fibers.

It is well known that del Pezzo surfaces of degree 9 − n one-to-one correspond to flat En bundles over an elliptic curve. In this paper, we construct ADE-bundles over a broader class of rational surfaces that we call ADE-surfaces, and extend the above correspondence to all flat G-bundles over an elliptic curve, where G is any simply laced, simple, compact and simply connected Lie group. In what follows, we will construct G-bundles for a non-simply laced Lie group G over these rational surfaces, and extend the above correspondence to non-simply laced cases.