Study on toric varieties, vector bundles on them and their subvarieties

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It is known that for an ample line bundle on a projective toric variety of dimension n the global sections of the line bundle twisted by itself over n-1 times defines an embedding of the toric variety into a projective space. In 1999 we proved that the ideal defining the embedded toric variety is generated by quadrics when the line bundle is twisted over n times. Ample line bundles on toric varieties are heavily related with convex polytopes, we investigated the convex polytopes of dimension 3 and lattice points in them by using computors, and we showed that the ideals of the toric varieties embedded by n-1 times twisted ample line bundles are generated by quadrics when the varieties are quotients of projective space by finite abelian groups.In 2000 we proved that the ideals defining the toric varieties of dimension n embedded by n-1 times twisted ample line bundles are generated by quadrics in general.We have studied the Futaki invariant which is an obstraction to the existence of an Einstein Kaehler metric on the tangent bundle over toric Fano manifold. In 1999 and 2000 we generalized it as the Bando-Calabi-Futaki character which is an obstraction to the existance of a Kaehler metric with constant sscalar curvature, and we showed that the Bando-Calabi-Futaki character is also an obstraction to semistability of algebraic varieties in the geometric invariant theory.We also studied the embedding problem of abelian surfaces into nonsingular toric varieties of dimension 4. We showed that only the product of elliptic curves can be embedded into the product of toric varieties of dimension 2.