In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if E is an ample vector bundle over a compact Khler manifold E , $ S^ kE\ts\det E $ is both Nakano-positive and dual-Nakano-positive for any E . Moreover, $ H^{n, q}(X, S^ kE\ts\det E)= H^{q, n}(X, S^ kE\ts\det E)= 0$ for any E . In particular, if E is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle $(S^ kE\ts\det E, S^ kh\ts\det h) $ is both Nakano-positive and dual-Nakano-positive for any E .

We study the rigidity results for self-shrinkers in Euclidean space by restriction of the image under the Gauss map. The geometric properties of the target manifolds carry into effect. In the self-shrinking hypersurface situation Theorem 3.1and Theorem 3.2 not only improve the previous results, but also are optimal. In higher codimensional case, using geometric properties of the Grassmannian manifolds (the target manifolds of the Gauss map) we give a rigidity theorem for self-shrinking graphs.

For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov partition function. Using the solution of the Nekrasov conjecture [33, 38, 3] and its refinement [34], we apply this result to give a generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with b+ = 1 in terms of modular forms. This formula was proved earlier in [19] more generally for simply connected 4-manifolds with b+ = 1, as- suming the Kotschick-Morgan conjecture, and it was also derived by physical arguments in [31].

We develop some results from [4] on the positivity of direct image bundles in the particular case of a trivial ¯bration over a one-dimensional base. We also apply the results to study variations of KÄahler metrics.

We give a local classification of generalized complex structures. About a point, a generalized complex structure is equivalent to a product of a symplectic manifold with a holomorphic Poisson manifold. We use a Nash-Moser type argument in the style of Conn’s linearization theorem.