We prove the local hard Lefschetz theorem and local Hodge–Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kübel, one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.

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 .

Consider an N-Boson system interacting via a two-body repulsive short-range potential $V$ in a three dimensional box  of side length $L$. We take the $limit N,L \to \infty $ while keeping the density $\rho = N/L^3 fixed and small. We prove a new lower bound for its ground state energy per particle $$ \frac{E(N,\Lambda)}{N} \ge 4 \pi a \rho [1- O(\rho^{1/3} | log \rho |^3) ] $$ as $\rho \to 0$, where $a$ is the scattering length of $V$.

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