Geometric and algebraic restrictions of differential forms

For a smooth manifold $M$ and the space $\Lambda ^p(M)$ of all differential $p$-forms on $M$ the restriction $\omega \vert_N$ of $\omega \in \Lambda ^p(M)$ to a smooth submanifold $N\subset M$ is well defined by the geometry of $N.$ If $N$ is any subset of $M$ then the forms $\alpha +d\beta ,$ $\alpha \in \Lambda ^{p}(M),$ $\beta \in \Lambda ^{p-1}(M),$ where $\alpha $ and $\beta $ annihilates any $p$ - tuple (and $p-1$ - tuple respectively) of vectors in $T_xM$, $x\in N,$ are called algebraically vanishing on $N$ or having zero algebraic restriction to $N$. Now the restriction (algebraic restriction) of $\omega \in \Lambda ^p(M)$ to $N$ is defined as an equivalence class of $\omega $ modulo forms with zero algebraic restriction to $N.$ We study germs of differential forms over singular varieties. The geometric restriction of differential forms to singular varieties is introduced and algebraic restrictions of differential forms with vanishing geometric restrictions, called residual algebraic restrictions, are investigated. Residues of plane curves-germs, hypersurfaces, Lagrangian varieties as well as the geometric and algebraic restriction via a mapping were calculated. This is a joint work with Goo Ishikawa.