Аннотация:
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.
