We study the quotient complexes of the de Rham complex on singular mappings; the complex of algebraic restrictions, the complex of geometric restrictions and the residual complex. Vanishing theorem for algebraic, geometric and residual cohomologies on quasi-homogeneous map-germs was proved. The finite order and symplectic zero-forms were characterized on parametric singularities. In this context the singular parametric Lagrangian surfaces were investigated, with the classification list of $\mathcal{A}$-simple Lagrangian singularities of $\mathbb{R}^2$ into $\mathbb{R}^4$.

中文翻译：

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奇异映射及其零形式

我们研究奇异映射上的de Rham复数的商复数；代数约束的复数，几何约束的复数和残差复数。证明了拟齐型图胚上代数，几何和残差同调的消失定理。有限阶和辛零形式以参数奇异性为特征。在这种情况下，研究了奇异参量拉格朗日曲面，并将$ \ mathcal {A} $-$ \ mathbb {R} ^ 2 $的简单Lagrangian奇点分类为$ \ mathbb {R} ^ 4 $的分类列表。