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Computing Coefficients in Certain Normally Nonsingular Expansions

In his paper Banagl derives explicit formulae for the coefficients in the linear combination (for which he coins the term normally nonsingular expansion) of the Goresky-MacPherson L-class in real codimension 4 of certain singular Schubert varieties. However, these formulae still contain non- trivial Kronecker products, namely Kronecker products of the pulled back cohomological Hirzebruch L-class of the normal bundle of some transversely intersecting smooth submanifold in an ambient Grassmannian (or cup products in which such L-classes appear) with the fundamental class of the underlying transverse intersection. The aim of this thesis is to develop general techniques to compute these concrete Kronecker products and therefore determine the coefficients in the related normally nonsingular expansions. In particular, we will determine the coefficients in the normally nonsingular expansion of $L_6(X_{3,2})$, i.e. the sixth Goresky-MacPherson L-class of the Schubert variety $X_{3,2}$ . A great deal of these techniques will be to transform certain linear bases into one another: One interesting result we obtain in this regard is the novel Epsilon-Algorithm, which – as it turns out later – is a weaker version of a well-known theorem in algebraic geometry that Griffiths-Harris [GH78] call Gauss-Bonnet (I).

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Year: 2023

Author : Tom Stalljohann

Markus Banagl