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2019-02-17  16:26 hrs.

Chern classes and transversality for singular spaces

Jörg Schürmann
University of Münster


Let $X$ and $Y$ be closed complex subvarieties in an ambient complex manifold $M$. We will explain the intersection formula $$c(X) \cdot c(Y)= c(TM)\cap c(X\cap Y)$$ for suitable notions of Chern classes and transversality for singular spaces. If $X$ and $Y$ intersect transversal in a Whitney stratified sense, this is true for the MacPherson Chern classes (of adopted constructible functions). If $X$ and $Y$ are "splayed" in the sense of Aluffi-Faber, then this formula holds for the Fulton-(Johnson-) Chern classes, and is conjectured for the MacPherson Chern classes.

We explain, that the version for the MacPherson Chern classes is true under a micro-local "non-characteristic" condition for the diagonal embedding of $M$ with respect to $X\times Y$. This notion of non-characteristic is weaker than the Whitney stratified transversality as well as the splayedness assumption.



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