Topological Gysin Coherence for Algebraic Characteristic Classes of Singular Spaces

Abstract Brasselet, the second author and Yokura introduced Hodge-theoretic Hirzebruch-type characteristic classes $$IT_{1, *}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , and conjectured that they are equal to the Goresky-MacPherson L -classes for pure-dimensional compact complex algebraic varieties. In this paper, we show that the framework of Gysin coherent characteristic classes of singular complex algebraic varieties developed by the first and third author in previous work applies to the characteristic classes $$IT_{1, *}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . In doing so, we prove the ambient version of the above conjecture for subvarieties in a Grassmannian. Since the homology of Schubert subvarieties injects into the homology of the ambient Grassmannian, this implies the conjecture for all Schubert varieties in a Grassmannian. We also study other algebraic characteristic classes such as Chern classes and Todd classes (or their variants for the intersection cohomology sheaves) within the framework of Gysin coherent characteristic classes.