Smooth Algebraic Transfer for Intersection Hirzebruch Classes

We prove a smooth algebraic transfer formula for intersection Hirzebruch classes of pure-dimensional complex algebraic varieties. If \(\pi\colon X\to Y\) is a smooth algebraic morphism of pure fiber dimension between pure-dimensional complex algebraic varieties, then \(\Ty(T_{X/Y})\cap \pi_{\BM}^*\IT_{y*}(Y)=\IT_{y*}(X)\), and specializing at \(y=1\) gives the \(L\)-class transfer formula \(L^*(T_{X/Y})\cap \pi_{\BM}^*\IT_{1*}(Y)=\IT_{1*}(X)\). The formula completes a basic pair of functorial operations for intersection Hirzebruch classes: it is the smooth-morphism counterpart to Banagl's algebraic Gysin formula for \(\IT_{1*}\) under upwardly normally nonsingular embeddings, and it establishes the smooth algebraic transfer formula anticipated in Banagl's work on \(L\)-homology bundle transfer for Witt spaces. Both base and total space may be singular; only the morphism is required to be smooth, so the formula computes the intersection Hirzebruch class of a smooth family directly from the class of its base together with the relative tangent correction. Beyond the class-level statement, we isolate the precise structural input governing its object-level refinement: we show that whether the canonical comparison \(\pi^*\IC_Y^H[r]\simeq\IC_X^H\) holds as an isomorphism of mixed Hodge modules reduces to a single compatibility, namely that shifted smooth pullback commutes with the \(j_!/j_*\) boundary package of the intersection Hodge module, equivalently with the associated Beilinson--MacPherson--Vilonen gluing data. This identifies exactly what must be supplied to upgrade Banagl's class-level identity in \(K_0(\MHM)\) to an object-level isomorphism. We also establish the smooth pullback formula for the full motivic Hirzebruch transformation \(\MHT_{y*}\), prove compatibility with composition of smooth morphisms, recover the vector bundle and zero-section cases, and combine the smooth theorem with Banagl's algebraic Gysin theorem to obtain a restricted lci-type composition principle. The results are illustrated through examples, including projective bundles over Schubert varieties.