Topological and Hodge L-classes of singular covering spaces and varieties with trivial canonical class

Abstract

The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincaré complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more generally, the topological L-class, for closed oriented stratified pseudomanifolds that can be equipped with a middle-perverse Verdier self-dual complex of sheaves, determined by Lagrangian sheaves along strata of odd codimension (so-called L-pseudomanifolds). This class of spaces contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We apply this result in proving the Brasselet-Schürmann-Yokura conjecture for normal complex projective 3-folds with at most canonical singularities, trivial canonical class and positive irregularity. The conjecture asserts the equality of topological and Hodge L-class for compact complex algebraic rational homology manifolds.

The uniqueness theorem for Gysin coherent characteristic classes of singular spaces

“…The conjecture holds for simplicial projective toric varieties as shown by Maxim and Schürmann [29, Corollary 1.2(iii)]. Furthermore, the conjecture was established by the first author in [5] for normal connected complex projective threefolds $$X$$ that have at worst canonical singularities, trivial canonical divisor, and $$\operatorname{dim} H^{1}(X; \mathcal {O}_{X}) > 0$$. Generalizing the above cases, Fernández de Bobadilla and Pallarés [17] proved the conjecture for all compact complex algebraic varieties that are rational homology manifolds.…”

“…Conversely, let us show that 𝑋 𝑎 (𝐸 * ) ∩ 𝑋 𝑏 (𝐹 * ) ⊂ 𝑆(𝑋 𝑎 ′ (𝐸 ′ * ) × 𝑋 𝑎 ′′ (𝐸 ′′ * )) in (5). Given 𝑃 ∈ 𝑋 𝑎 (𝐸 * ) ∩ 𝑋 𝑏 (𝐹 * ), we set 𝑃 ′ = 𝑃 ∩ 𝐸 𝑛 ′ and 𝑃 ′′ = 𝑃 ∩ 𝐹 𝑛 ′′ , and note that dim ℂ (𝑃 ′ ) = dim ℂ (𝑃 ∩ 𝐸 𝑛 ′ ) ⩾ dim ℂ (𝑃 ∩ 𝐸 𝑎 𝑘+1−𝑘 ′ +𝑘 ′ ) ⩾ 𝑘 ′ because 𝑃 ∈ 𝑋 𝑎 (𝐸 * ) (where we note that 𝐸 𝑎 𝑘+1−𝑘 ′ +𝑘 ′ ⊂ 𝐸 𝑛 ′ holds because 𝑎 𝑘 ′′ +1 ⩽ 𝑚 ′ implies that 𝑎 𝑘+1−𝑘 ′ + 𝑘 ′ ⩽ 𝑛 ′ ), and dim ℂ (𝑃 ′′ ) = dim ℂ (𝑃 ∩ 𝐹 𝑛 ′′ ) = dim ℂ (𝑃 ∩ 𝐹 𝑏 𝑘+1−𝑘 ′′ +𝑘 ′′ ) ⩾ 𝑘 ′′ because 𝑃 ∈ 𝑋 𝑏 (𝐹 * ) (where we note that 𝐹 𝑏 𝑘+1−𝑘 ′′ +𝑘 ′′ = 𝐹 𝑛 ′′ holds because 𝑏 𝑘 ′ +1 = 𝑚 ′′ implies that 𝑏 𝑘+1−𝑘 ′′ + 𝑘 ′′ = 𝑛 ′′ ).…”

“…A different axiomatization involving codimension 0 restrictions was used by Matsui in [28, p. 61] to investigate ambient intersection formulae for Goresky–MacPherson $$L$$‐classes. In [5], the first author analyzed the behavior of $$L$$‐classes under Gysin transfers associated to finite degree covers.…”

“…In fact, using that 𝑎 ⩽ [𝑚 ′ × 𝑘 ′ ] ⊔ [𝑚 ′′ × 𝑘 ′′ ] and 𝑏 ⩽ [𝑚 ′′ × 𝑘 ′′ ] ⊔ [𝑚 ′ × 𝑘 ′ ], we then obtain𝑋 𝑎 (𝐸 * ) ∩ 𝑋 𝑏 (𝐹 * ) = 𝑆(𝑋 𝑎 ′ (𝐸 ′ * ) × 𝑋 𝑎 ′′ (𝐸 ′′where we have also used that 𝑆 is injective.) In the following, we show(5), where we write𝑏 ′ = [𝑚 ′ × 𝑘 ′ ], 𝑏 ′′ = [𝑚 ′′ × 𝑘 ′′ ], and 𝑏 = [𝑚 ′′ × 𝑘 ′′ ] ⊔ [𝑚 ′ × 𝑘 ′ ] (the proofof (6) is similar). Let us first show the inclusion 𝑆(𝑋 𝑎 ′ (𝐸 ′ * ) × 𝑋 𝑎 ′′ (𝐸 ′′ * )) ⊂ 𝑋 𝑎 (𝐸 * ) ∩ 𝑋 𝑏 (𝐹 * ) in (5).…”

Gysin Restriction of Topological and Hodge-Theoretic Characteristic Classes for Singular Spaces

“…The behavior of the L-class for singular spaces under Gysin transfers associated to finite degree covers is already completely understood. In [5], we showed that for a closed oriented Whitney stratified pseudomanifold X admitting Lagrangian structures along strata of odd codimension (e.g. X Witt) and p : X ′ → X an orientation preserving topological covering map of finite degree, the L-class of X transfers to the L-class of the cover, i.e.…”