Lefschetz properties for noncompact arithmetic ball quotients. II
For congruence subgroups \(\Gamma\) of SU\((h)\), where \((V,h)\) is an \((n+1)\)-dimensional Hermitian space constructed over an imaginary quadratic field, the author proves injectivity of the following induced morphism. Let \(W\) be an \((m+1)\)-dimensional subspace of \(V\) on which \(h\) is non-degenerate and indefinite, so the standard embedding of \(H := \) SU\((h|_W)\) into SU\((h)\) gives an embedding of the complex \(m\)-ball into the complex \(n\)-ball. Let \(M\) be the orbit space of the action of \(\Gamma\) on the complex \(n\)-ball, and let \(M_H\) be the orbit space of the action of \(\Gamma \cap H\) on the complex \(m\)-ball. Then the above embedding passes to a map \(M_H \to M\) with finite fibers. The author's main theorem states that for each \(m < n\), there exists a finite collection of \((m+1)\)-dimensional subspaces \(W_j\) in \(V\) such that the morphism \[ H^i(M) \to \bigoplus_j H^i(M_{H_j}) \] induced on cohomology with complex coefficients by the above described maps is injective in degrees \(i < m\). This improves a theorem from a previous paper of the author [J. Reine Angew. Math. 730, 163--198 (2017; Zbl 1422.11125)], where this injectivity was proven only for \(i < m-1\). Furthermore, the present paper proves injectivity also for \(i = m\) when restricting to so-called interior cohomology: The image of compactly supported cohomology, which includes the cuspidal cohomology. The arguments for the latter statement at \(i = m\) were present already in the author's previous paper, but not used there for this purpose. Finally, the author builds on the insights gained through this main theorem, and establishes a non-vanishing theorem for cup products involving the action of Hecke operators on the above defined cohomology modules \(H^i(M)\).
scientific article; zbMATH DE number 6690953
scientific article