Complex Analytic and Differential Geometry 2025, 28 - Direct Image Theorem

Complete analytic schemes X, Y with F: X → Y a proper analytic morphism. If S is a coherent O_X-module, the direct images R^qR_*S are coherent O_Y. For locally convex topological vector spaces E, F their topological tensor product is the Hausdorff completion of E ⊗ F, with the family of semi-norms p ⊗_πq associated to fundamental families of semi-norms on E, F. It preserves epimorphisms and monomorphisms. For f: E → F of is unclear if f(x) = Σλ_jξ_j(x)y_j with a sequence of scalars (λ_j) so that Σ|λ_j| < ∞, ξ_j ∈ E' an equicontinuous sequence of linear forms and y_j a bounded sequence. Banach spaces E, F, G with f: E → F nuclear,

Note B(E) as the Banach space for some induced form in the index. A Frechet space E is nuclear if the topology of E is defined by an increasing sequence of semi-norms p_j with canonical B(E)_pj+1 → B(E)_pj is nuclear. A nuclear space E with morphism f: E→F is nuclear iff f admits a factorization E → M → F through a Banach space M. For E, F, G Banach spaces, if f: E → F is nuclear, then f is factorized through a Hilbert space H as a morphism E → H → F. If g: F → G another nuclear morphism. If Im(g ○ f) is contained in a closed subspace T of G, then g ○ f is nuclear. If ker(g ○ f) contains a closed subspace S of E, then (g ○ f)˜: E/S → G is nuclear.

A coherent analytic sheaf on a complex analytic scheme (X, O_X), then F(X) is a nuclear space. For coherent sheaves F, G on complex analytic schemes X, Y, there is a canonical isomorphism between F □ F(X × Y) ≅ F(X) ^⊗ G(Y). For coherent sheaves F, G over complex analytic schemes X, Y with projection π: X × T → X and H^○(Y, G) Hausdorff, then if X is Stein, then H^q(X × Y, F □ G) ≅ F(X) ^⊗ H^q(Y, G). For all open sets U in X (R^qπ_*(F □ G))(U) = F(U) ^⊗ H^q(Y, G). If H^q(Y, G) is finite-dim then R^qπ_*(F □ G) = F ⊗ H^q(Y, G).

A Frechet / nuclear A-module E is a Frechet / nuclear space E with A-module structure and multiplication A × E → E continuous. E is nuclearly free if E is of the form A ^⊗ V where V is a nuclear Frechet space. Every nuclear A-module E admits a direct nuclearly free resolution. For nuclearly free, non-direct resolution L_○ of F there is a canonical isomorphism Tor_q^A(E, F) ≅ H_q(E ^⊗_A L_○). Two nuclear A-modules E, F are transverse if E ^⊗_A F is Hausdorff and if Tor_q^A(E, F) = 0 for q ≥ 1. Stein spaces X, Y with U' ⊂ U ⋐ X, V ⋐ Y Stein open, and a coherent sheaf F over X × Y, then O(U') and F(U × V) are transverse over O(U).

A fully nuclear algebra A, with complexes of fully A-transverse nuclear A-modules E^○, F^○and a morphism of complexes f^○ between them and f^q A-subnuclear. If the complexes are bounded on the right and H^q(f^○) is an isomorphism for each q. For all t < 1 ∃ complex L^○ of finitely generated free A_t-modules and a complex morphism h^○ onto E^○_t onto it, inducing an isomorphism on cohomology.