Geometric Formalities Along the Chern-Ricci Flow

Abstract

In this paper, we study how the notions of geometric formality according to Kotschick and other geometric formalities adapted to the Hermitian setting evolve under the action of the Chern-Ricci flow on class VII surfaces, including Hopf and Inoue surfaces, and on Kodaira surfaces.ker ∂∂ im ∂+im ∂ [Aep65] cohomologies provide further cohomological invariants on complex manifolds.and we study the possible algebra structure on the space of (de Rham, Dolbeault, Bott-Chern, Aeppli) harmonic forms with respect to ω(t) varying t.We study in details geometric formality according to Kotschick for a whole class of surfaces evolving by the Chern-Ricci flow, i.e. compact complex non-Kähler surfaces with Kodaira dimension Kod(X) = −∞ and first Betti number b 1 (X) = 1, known as class VII of the Enriques-Kodaira classification. In particular, we first rule out class VII surfaces with second Betti number b 2 > 0 by applying arguments as in [Kot01]. Then, we exploit the structure of quotients of Lie groups with invariant complex and Hermitian structure on the only class VII surfaces with b 2 = 0, that is Hopf and Inoue surfaces see [Bog82,Kod64,LYZ94,Tel94], in order to reduce the description of harmonic forms and the equation of the Chern-Ricci flow of such surfaces at the level of invariant forms and thus make explicit computations. We obtain the following.Theorem 3.1. On class VII surfaces, the Chern-Ricci flow preserves the notion of geometric formality according to Kotschick starting at initial invariant metrics.We also study the evolution of geometric formality according to Kotschick on other compact complex non-Kähler surfaces that are diffeomorphic to solvmanifolds, e.g. Kodaira surfaces. Since any complex structures on such surfaces is left-invariant, see [Has05, Theorem 1], we focus on invariant forms also in this case.Proposition 3.2. On Kodaira surfaces, geometric formality according to Kotschick is preserved by the Chern-Ricci flow starting at initial invariant metrics.We note that, also in this case, it is possible to rule out primary Kodaira surfaces by the obstructions in [Kot01] or [Has89], and therefore we focus on secondary Kodaira surfaces with initial invariant metrics.Regarding Dolbeault and Bott-Chern geometric formalities evolving by the Chern-Ricci flow, by applying the analogous procedure on Hopf, Inoue, and Kodaira surfaces, we have reached results as follows. We also checked how the algebraic structures of Aeppli cohomology and its harmonic representatives are modified along the Chern-Ricci flow.