Complex and Kähler structures on Compact Solvmanifolds

Abstract

We discuss our recent results on the existence and classification problem of complex and Kähler structures on compact solvmanifolds. In particular, we determine in this paper all the complex surfaces which are diffeomorphic to compact solvmanifolds (and compact homogeneous manifolds in general).

Exotic smooth structures and symplectic forms on closed manifolds

“…To the authors knowledge, the final clean proof was given in [Ha,Ha1]. In [C,C1,ABCKT] some properties of the Albanese map of Kähler K(π, 1)-manifolds with solvable π were established.…”

“…As a result of efforts of [A,AN,Ha,Ha1,TK] together with developments from algebraic geometry (see [BC,C1]) the final (affirmative) solution of this problem was recently achieved (See Section 3). To the authors knowledge, the final clean proof was given in [Ha,Ha1].…”

On deformations of compact balanced manifolds

“…In fact it turns out that N can also be presented as a compact quotient of a 6-dimensional simply connected completely solvable Lie group by a uniform discrete subgroup (see [5] and [11,Example 5]). Let us consider the complex structure on N defined by saying that the following ϕ j = e j + ie j+3 , j = 1, 2, 3, are the complex (1, 0)-forms.…”

Non-Kähler solvmanifolds with generalized Kähler structure

“…admits a compact quotient by a uniform discrete subgroup of the form Γ 1 = Z ϕ 1 Z 3 , since it can be identified with the Inoue surface M 4 of type S 0 [22], described as in [17,32]. More precisely, the action ϕ 1 can be given by assigning a matrix ϕ 1 (1) = (m jk ) ∈ SL(3, Z), with two conjugate eigenvalues α, α and a irrational eigenvalue c > 1 such that |α| 2 c = 1 and considering the product on R (R × C) defined by…”

“…The complex surface S 0 can be also obtained as a 4-dimensional solvmanifold [17]. Moreover, since by Hattori's theorem [19] its de Rham cohomology is given by the invariant one, it is easy to check that S 0 is formal.…”