Locally Conformally Symplectic Structures on Compact Non-Kähler Complex Surfaces

Abstract

Abstract. We prove that every compact complex surface with odd first Betti number admits a locally conformally symplectic 2-form which tames the underlying almost complex structure.

Morse-Novikov cohomology of locally conformally Kähler surfaces

“…It was shown in [AD,Lemma 4.1] that on a compact complex surface (M, J) with b 1 (M ) = 1, the sign of deg g L η does not depend on the choice of the Gauduchon metric on M . In the case of (S 0 , J), we can choose as Gauduchon metric Tricerri's metric ω, with its corresponding Lee form θ (see also Remark 3.10).…”

Conformal symplectic geometry of cotangent bundles

“…We have the following version of Moser's Theorem (first proved in [4]). 3 The reader experienced with contact geometry will be familiar with this situation: the contact vector field X H associated to a Hamiltonian H ∈ C ∞ M depends on a choice of contact form, but the correct way to define the relationship without reference to a contact form is to take contact Hamiltonians H ∈ Ω 0 (M, T M/ξ). Theorem 2.10.…”

“…It is an open question whether we can in general construct conformal symplectic structures where [η] is a prescribed 1-form, either in the case where [η] is small, or when [η] : π 1 M → R has non-discrete image. Note that in [3], there is one example that shows that after fixing a complex structure J, the set of [η] for which there exists a conformal symplectic structure compatible with J consists of a single point. Remark 2.17.…”

Locally conformally symplectic and Kähler geometry

“…Thus our phase space consists of T * Q with an open covering {U α } and a symplectic form ω α can on each U α such that, on U α ∩U β = / 0, (7) ω α can = µ β α ω β can .…”

“…The blow-up of a locally conformally Kähler manifold was studied in [106,121,133]. An interesting contact point between locally conformally symplectic and Kähler geometry appears in the papers [7,8]. The authors consider locally conformally symplectic structures (ω, ϑ ) on compact complex surfaces (M, J) such that ω tames J, i. e. the (1, 1)-part of ω is positive definite.…”