Formality and the Lefschetz property in symplectic and cosymplectic geometry

Abstract

We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the b 1 = 1 case (in the coKähler/cosymplectic situation). MSC: 53C15, 55S30, 53D35, 55P62, 57R17.

Structure of locally conformally symplectic Lie algebras and solvmanifolds

“…If ω 14 ‰ 0, the automorphism (54) with a "´ω 24 ω 14 and b " c " 0 gives ϑ 1 " ϑ and Ω 1 " ω 12 pe 12´1 2 e 34 q`ω 14 e 14 . Using (55), according to the sign of ω 12 , with s " 1 ?˘ω 12 gives ϑ 2 "´1 2 e 4 and Ω 2 "˘pe 12´1 2 e 34 q`ω 14…”

“…We show that, in case µ R t´1, 0, 1u, the lcs structures with Lee forms ϑ 1 -´pµ`1qe 4 and ϑ 2 -´2µe 4 are not equivalent. Indeed, the Gröbner basis for the ideal generated by the conditions that A " pa jk q with respect to the basis te j u is a morphism of the Lie algebra and it sends ϑ 1 to ϑ 2 is B " pa 12 , a 23 a 31`a21 a 33 , a 13 a 33 , a 11 a 44´a11 , a 12 a 44´2 a 12 µ´2a 13 µ`a 12`a13 , a 13 a 44´2 a 13 µ`a 13 , a 21 a 44´a21 µ, a 22 a 44´a22 µ´a 23 µ, a 23 a 44´a23 µ, a 31 a 44´a31 µ´a 21 , a 32 a 44´a32 µ´a 33 µ`a 22`a23 , a 33 a 44´a33 µ`a 23 , a 11 µ´a 11 , a 44 µ`a 44´2 µ, a 14 , a 24 , a 34 q .…”

“…b " 4ω 14 pϑ 4`1 q´ω 24 p2ϑ 4`1 q p2ϑ qω12 and a " 0 leaves ϑ invariant, while giving Ω 1 " ω 12 pe 12´p ϑ 4`1 qe 34 q. We consider next the automorphism s according to the sign of ω 12 .…”

“…?˘p ω 13 ω 24´λ ω 2 12 q ω 13 and b " ω 13 ω 13 ω 24´λ ω 2 12 gives the normal form (λ R t 1 2 , 1u) "…”

Locally conformally symplectic and Kähler geometry

“…At the end of the thirties of the twentieth century, the physical community was able to catch a breath after the establishment of two very important theories, namely General Relativity by Einstein and Quantum Mechanics by Bohr, Heisenberg and Schrödinger among others 19 . By the end of the seventies the four foundamental forces of Physics, Gravitation, Electromagnetism, Weak and Strong interactions, had been completely described: while General Relativity took care of Gravitation, Quantum Mechanics was able to explain the other three.…”