Abstract
For every positive integer r, we introduce two new cohomologies,
that we call
E
r
{E_{r}}
-Bott–Chern and
E
r
{E_{r}}
-Aeppli, on compact complex manifolds. When
r
=
1
{r\kern-1.0pt=\kern-1.0pt1}
, they coincide with the usual Bott–Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when
r
≥
2
{r\geq 2}
. They provide analogues in the Bott–Chern–Aeppli context of the
E
r
{E_{r}}
-cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page-
(
r
-
1
)
{(r-1)}
-
∂
∂
¯
{\partial\bar{\partial}}
-manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott–Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.
Logarithmic Cartan geometry on complex manifolds
