5 problems

Let $(M^n,g)$ be a compact Hermitian manifold of complex dimension $n\geq 2$. For a real number $t$, let $\nabla^{(t)}=(1-t)\nabla+t\nabla^b$ be the $t$-Gauduchon connection, let…

Let $(M^n,g)$ be a compact Hermitian manifold, and let $D^r$ and $D^{r'}$ be Gauduchon connections for real parameters $r$ and $r'$. A connection is Kähler-like when its curvature…

Let $M$ be a compact complex manifold endowed with a Hermitian metric, and let $abla^{varepsilon}$ be a canonical connection in the Gauduchon family, distinct from the Strominger–…

Compact $\nabla^s$-flatness conjecture. If a compact $\nabla^s$-flat Hermitian manifold satisfies $s\neq 0,2$, then it must be Ke4hler and hence a finite cover of a flat complex to…

Let $(M^n,g)$ be a compact Hermitian manifold, and let $\nabla^s$ denote its $s$-Gauduchon connection. A connection is flat when its curvature tensor vanishes identically. Flat Gau…