these question in some examples (without any claim of providing a complete picture), and Section \[sec:D\] discusses some conclusions and conjectures as well as the Donaldson geometric flow approach to the uniqueness problem for hyperkähler surfaces.

The space of symplectic forms {#sec:SYMP}
=============================

\[para:equiv\]Let $M$ be a manifold without boundary. Consider the following statements for two symplectic forms ${{\omega}}_0$, ${{\omega}}_1$ on $M$.

(a)

:   ${{\omega}}_0$ and ${{\omega}}_1$ are connected by a path of cohomologous symplectic forms.

(b)

:   ${{\omega}}_0$ and ${{\omega}}_1$ are connected by a path of symplectic forms.

(c)

:   ${{\omega}}_0$ and ${{\omega}}_1$ are connected by a path of nondegenerate $2$-forms.

(d)

:   ${{\omega}}_0$ and ${{\omega}}_1$ have the same first Chern class in $H^2(M;{{\mathbb Z}})$.

(A)

:   There exists a diffeomorphism ${{\varphi}}$ of $M$ such that ${{\omega}}_0={{\varphi}}^*{{\omega}}_1$.

(B)

:   There exists a diffeomorphism ${{\varphi}}$ of $M$ such that ${{\omega}}_0$ and ${{\varphi}}^*{{\omega}}_1$ are connected by a path of symplectic forms.

(C)

:   There exists a diffeomorphism ${{\varphi}}$ of $M$ such that ${{\omega}}_0$ and ${{\varphi}}^*{{\omega}}_1$ are connected by a path of nondegenerate $2$-forms.

(D)

:   There exists a diffeomorphism ${{\varphi}}$ of $M$ such that $c_1({{\omega}}_0)={{\varphi}}^*c_1({{\omega}}_1)$.

These statements define equivalence relations. Two symplectic forms ${{\omega}}_0$ and ${{\omega}}_1$ are called [**isotopic**]{} if they 