Symplectic taming of an almost complex structure 
by a symplectic
form 
means that
 |
for every nonzero tangent vector 
.
Compatibility requires in addition 
. A compatible pair defines the Riemannian
metric 
.
A Kähler form is compatible with its complex
structure, but neither taming nor compatibility requires that an almost
complex structure be integrable. The Donaldson
tamed-to-compatible conjecture asks whether existence of a taming form in real dimension four forces
existence of a compatible one.
