An almost complex structure on a smooth manifold 
is a smooth bundle endomorphism 
of its tangent bundle such
that 
. It makes each tangent
space into a complex vector space, so
an almost complex manifold has even real dimension. An almost complex structure comes from a complex manifold precisely when it is integrable.
Every Kähler manifold carries an almost
complex structure compatible with both its Riemannian
metric and symplectic form.
Almost Complex Structure
See also
Complex Manifold, Complex Structure, Donaldson Tamed-to-Compatible Conjecture, Kähler Manifold, Symplectic Taming, Tangent Bundle, Unitary ManifoldExplore with Wolfram|Alpha
References
McDuff, D. and Salamon, D. Introduction to Symplectic Topology, 2nd ed. Oxford, England: Oxford University Press, 1998.Cite this as:
Weisstein, Eric W. "Almost Complex Structure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlmostComplexStructure.html