A Kähler potential is a real-valued function $f$ defined on some coordinate patch of a Hermitean manifold such that the metric of the manifold is given by the expression $$ g_{ij*} = {\partial^2 f \over dz^i d{\overline z}^j} . $$ It turns out that, for every Káhler manifold, there will exist a coordinate neighborhood of any given point in which the metric can be expresses in terms of a potential this way.

As an elementary example of a Kähler potential, we may consider $f(z,{\overline z}) = z {\overline z}$ This potential gives rise to the flat metric $ds^2 = dz d{\overline z}$

Kähler potentials have applications in physics. For example, this function $f(x) = \log(x) + g(x)$ relates to the motion of certain subatomic particles called gauginos.

Bibliography

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T. Barreiro, B. de Carlos & E. J. Copeland, ``On non-perturbative corrections to the Kähler potential'' Physical Review D57 (1998): 7354 - 7360