Note on $N=4\mathrm{}$ supersymmetric mechanics on Kähler manifolds

The geometric models of $N=4\mathrm{}$ supersymmetric mechanics with $(2d\cdot {2d)}_C$-dimensional phase space are proposed, which can be viewed as one-dimensional counterparts of the two-dimensional $N=2\mathrm{}$ supersymmetric sigma models of Alvarez-Gaumé and Freedman. The related construction of supersymmetric mechanics whose phase space is a Kähler supermanifold is considered. Also, its relation with antisymplectic geometry is discussed.