Someone thought they could achieve anti-gravity thus: they noticed the power of a gyroscope to resist rotational twisting. They arranged a rotating object with arms that would swing out (centrifugal force). But they had little gyroscopes on the ends of the arms such that they would fight the change-in-position from the arms' swing. To these gyroscopes such a change upwards and outwards in the arms on which they were mounted meant being twisted out of position, so they resisted the usual swing-out-and-up of the arms.
So they thought by using gyroscopes to defeat centrifugal effect on arms of rotating object, the upwards-movement that the arms would have had, would be channeled into upwards movement of the whole structure.
However, the laws of action-reaction and conservation of momentum, conservation of angular momentum, conservation of energy-mass etc. mean their logic was apparantly misguided.
However, suppose you had a huge very thin super-rigid disc 10-times the diameter of the Earth spinning above the Earth. It's rigidity would un-curve the "space-time-curvature" of the Earth because normally the different parts of the disc would move at different angles towards the Earth under gravity. But different angles of movement from different parts of one flat disc means TWIST; and the gyroscopic efect of spinning the disc would effectively un-curve the Earth's field? Probably still same problems with logic as first example.
However; it seems possible that if there is some merit in the above; what you need to do to make a hoverboard is this:
You create a thin layer on the base of the board which is fantastically rigid and true-flat, even flatter than the Earth's space-time curvature across the board length. The surface consists of zillions of atoms in a lattice of perfect torsional rigidity where each atom has a perfectly matching spin-direction and speed to its neighbours.
To create this effect requires a "spin-laser"; i.e. a system for synchronising atomic spins to create a kind of global single spin-quantum-state across the whole board surface.
Such a surface might locally "uncurve Earth's field" and allow "surfing" on the expansion of the Universe wave etc.
Other possibility is that gravity is an attraction in principle due to perspective- like parrallel lines appear to attract into the distance. So may need a "backwards in time" surface made of positrons or something.