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In this paper, we consider the $V$-soliton equation which is a degeneratefully nonlinear equation introduced by La Nave and Tian in their work onK\"ahler-Ricci flow on symplectic quotients. One can apply the interpretationto study finite time singularities of the K\"ahler-Ricci flow. As in the caseof K\"ahler-Einstein metrics, we can also reduce the $V$-soliton equation to ascalar equation on K\"ahler potentials, which is of Monge-Amp\`ere type. Weformulate some preliminary estimates for such a scalar equation on a compactK\"ahler manifold $M$.

Scalar V-soliton equation and K\"ahler-Ricci flow on symplectic quotients