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On a compact complex manifold $(M, J)$ endowed with a holomorphic Poissontensor $\pi_J$ and a deRham class $\alpha\in H^2(M, \mathbb R)$, we study thespace of generalized K\"ahler (GK) structures defined by a symplectic form$F\in \alpha$ and whose holomorphic Poisson tensor is $\pi_J$. We define anotion of generalized K\"ahler class of such structures, and use the moment mapframework of Boulanger and Goto to extend the Calabi program to GK geometry. Weobtain generalizations of the Futaki--Mabuchi extremal vector field andCalabi--Lichnerowicz--Matsushima result for the Lie algebra of the group ofautomorphisms of $(M, J, \pi_J)$. We define a closed $1$-form on a GK class,which yields a generalization of the Mabuchi energy and thus a variationalcharacterization of GK structures of constant scalar curvature. Next weintroduce a formal Riemannian metric on a given GK class, generalizing thefundamental construction of Mabuchi--Semmes--Donaldson. We show that thismetric has nonpositive sectional curvature, and that the Mabuchi energy isconvex along geodesics, leading to a conditional uniqueness result for constantscalar curvature GK structures. We finally examine the toric case, proving theuniqueness of extremal generalized K\"ahler structures and showing that theirexistence is obstructed by the uniform relative K-stability of thecorresponding Delzant polytope. Using the resolution of theYau--Tian--Donaldson conjecture in the toric case by Chen--Cheng and He, weshow in some settings that this condition suffices for existence and thusconstruct new examples.

The Riemannian and symplectic geometry of the space of generalized K\"ahler structures