We introduce a notion of K-polystability for compact Kähler holomorphic Poisson manifolds. On the one hand, this notion of stability is well-adapted to constructions of moduli spaces. For instance, when the underlying manifold is K-polystable with reductive reduced automorphism group, Poisson K-stability is equivalent to geometric invariant theoretic stability in the space of Poisson bivectors, but there also exist K-unstable varieties that become stable after incorporating a Poisson structure. On the other hand, the Poisson K-stability condition interacts well with generalized Käher metrics – the background geometry of (2,2) supersymmetric string theory. In particular, we conjecture that Poisson K-polystability characterizes the existence of constant scalar curvature symplectic generalized Kähler structures with a sufficiently small Poisson tensor – a natural extension of the Yau–Tian–Donaldson (YTD) conjecture. Our main result is a proof of the existence part of this “semiclassical YTD conjecture” for Poisson structures on Kähler–Einstein Fano manifolds, using infinite-dimensional momentum map techniques. In this way, we obtain the existence of many new examples of symplectic generalized Kähler structure of constant scalar curvature, and prove the conjecture completely in the case of the projective plane.

We prove that for any smooth polarized complex n-dimensional manifold (X, L_X) which admits an extremal Kähler metric in c_1(L_X), and for any integer k large enough (in terms of a bound depending on (X, L_X)), the (n+k+1)-dimensional complex cone 𝒴:= (L_X ⊗𝒪_ℙ^k(1))^× with section X ×ℙ^k admits a scalar-flat Kähler cone metric. Equivalently, the unweighted Sasaki join of a smooth compact quasi-regular extremal Sasaki manifold with a regular Sasaki sphere 𝕊^2k+1 of sufficiently large dimension (2k+1) admits a Sasaki metric of constant (positive) scalar curvature. This gives an affirmative answer to an asymptotic version of a question raised by Boyer–Huang–Legendre–Tønnesen-Friedman in arXiv:1906.04827.

We study compact complex 3-dimensional non-Kähler Bismut Ricci flat pluriclosed Hermitian manifolds (BHE) via their dimensional reduction to a special Kähler geometry in complex dimension 2, recently obtained by Barbaro, Streets and the first and third authors. We show that in the quasi-regular case, the reduced geometry satisfies a 6th order non-linear PDE which has infinite dimensional momentum map interpretation, similar to the much studied Kähler metrics of constant scalar curvature (cscK). We use this to associate to the reduced manifold or orbifold Mabuchi and Calabi functionals, as well as to obtain obstructions for the existence of solutions in terms of the authomorphism group, paralleling results by Futaki and Calabi-Lichnerowicz-Matsushima in the cscK case. This is used to characterize the Samelson locally homogeneous BHE geometries in complex dimension 3 as the only non-Kähler BHE 3-folds with 2-dimensional Bott-Chern (1,1)-cohomology group, for which the reduced space is a smooth Kähler surface. We also discuss explicit solutions of the PDE on orthotoric Kähler orbifold surfaces, extending examples found by Couzens-Gauntlett-Martelli-Sparks in the framework of supersymmetric AdS_3 × Y_7 type IIB supergravity. Our construction yields infinitely many non-Kähler BHE structures on S^3× S^3 and S^1× S^2 × S^3, which are not locally isometric to a Samelson geometry. These appear to be the first such examples.

Building on works of Boulanger [J. Symplectic Geom. 17 (2019), pp. 973–1019] and Goto [arXiv:2105.13654, 2021; and J. Symplectic Geom. 18 (2020), pp. 147–190], we show that Goto’s scalar curvature is the moment map for an action of generalized Hamiltonian automorphisms of the associated Courant algebroid, constrained by the choice of an adapted volume form. We derive an explicit formula for Goto’s scalar curvature, and show that it is constant for generalized Kähler-Ricci solitons. Restricting to the generically symplectic type case, we realize the generalized Kähler class as the complexified orbit of the Hamiltonian action above. This leads to a natural extension of Mabuchi’s metric and K K -energy, implying a conditional uniqueness result. Finally, in this setting we derive a Calabi-Matsushima-Lichnerowicz obstruction and a Futaki invariant.

We establish a local equivalence between toric steady Kähler-Ricci solitons and A-type toric generalized Kähler-Ricci solitons (GKRS). Under natural global conditions we show this equivalence extends to complete GKRS, yielding a general construction of new examples in all dimensions. We show that in four dimensions, all GKRS are either described by the generalized Kähler Gibbons-Hawking ansatz, or have split tangent bundle, or are A-type toric. This yields a local classification in four dimensions, together with a conjecturally exhaustive construction of complete symplectic-type examples.

We show that if X is a smooth Fano manifold which caries a Kähler Ricci soliton, then the canonical cone of the product of X with a complex projective space of sufficiently large dimension is a Calabi–Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and Nikagawa. This result is obtained by the openness of the set of weight functions v over the momentum polytope of a given smooth Fano manifold, for which a v-soliton exists. We discuss other ramifications of this approach, including a Licherowicz type obstruction to the existence of a Kähler Ricci soliton and a Fujita type volume bound for the existence of a v-soliton.

On a compact complex manifold $(M, J)$ endowed with a holomorphic Poisson tensor $\pi_J$ and a deRham class $\alpha\in H^2(M, \mathbb R)$, we study the space of generalized K\"ahler (GK) structures defined by a symplectic form $F\in \alpha$ and whose holomorphic Poisson tensor is $\pi_J$. We define a notion of generalized K\"ahler class of such structures, and use the moment map framework of Boulanger and Goto to extend the Calabi program to GK geometry. We obtain generalizations of the Futaki--Mabuchi extremal vector field and Calabi--Lichnerowicz--Matsushima result for the Lie algebra of the group of automorphisms 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 variational characterization of GK structures of constant scalar curvature. Next we introduce a formal Riemannian metric on a given GK class, generalizing the fundamental construction of Mabuchi--Semmes--Donaldson. We show that this metric has nonpositive sectional curvature, and that the Mabuchi energy is convex along geodesics, leading to a conditional uniqueness result for constant scalar curvature GK structures. We finally examine the toric case, proving the uniqueness of extremal generalized K\"ahler structures and showing that their existence is obstructed by the uniform relative K-stability of the corresponding Delzant polytope. Using the resolution of the Yau--Tian--Donaldson conjecture in the toric case by Chen--Cheng and He, we show in some settings that this condition suffices for existence and thus construct new examples.

For each partition of the positive integer $n= \ell +\sum_{j=1}^\ell d_j$, where $\ell\ge 1$ and $d_j \ge 0$ are integers, we construct a continuous $(\ell-1)$-parameter family of explicit complete gradient steady K\"ahler--Ricci solitons on $\mathbb{C}^n$ admitting a hamiltonian $2$-form of order $\ell$ and symmetry group ${\rm U}(d_1+ 1) \times \cdots \times {\rm U}(d_{\ell} + 1)$. For $\ell=1, \, d_1=n-1, \, d_2=\cdots = d_{\ell}=0$ we obtain Cao's example [17] whereas for other partitions the metrics are new. Furthermore, when $n=2, \, \ell=2, \, d_1=d_2=0$ we obtain complete gradient steady K\"ahler--Ricci solitons on $\mathbb{C}^2$ which have positive sectional curvature but are not isometric to Cao's ${\rm U}(2)$-invariant example. This disproves a conjecture by Cao. We also present a construction yielding explicit families of complete gradient steady K\"ahler-Ricci solitons on $\mathbb{C}^n$ containing higher dimensional extensions of the Taub-NUT Ricci-flat K\"ahler metric on $\mathbb{C}^2$. When $n\ge 3$, the complete Ricci-flat K\"ahler metrics, and when $n\ge 2$, their deformations to complete gradient steady K\"ahler Ricci solitons seem not to have been observed before our work.

We derive a canonical symmetry reduction associated to a compact non-Kähler Bismut-Hermitian-Einstein manifold. In real dimension 6, the transverse geometry is conformally Kähler, and we give a complete description in terms of a single scalar PDE for the underlying Kähler structure. In the case when the soliton potential is constant, we show that that the Bott–Chern number h^1,1_BC≥ 2 , and that equality holds if and only if the metric is Bismut-flat, and hence a quotient of either SU (2) ×ℝ×ℂ or SU (2) × SU (2) .

Using the Yau-Tian-Donaldson type correspondence for $v$-solitons established by Han-Li, we show that a smooth complex $n$-dimensional Fano variety admits a Mabuchi soliton provided it admits an extremal Kähler metric whose scalar curvature is strictly less than $2(n+1)$. Combined with previous observations by Mabuchi and Nakamura in the other direction, this gives a characterization of the existence of Mabuchi solitons in terms of the existence of extremal Kähler metrics on Fano manifolds. An extension of this correspondence to $v$-solitons is also obtained.

Using the Yau–Tian–Donaldson type correspondence for v $v$ -solitons established by Han–Li, we show that a smooth complex n $n$ -dimensional Fano variety admits a Mabuchi soliton provided it admits an extremal Kähler metric whose scalar curvature is strictly less than 2 ( n + 1 ) $2(n+1)$ . Combined with previous observations by Mabuchi and Nakamura in the other direction, this gives a characterization of the existence of Mabuchi solitons in terms of the existence of extremal Kähler metrics on Fano manifolds. An extension of this correspondence to v $v$ -solitons is also obtained.

Building on works of Boulanger and Goto, we show that Goto's scalar curvature is the moment map for an action of generalized Hamiltonian automorphisms of the associated Courant algebroid, constrained by the choice of an adapted volume form. We derive an explicit formula for Goto's scalar curvature, and show that it is constant for generalized Kähler-Ricci solitons. Restricting to the generically symplectic type case, we realize the generalized Kähler class as the complexified orbit of the Hamiltonian action above. This leads to a natural extension of Mabuchi's metric and K-energy, implying a conditional uniqueness result. Finally, in this setting we derive a Calabi-Matsushima-Lichnerowicz obstruction and a Futaki invariant.

We study the set of deRham classes of Lee 1-forms of the locally conformally symplectic (LCS) structures taming the complex structure of a compact complex surface in the Kodaira class VII, and show that the existence of non-trivial upper/lower bounds with respect to the degree function correspond respectively to the existence of certain negative/non-negative PSH functions on the universal cover. We use this to prove that the set of Lee deRham classes of taming LCS is connected, as well as to obtain an explicit negative upper bound for this set on the hyperbolic Kato surfaces. This leads to a complete description of the sets of Lee classes on the known examples of class VII complex surfaces, and to a new obstruction to the existence of bi-hermitian structures on the hyperbolic Kato surfaces of the intermediate type. Our results also reveal a link between bounds of the set of Lee classes and non-trivial logarithmic holomorphic 1-forms with values in a flat holomorphic line bundle.

We show that a compact weighted extremal Kahler manifold (as defined by the third named author) has coercive weighted Mabuchi energy with respect to a maximal complex torus in the reduced group of complex automorphisms. This provides a vast extension and a unification of a number of results concerning Kahler metrics satisfying special curvature conditions, including constant scalar curvature Kahler metrics, extremal Kahler metrics, Kahler-Ricci solitons and their weighted extensions. Our result implies the strict positivity of the weighted Donaldson-Futaki invariant of any non-product equivariant smooth K\"ahler test configuration with reduced central fibre, a property also known as weighted K-polystability on such test configurations. For a class of fibre-bundles, we use our result in conjunction with the recent results of Chen-Cheng, He, and Han-Li in order to characterize the existence of extremal Kahler metrics and Calabi-Yau cones associated to the total space, in terms of the coercivity of the weighted Mabuchi energy of the fibre. In particular, this yields an existence result for Sasaki-Einstein metrics on Fano toric fibrations, extending the results of Futaki-Ono-Wang in the toric Fano case, and of Mabuchi-Nakagawa in the case of Fano projective line bundles.

We study the existence of weighted extremal K\ahler metrics in the sense of Apostolov-Calderbank-Gauduchon-Legendre and Lahdili on the total space of an admissible projective bundle over a Hodge K\ahler manifold of constant scalar curvature. Admissible projective bundles have been defined by Apostolov-Calderbank-Gauduchon-T{\o}nnesen-Friedman, and they include the projective line bundles (Hwang-Singer) and their blow-downs (Koiso-Sakane), thus providing a most general setting for extending the existence theory for extremal K\ahler metrics pioneered by a seminal construction of Calabi. We obtain a general existence result for weighted extremal metrics on admissible manifolds, which yields many new examples of conformally K\ahler, Einstein-Maxwell metrics of complex dimension $m>2$, thus extending the recent constructions of LeBrun and Koca-T{\o}nnesen-Friedman to higher dimensions. For each admissible K\ahler class on an admissible projective bundle, we associate an explicit function of one variable and show that if it is positive on the interval $(-1,1)$, then there exists a weighted extremal K\ahler metric in the given class, whereas if it is strictly negative somewhere in $(-1,1)$, there is no K\ahler metrics of constant weighted scalar curvature in that class. We also relate the positivity of the function to a notion of weighted K-stability, thus establishing a Yau-Tian-Donaldson type correspondence for the existence of K\ahler metrics of constant weighted scalar curvature in the rational admissible K\ahler classes on an admissible projective bundle. Weighted extremal orthotoric metrics are examined in an appendix.

These lecture notes are written for a PhD mini-course I gave at the CIRM in Luminy in 2019. Their intended purpose was to present, in the context of smooth toric varieties, a relatively self-contained and elementary introduction to the theory of extremal Kähler metrics pioneered by E. Calabi in the 1980's and extensively developed in recent years. The framework of toric manifolds, used in both symplectic and algebraic geometry, offers a fertile testing ground for the general theory of extremal Kähler metrics and provides an important class of smooth complex varieties for which the existence theory is now understood in terms of a stability condition of the corresponding Delzant polytope. The notes do not contain any original material nor do they take into account some more recent developments, such as the non-Archimedean approach to the Calabi problem. I am making them available on the arXiv because I continue to get questions about how they can be cited.

Under broad hypotheses we derive a scalar reduction of the generalized Kähler–Ricci soliton system. We realize solutions as critical points of a functional, analogous to the classical Aubin energy, defined on an orbit of the natural Hamiltonian action of diffeomorphisms, thought of as a generalized Kähler class. This functional is convex on a large set of paths in this space, and using this we show rigidity of solitons in their generalized Kähler class. As an application we prove uniqueness of the generalized Kähler–Ricci solitons on Hopf surfaces constructed in Streets and Ustinovskiy [Commun. Pure Appl. Math. 74(9), 1896–1914 (2020)], finishing the classification in complex dimension 2.

We formulate an extension of the Calabi conjecture to the setting of generalized Kähler geometry. We show a transgression formula for the Bismut Ricci curvature in this setting, which requires a new local Goto/Kodaira-Spencer deformation result, and use it to show that solutions of the generalized Calabi-Yau equation on compact manifolds are classically Kähler, Calabi-Yau, and furthermore unique in their generalized Kähler class. We show that the generalized Kähler-Ricci flow is naturally adapted to this conjecture, and exhibit a number of a priori estimates and monotonicity formulas which suggest global existence and convergence. For initial data in the generalized Kähler class of a Kähler Calabi-Yau structure we prove the flow exists globally and converges to this unique fixed point. This has applications to understanding the space of generalized Kähler structures, and as a special case yields the topological structure of natural classes of Hamiltonian symplectomorphisms on hyperKähler manifolds. In the case of commuting-type generalized Kähler structures we establish global existence and convergence with arbitrary initial data to a Kähler, Calabi-Yau metric, which yields a new d d^c-lemma for these structures.

We study the generalized Kähler-Ricci flow with initial data of symplectic type, and show that this condition is preserved. In the case of a Fano background with toric symmetry, we establish global existence of the normalized flow. We derive an extension of Perelman’s entropy functional to this setting, which yields convergence of nonsingular solutions at infinity. Furthermore, we derive an extension of Mabuchi’s K-energy to this setting, which yields weak convergence of the flow.

Abstract We formulate a Calabi–Yau-type conjecture in generalized Kähler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized Kähler structures generalizing the notion of Kähler class, we conjecture unique solvability of Gualtieri’s Calabi–Yau equation within this class. We establish the uniqueness, and moreover show that all such solutions are actually hyper-Kähler metrics. We furthermore establish a GIT framework for this problem, interpreting solutions of this equation as zeroes of a moment map associated to a Hamiltonian action and finding a Kempf–Ness functional. Lastly we indicate the naturality of generalized Kähler–Ricci flow in this setting, showing that it evolves within the given Hamiltonian deformation class, and that the Kempf–Ness functional is monotone, so that the only possible fixed points for the flow are hyper-Kähler metrics. On a hyper-Kähler background, we establish global existence and weak convergence of the flow.