arXiv subjects

Yury Ustinovskiy

Publications and source records attributed to Yury Ustinovskiy.

16 recordsLinked to original sources

arXiv2026

Formal structure of scalar curvature in generalized Kähler geometry

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.

math.DG↗

arXiv2025

Toric geometry of generalized Kähler-Ricci solitons

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.

math.DG↗

arXiv2024

The generalized Kähler Calabi-Yau problem

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.

math.DG↗

arXiv2023

The Riemannian and symplectic geometry of the space of generalized Kähler structures

On a compact complex manifold $(M, J)$ endowed with a holomorphic Poisson tensor $π_J$ and a deRham class $α\in H^2(M, \mathbb R)$, we study the space of generalized Kähler (GK) structures defined by a symplectic form $F\in α$ and whose holomorphic Poisson tensor is $π_J$. We define a notion of generalized Kähler 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 $(M, J, π_J)$. 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 Kähler 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.

math.DG↗

arXiv2022

On geometry of steady toric Kähler-Ricci solitons

In this paper we study the gradient steady Kähler-Ricci soliton metrics on non-compact toric manifolds. We show that the orbit space of the free locus of such a manifold carries a natural Hessian structure with a nonnegative Bakry-Émery tensor. We generalize Calabi's classical rigidity result and use this to prove that any complete $\mathbf T^n$-invariant gradient steady Kähler-Ricci soliton with a free torus action must be a flat $(\mathbb C^*)^n$.

math.DG↗

arXiv2022

The Gibbons-Hawking ansatz in generalized Kähler geometry

We derive a local ansatz for generalized Kähler surfaces with nondegenerate Poisson structure and a biholomorphic $S^1$ action which generalizes the classic Gibbons-Hawking ansatz for invariant hyperKähler manifolds, and allows for the choice of one arbitrary function. By imposing the generalized Kähler-Ricci soliton equation, or equivalently the equations of type IIB string theory, the construction becomes rigid, and we classify all complete solutions with the smallest possible symmetry group.

math.DG↗

arXiv2022

Generalized Kähler-Ricci flow on toric Fano varieties

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.

math.DG↗

arXiv2021

Variational structure and uniqueness of generalized Kähler-Ricci solitons

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 the orbit of a 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 arXiv:1907.03819, finishing the classification in complex dimension $2$.

math.DG↗

arXiv2019

Classification of generalized Kähler-Ricci solitons on complex surfaces

Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in arXiv:1802.00170. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first examples of generalized Kähler structures with nonvanishing Poisson structure on non-standard Hopf surfaces, completing the existence question for such structures. Moreover this gives a complete answer to the existence question for generalized Kähler-Ricci solitons on compact complex surfaces. In the setting of generalized Kähler geometry with vanishing Poisson structure, we show that these solitons are unique. We show that these solitons are global attractors for the generalized Kähler-Ricci flow among metrics with maximal symmetry.

math.DG↗

arXiv2019

On the Structure of Hermitian Manifolds with Semipositive Griffiths Curvature

In this paper we establish partial structure results on the geometry of compact Hermitian manifolds of semipositive Griffiths curvature. We show that after appropriate arbitrary small deformation of the initial metric, the null spaces of the Chern-Ricci two-form generate a holomorphic, integrable distribution. This distribution induces an isometric, holomorphic, almost free action of a complex Lie group on the universal cover of the manifold. Our proof combines the strong maximum principle for the Hermitian Curvature Flow (HCF), new results on the interplay of the HCF and the torsion-twisted connection, and observations on the geometry of the torsion-twisted connection on a general Hermitian manifold.

math.DG↗

arXiv2017

Lie-Algebraic Curvature Conditions preserved by the Hermitian Curvature Flow

The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF) on an Hermitian manifold $(M,g,J)$ preserves many natural curvature positivity conditions. Following Wilking, for an $Ad\,{GL(T^{1,0}M)}$-invariant subset $S\subset End(T^{1,0}M)$ and a ncie function $F\colon End(T^{1,0}M)\to\mathbb R$ we construct a convex set of curvature operators $C(S,F)$, which is invariant under the HCF. Varying $S$ and $F$, we prove that the HCF preserves Griffiths positivity, Dual-Nakano positivity, positivity of holomorphic orthogonal bisectional curvature, lower bounds on the second scalar curvature. As an application, we prove that periodic solutions to the HCF can exist only on manifolds $M$ with the trivial canonical bundle on the universal cover $\widetilde{M}$.

math.CV↗

arXiv2017

Hermitian curvature flow on complex homogeneous manifolds

In this paper we study a version of the Hermitian curvature flow (HCF). We focus on complex homogeneous manifolds equipped with induced metrics. We prove that this finite-dimensional space of metrics is invariant under the HCF and write down the corresponding ODE on the space of Hermitian forms on the underlying Lie algebra. Using these computations we construct HCF-Einstein metrics on $G$-homogeneous manifolds, where $G$ is a complexification of a compact simple Lie group. We conjecture that under the HCF any induced metric on such a manifold pinches towards the HCF-Einstein metric. For a nilpotent or solvable complex Lie group equipped with an induced metric we investigate the blow-up behavior of the HCF.

math.DG↗

arXiv2016

The Hermitian curvature flow on manifolds with non-negative Griffiths curvature

In this paper we study a particular version of the Hermitian curvature flow (HCF) over a compact complex Hermitian manifold $(M,g,J)$. We prove that if the initial metric has Griffiths positive (non-negative) Chern curvature $Ω$, then this property is preserved along the flow. On a manifold with Griffiths non-negative Chern curvature the HCF has nice regularization properties, in particular, for any $t>0$ the zero set of $Ω(ξ,\barξ,η,\barη)$ becomes invariant under certain torsion-twisted parallel transport.

math.CV↗

arXiv2016

On face numbers of flag simplicial complexes

Denham, Suciu and Panov, Ray computed ranks of homotopy groups and Poincaré series of a moment-angle-complex $\mathcal Z(\mathcal K$) / Davis-Januzskiewicz space $DJ(\mathcal K)$ associated to a flag simplicial complex $\mathcal K$. In this note we revisit these results and interpret them as polynomial bounds on the face numbers of an arbitrary simplicial flag complex.

math.CO↗

arXiv2016

Projective toric generators in the unitary cobordism ring

By the classical result of Milnor and Novikov, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators: $Ω^U_*\simeq \mathbb Z[a_1,a_2,\dots]$, ${\rm deg}(a_i)=2i$. In this paper we solve a well-known problem of constructing geometric representatives for $a_i$ among smooth projective toric varieties, $a_n=[X^{n}], \dim_\mathbb C X^{n}=n$. Our proof uses a family of equivariant modifications (birational isomorphisms) $B_k(X)\to X$ of an arbitrary smooth complex manifold $X$ of (complex) dimension $n$ ($n\geq 2$, $k=0,\dots,n-2$). The key fact is that the change of the Milnor number under these modifications depends only on the dimension $n$ and the number $k$ and does not depend on the manifold $X$ itself.

math.AT↗

arXiv2016

Geometry of compact complex manifolds with maximal torus action

In this present paper we study geometry of compact complex manifolds equipped with a \emph{maximal} torus $T=(S^1)^k$ action. We give two equivalent constructions providing examples of such manifolds given a simplicial fan $Σ$ and a compelx subgroup $H\subset T_\mathbb C=(\mathbb C^*)^k$. On every manifold $M$ we define the canonical holomorphic foliation $\mathcal F$ and under additional restrictions construct transverse-Kähler form $ω_\mathcal F$. As an application of these constructions, we prove some results on geometry of manifolds~$M$ regarding its analytic subsets.

math.CV↗