arXiv subjects
Publications and source records attributed to Giovanni Gentili.
We prove that on a compact locally flat HKT manifold the quaternionic Monge--Ampère equation always has a unique continuous solution. This provides new evidence for the Conjecture of Alesker and Verbitsky.
We study the holonomy of the Obata connection on Joyce hypercomplex manifolds. For all such group manifolds except $\mathrm{SU}(2n+1)$, we show that the holonomy group is strictly contained in the quaternionic general linear group. The case of $\mathrm{SU}(2n+1)$ is more subtle: for every $n>1$, we show that there exist infinitely many Joyce hypercomplex structures with Obata holonomy strictly contained in $\mathrm{GL}(n(n+1),\mathbb{H})$. On the other hand, Soldatenkov showed that $\mathrm{SU}(3)$ has Obata holonomy equal to $\mathrm{GL}(2,\mathbb{H})$ \cite{Sol}, and we present here a new example on $\mathrm{SU}(5)$ with holonomy equal to $\mathrm{GL}(6,\mathbb{H})$. Finally, we investigate Joyce hypercomplex manifolds whose restricted holonomy lie in $\mathrm{SL}(n, \mathbb{H})$, yielding new compact examples of twisted Calabi-Yau manifolds.
We prove that a large class of $2$-step solvable Lie algebras equipped with a complex structure $J$ admits a Levi-Malcev type decomposition, adapted to $J$. As an application, we prove that the Fino--Vezzoni conjecture holds true for $2$-step solvable unimodular Lie algebras. Finally, we give a structural characterisation of $2$-step, unimodular, completely solvable Lie algebras admitting an SKT metric.
In this work, we introduce a family of almost hyperhermitian structures that we call semi-integrable. We subdivide them into four disjoint classes and show that each class is non-empty. Finally, we construct semi-integrable almost hyperhermitian structures on all reductive Lie algebras of compact type and dimension $4n$, with $n\geq 2$.
We investigate the existence and geometric properties of special hyperhermitian metrics. First of all, we characterise hypercomplex structures with Obata holonomy in $\mathrm{SL}(n, \mathbb{H})$ in terms of the existence of quaternionic Gauduchon metrics together with the vanishing of a hypercomplex cohomological invariant. In view of this, the quaternionic Gauduchon and quaternionic balanced conditions are investigated at length: we describe their properties and determine criteria for their existence. Furthermore, we prove an incompatibility result concerning strong HKT and balanced hyperhermitian metrics, confirming an open conjecture by Fino and Vezzoni in the hypercomplex framework. Finally, we introduce an Einstein-type condition, determining basic properties, obstructions and providing examples. In particular, we show that Joyce's manifolds always admit such type of metrics.
We revisit the second order estimate for solutions to the quaternionic Calabi-Yau problem on hyperkähler manifolds, originally established by Dinew and Sroka. In this note, we present a simplified argument to derive the estimate.
We prove the openness of the balanced HKT cone within the cone of HKT structures on a compact hypercomplex manifold $(M,I,J,K)$. We also study the Lie algebra of hyperholomorphic vector fields of type (1,0) with respect to $I$, with particular emphasis on the case when there exists a compatible balanced HKT metric. These fields exhibit a strict interplay with the balanced HKT structure, for instance, we prove a harmonicity property for (1,0)-forms dual to hyperholomorphic vector fields. We also show non-existence of hyperholomorphic (1,0)-vector fields on some hypercomplex manifolds admitting a HKT--Einstein metric.
We consider a general class of elliptic equations on hypercomplex manifolds which includes the quaternionic Monge-Ampère equation, the quaternionic Hessian equation and the Monge-Ampère equation for quaternionic $(n-1)$-plurisubharmonic functions. We prove that under suitable assumptions the solutions to these equations on hyperkähler manifolds satisfy a $C^{2,α}$ a priori estimate.
Let $(M,I,J,K,Ω)$ be a compact HKT manifold and denote with $\partial$ the conjugate Dolbeault operator with respect to $I$, $\partial_J:=J^{-1}\overline\partial J$, $\partial^Λ:=[\partial,Λ]$ where $Λ$ is the adjoint of $L:=Ω\wedge-$. Under suitable assumptions, we study Hodge theory for the complexes $(A^{\bullet,0},\partial,\partial_J)$ and $(A^{\bullet,0},\partial,\partial^Λ)$ showing a similar behavior to Kähler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT $\mathrm{SL}(n,\mathbb{H})$-manifold the differential graded algebra $(A^{\bullet,0},\partial)$ is formal and this will lead to an obstruction for the existence of an HKT $\mathrm{SL}(n,\mathbb{H})$-structure $(I,J,K,Ω)$ on a compact complex manifold $(M,I)$. Finally, balanced HKT structures on solvmanifolds are studied.
We show that the parabolic quaternionic Monge-Ampère equation on a compact hyperkähler manifold has always a long-time solution which once normalized converges smoothly to a solution of the quaternionic Monge-Ampère equation. This is the same setting in which Dinew and Sroka prove the conjecture of Alesker and Verbitsky. We also introduce an analogue of the Chern-Ricci flow in hyperhermitian manifolds.
Our recent work about fully non-linear elliptic equations on compact manifolds with a flat hyperkähler metric is hereby extended to the parabolic setting. This approach will help us to study some problems arising from hyperhermitian geometry.
Mainly motivated by a conjecture of Alesker and Verbitsky, we study a class of fully non-linear elliptic equations on certain compact hyperhermitian manifolds. By adapting the approach of Székelyhidi to the hypercomplex setting, we prove some a priori estimates for solutions to such equations under the assumption of existence of $\mathcal{C}$-subsolutions. In the estimate of the quaternionic Laplacian, we need to further assume the existence of a flat hyperkähler metric. As an application of our results we prove that the quaternionic analogue of the Hessian equation and Monge-Ampère equation for $(n-1)$-plurisubharmonic functions can always be solved on compact flat hyperkähler manifolds.
We study the quaternionic Monge-Ampère equation on HKT manifolds admitting an HKT foliation having corank 4. We show that in this setting the quaternionic Monge-Ampère equation has always a unique solution for every basic datum. This approach includes the study of the equation on SU(3).
Mainly motivated by a conjecture of Alesker and Verbitsky, we study a class of elliptic equations on compact hyperhermitian manifolds. By adapting the approach of Székelyhidi to the hypercomplex setting, we prove some a priori estimates for solutions to such equations. In the estimate of the Laplacian we assume the existence of a flat hyperkähler metric. As an application of our results we prove that the quaternionic analogue of the Hessian equation can always be solved on compact flat hyperkähler manifolds.
We consider the natural generalization of the parabolic Monge-Ampère equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex manifold is locally flat and admits a hyperkähler metric, then the equation has a long-time solution whose normalization converges to a solution of the quaternionic Monge-Ampère equation introduced by Alesker and Verbitsky. The result gives an alternative proof of a theorem of Alesker.
We study the quaternionic Calabi-Yau problem in HKT geometry introduced by Alesker and Verbitsky on 8-dimensional 2-step nilmanifolds with an abelian hypercomplex structure. We show that the quaternionic Monge-Ampère equation on these manifolds can always be solved for every data which is invariant by the action of a 3-dimensional torus.