arXiv subjects
Publications and source records attributed to Luigi Vezzoni.
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 geometric heterotic G$_2$-system on 7-dimensional 2-step nilmanifolds $M=Γ\backslash N$ endowed with principal torus bundles and with a prescribed flat tangent bundle instanton. We first prove that every invariant G$_2$-structure solving the system must be coclosed when the dimension of the commutator of $N$ is $1$ or $2$, and under an additional calibration assumption when the dimension is $3$. Then, we discuss the existence of solutions for all possible isomorphism classes of 7-dimensional 2-step nilpotent Lie algebras, and we provide examples with constant dilaton function.
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 long-time existence of the pluriclosed flow on certain compact quotients of Lie groups for non-invariant initial data, as well as on some holomorphic principal torus bundles over nonpositively curved Kähler manifolds. In particular, our results cover the cases of nilmanifolds and almost-abelian solvmanifolds, and provide a new proof of the long-time existence of the pluriclosed flow on certain complex surfaces, originally established by Garcia-Fernandez, Jordan, and Streets. These results follow from a general theorem on holomorphic submersions, which is of independent interest and, in particular, also implies the long-time existence of the pluriclosed flow on Oeljeklaus-Toma manifolds, as proved by Streets and Wang.
We conjecture that on any compact balanced manifold $(M, ω_B)$ with $c_{1}(M)=0$, the pluriclosed flow admits long-time solutions $ω_{t}$ for every initial pluriclosed metric, and that $ω_{t}$ converges smoothly to a Kähler metric as $t \to \infty$. We verify that this phenomenon occurs when $M$ is a compact quotient of a Lie group by a discrete subgroup, the background metric $ω_{B}$ is invariant with vanishing Chern--Ricci form, and the initial metric $ω_{0}$ is invariant. In particular, this provides new evidences for the Fino-Vezzoni conjecture.
We establish a general result ensuring a $C^1$ a priori bound for smooth curves of Hermitian metrics. As a main application, we obtain a new regularity result for Hermitian curvature flows, and in particular for the second Chern-Ricci flow.
In the paper we study the Bott-Chern and Aeppli cohomologies of the twistor space of a compact self-dual 4-manifold and we characterize the validity of the $\partial \overline \partial$-lemma. We also compute explicitly the Dolbeault cohomology of the twistor space $Z$ of the flat $4$-dimensional torus, which is known to not satisfy the $\partial\overline{\partial}$ lemma.
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 study the flow of Hermitian metrics governed by the second Chern-Ricci form on a compact complex manifolds. The flow belongs to the family of Hermitian curvature flows introduced by Streets and Tian and it was considered by Lee in order to study compact Hermitian manifolds with almost negative Chern bisectional curvature. We show a regularity result on compact complex parallelizable manifolds and we prove that Chern-flat metrics are dynamically stable.
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.
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.
We prove that the parabolic flow of conformally balanced metrics introduced by Phong, Picard and Zhang in "A flow of conformally balanced metrics with Kähler fixed points", is stable around Calabi-Yau metrics. The result shows that the flow can converge on a Kähler manifold even if the initial metric is not conformally Kähler.
We investigate the pluriclosed flow on Oeljeklaus-Toma manifolds. We parametrize left-invariant pluriclosed metrics on Oeljeklaus-Toma manifolds and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution $ω_t$ which once normalized collapses to a torus in the Gromov-Hausdorff sense. Moreover the lift of $\tfrac{1}{1+t}ω_t$ to the universal covering of the manifold converges in the Cheeger-Gromov sense to $(\mathbb H^r\times\mathbb C^s, \tildeω_{\infty})$ where $\tildeω_{\infty}$ is an algebraic soliton.
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).
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 construct new smooth solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for $13 \leq k \leq 22$ and $14\leq r\leq 22$, the smooth manifolds $S^1\times \sharp_k(S^2\times S^3)$ and $\sharp_r (S^2 \times S^4) \sharp_{r+1} (S^3 \times S^3)$, have a complex structure with trivial canonical bundle and admit a solution to the Hull-Strominger system.
We show that the existence of a left-invariant pluriclosed Hermitian metric on a unimodular Lie group with a left-invariant abelian complex structure forces the group to be $2$-step nilpotent. Moreover, we prove that the pluriclosed flow starting from a left-invariant Hermitian metric on a $2$-step nilpotent Lie group preserves the Strominger Kähler-like condition.
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.