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Paul-Andi Nagy

Publications and source records attributed to Paul-Andi Nagy.

At least 19 recordsLinked to original sources
arXiv2026

Third order Einstein deformations for Kaehler-Einstein metrics

For compact Kähler manifolds $(M,g,J)$ with negative scalar curvature we study the existence problem for non-trivial Einstein deformations of $g$, that is small time curves $g_t$ of Einstein metrics with $g_0=g$. No asssumption on the complex structure $J$ is made; also we do not assume that the metrics $g_t$ are Kähler w.r.t. $J$. We determine explicitly the obstruction to third order Einstein deformation for $g$; that is we fully solve the equations $(\Ric^{g_t})^{(k)}(0)=0$ for $1 \leq k \le 3$ in terms of the Taylor expansion $g^{-1}g_t=\id+th_1+\tfrac{t^2}{2!}h_2+\tfrac{t^3}{3!}h_3+o(t^4)$ at $t=0$. Up to a suitable gauge transformation we show that third order integrability for the Einstein equation amounts to Maurer-Cartan type equations and polynomial identities relating the coefficients $h_3,h_2,h_1$. This result is interpreted in terms of the underlying complex geometry of $M$ by means of the Cayley transform of the metric $g$; the Cayley transform is also used for formulating conjectures for the higher order Einstein deformation problem.

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arXiv2026

Einstein deformations of Kähler Einstein metrics

We study Einstein deformations of negative Kähler Einstein metrics. We relate the second order Einstein deformation theory of negative Kähler-Einstein metrics to the complex geometry of the underlying Kähler manifold. After suitable gauge normalisation we show that the Taylor expansion to order two of an Einstein deformation tangent to $h_1$ in the infinitesimal deformation space is fully determined by $h_1^2$ and the divergence of the Kodaira-Spencer bracket $[h_1,h_1]^c$. This substantially refines and extends recent results of Nagy-Semmelmann which state that Einstein deformations for negative Kähler-Einstein metrics are unobstructed to second order.

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arXiv2025

Higher order obstructions to Riccati-type equations

We develop new techniques in order to deal with Riccati-type equations, subject to a further algebraic constraint, on Riemannian manifolds $(M^3,g)$. We find that the obstruction to solve the aforementioned equation has order $4$ in the metric coefficients and is fully described by an homogeneous polynomial in $\mathrm{Sym}^{16}TM$. Techniques from real algebraic geometry, reminiscent of those used for the "PositiveStellen-Satz " problem, allow determining the geometry in terms of the connection coefficients and a class of Hessian-type equations. Analysis of the latter shows flatness for the metric $g$; in particular we complete the classification of asymptotically harmonic manifolds of dimension $3$, establishing those are either flat or real hyperbolic spaces.

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arXiv2025

Conformal foliations, Kähler twists and the Weinstein construction

We classify both local and global Kähler structures admitting totally geodesic homothetic foliations with complex leaves. The main building blocks are related to Swann's twists and are obtained by applying Weinstein's method of constructing symplectic bundles to Kähler data. As a byproduct we obtain new classes of: holomorphic harmonic morphisms with fibres of arbitrary dimension from compact Kähler manifolds; non-Kähler balanced metrics conformal to Kähler ones (but compatible with different complex structures). Some classes of non-Einstein constant scalar curvature Kähler metrics are also obtained in this way.

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arXiv2025

3-symmetric spaces, Ricci solitons, and homogeneous structures

The full classification of Riemannian $3$-symmetric spaces is presented. Up to Riemannian products the main building blocks consist in (possibly symmetric) spaces with semisimple isometry group, nilpotent Lie groups of step at most $2$ and spaces of type III and IV. For the most interesting family of examples, the Type III spaces, we produce an explicit description including results concerning the moduli space of all $3$-symmetric metrics living on a given Type III space. Each moduli space contains a unique distinguished point corresponding to an (almost-Kähler) expanding Ricci soliton metric. For certain classes of 3-symmetric metrics there are many different groups acting transitively and isometrically on a fixed Riemannian 3-symmetric space. The construction of expanding Ricci solitons on spaces of Type III is also shown to generalize to \emph{any} effective representation of a simple Lie group of non-compact type, yielding a very general construction of homogeneous Ricci solitons. We also give a procedure to compute the isometry group of any Ambrose--Singer space.

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arXiv2024

Second order Einstein deformations

We study the integrability to second order of infinitesimal Einstein deformations on compact Riemannian and in particular on Kähler manifolds. We find a new way of expressing the necessary and sufficient condition for integrability to second order, which also gives a very clear and compact way of writing the Koiso obstruction. As an application we consider the Kähler case, where the condition can be further simplified and in complex dimension $3$ turns out to be purely algebraic. One of our main results is the complete and explicit description of infinitesimal Einstein deformation integrable to second order on the complex $2$-plane Grassmannian, which also has a quaternion Kähler structure. As a striking consequence we find that the symmetric Einstein metric on the Grassmannian $ \mathrm{Gr}_2(\bbC^{n+2})$ for $n$ odd is rigid.

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arXiv2023

The $G_2$ geometry of $3$-Sasaki structures

We initiate a systematic study of the deformation theory of the second Einstein metric $g_{1/\sqrt{5}}$ respectively the proper nearly $G_2$ structure $φ_{1/\sqrt{5}}$ of a $3$-Sasaki manifold $(M^7,g)$. We show that infinitesimal Einstein deformations for $g_{1/\sqrt{5}}$ coincide with infinitesimal $G_2$ deformations for $φ_{1/\sqrt{5}}$. The latter are showed to be parametrised by eigenfunctions of the basic Laplacian of $g$, with eigenvalue twice the Einstein constant of the base $4$-dimensional orbifold, via an explicit differential operator. In terms of this parametrisation we determine those infinitesimal $G_2$ deformations which are unobstructed to second order.

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arXiv2023

Eigenvalue estimates for 3-Sasaki structures

We obtain new lower bounds for the first non-zero eigenvalue of the scalar sub-Laplacian for 3-Sasaki metrics, improving Lichnerowicz-Obata type estimates by Ivanov et al. The limiting eigenspace is fully decribed in terms of the automorphism algebra. Our results can be thought of as an analogue of the Lichnerowicz-Matsushima estimate for Kähler-Einstein metrics. In dimension 7, if the automorphism algebra is non-vanishing, we also compute the second eigenvalue for the sub-Laplacian and construct explicit eigenfunctions. In addition, for all metrics in the canonical variation of the 3-Sasaki metric we give a lower bound for the spectrum of the Riemannian Laplace operator, depending only on scalar curvature and dimension. We also strengthen a result pertaining to the growth rate of harmonic functions, due to Conlon, Hein and Sun, in the case of hyperkähler cones. In this setup we also describe the space of holomorphic functions.

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arXiv2020

Deformations of nearly $G_2$-structures

We describe the second order obstruction to deformation for nearly $G_2$ structures on compact manifolds. Building on work of B.Alexandrov and U.Semmelmann this allows proving rigidity under deformation for the proper nearly $G_2$ structure on the Aloff-Wallach space $N(1,1)$.

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arXiv2020

Conformal Killing forms in Kaehler geometry

For Kaehler manifolds we explicitly determine the solution to the conformal Killing form equation in middle degree. In particular, we complete the classification of conformal Killing forms on compact Kaehler manifolds. We give the first examples of conformal Killing forms on Kaehler manifolds not coming from Hamiltonian 2-forms. These are supported by Calabi type manifolds over a Kaehler Einstein base. In this set up we also give structure results and examples for the closely related class of Hermitian Killing forms.

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arXiv2019

Toric Nearly Kähler manifolds

We show that 6-dimensional strict nearly Kähler manifolds admitting effective $\mathbb{T}^3$ actions by automorphisms are completely characterized in the neigbourhood of each point by a function on $\mathbb{R}^3$ satisfying a certain Monge-Ampère type equation.

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arXiv2019

Complex Riemannian Foliations of open Kähler manifolds

Classification results for complex Riemannian foliations are obtained. For open subsets of irreducible Hermitian symmetric spaces of compact type, where one has explicit control over the curvature tensor, we completely classify such foliations by studying the infinitesimal model associated to the canonical connection. We also establish results for symmetric spaces of non-compact type and a general rigidity result for any irreducible Kähler manifold.

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arXiv2012

Systems of symplectic forms on four-manifolds

We study almost Hermitian 4-manifolds with holonomy algebra, for the canonical Hermitian connection, of dimension at most one. We show how Riemannian 4-manifolds admitting five orthonormal symplectic forms fit therein and classify them. In this set-up we also fully describe almost Kaehler 4-manifolds.

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arXiv2011

Complex homothetic foliations on Kähler manifolds

We obtain a local classification of complex homothetic foliations on Kaehler manifolds by complex curves. This is used to construct almost Kaehler, Ricci-flat metrics subject to additional curvature properties.

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arXiv2011

A splitting theorem for higher order parallel immersions

We consider isometric immersions into space forms having the second fundamental form parallel at order k. We show that this class of immersions consists of local products, in a suitably defined sense, of parallel immersions and normally flat immersions of flat spaces.

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arXiv2008

On the cohomology algebra of some classes of geometrically formal manifolds

We investigate harmonic forms of geometrically formal metrics, which are defined as those having the exterior product of any two harmonic forms still harmonic. We prove that a formal Sasakian metric can exist only on a real cohomology sphere and that holomorphic forms of a formal Kähler metric are parallel w.r.t. the Levi-Civita connection. In the general Riemannian case a formal metric with maximal second Betti number is shown to be flat. Finally we prove that a six-dimensional manifold with $b_1 \neq 1, b_2 \geqslant 2$ and not having the cohomology algebra of $\mathbb{T}^3 \times S^3$ carries a symplectic structure as soon as it admits a formal metric.

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arXiv2008

Prolongations of Lie algebras and applications

We study the skew-symmetric prolongation of a Lie subalgebra $\g \subseteq \mathfrak{so}(n)$, in other words the intersection $Λ^3 \cap (Λ^1 \otimes \g)$.We compute this space in full generality. Applications include uniqueness results for connections with skew-symmetric torsion and also the proof of the Euclidean version of a conjecture posed in \cite{ofarill} concerning a class of Plücker-type embeddings. We also derive a classification of the metric k-Lie algebras (or Filipov algebras), in positive signature and finite dimension. Prolongations of Lie algebras can also be used to finish the classification, started in \cite{datri}, of manifolds admitting Killing frames, or equivalently flat connections with 3-form torsion. Next we study specific properties of invariant 4-forms of a given metric representation and apply these considerations to classify the holonomy representation of metric connections with vectorial torsion, that is with torsion contained in $Λ^1 \subseteq Λ^1 \otimes Λ^2$.

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