arXiv subjects

Leonardo F. Cavenaghi

Publications and source records attributed to Leonardo F. Cavenaghi.

At least 19 recordsLinked to original sources

arXiv2026

On the atomic decomposition of complete intersection in flag varieties

We develop a root-theoretic localization formalism for genus-zero Gromov--Witten invariants of flag varieties and of smooth zero loci of globally generated homogeneous vector bundles. The resulting invariants are expressed as finite sums over decorated trees whose contributions are determined by the root system of the ambient flag variety and by the torus weights of the defining bundle. We connect these computations with the Theory of Hodge Atoms of Katzarkov--Kontsevich--Pantev--Yu via the matrix of small quantum multiplication by first Chern class. Let $X$ be a Fano fourfold with $h^{3,1}(X)=1$ arising as a hyperplane section of a suitable Fano fivefold. Its cohomology decomposes into a monodromy-fixed part and the middle vanishing cohomology. This decomposition is preserved by quantum multiplication by the first Chern class. Moreover, it acts by a scalar on the vanishing summand. We show, for the monodromy-fixed block, that (a) if every eigenvalue has algebraic multiplicity at most two and $X$ is Hodge general, then $X$ is irrational; (b) if every eigenvalue has Jordan defect at most one and $X$ is rational, then every weak factorization contains a smooth surface center whose minimal model is a projective K3 surface. Many applications are presented. Using the recently introduced quantum connection approach to irrationality, we obtain from the computations here presented the irrationality of the very general smooth complete intersection of a hyperplane and a quadric in $\operatorname{LG}(3,6)$; the very general smooth quadric section in $G_2/P_2$; and the very general smooth complete intersection of five hyperplanes and a quadric in the spinor 10-fold.

math.AG↗

arXiv2026

On the generalized cohomologies of almost abelian solvmanifolds

Generalized complex structures give rise to generalized Dolbeault, Bott--Chern and Aeppli cohomologies. We study these cohomologies for invariant generalized complex structures on almost abelian Lie algebras $\mathfrak g_A=\mathbb R e_0\ltimes_A\mathfrak h$, where $\mathfrak h$ is an abelian ideal of codimension one, and on almost abelian solvmanifolds $Γ\backslash G_A$. We first show that the invariant generalized Dolbeault cohomology can be identified with the Chevalley--Eilenberg cohomology of the complex Lie algebra $\mathcal L$, the $+{\mathbf i}$-eigenspace of the structure, with coefficients in a character determined by the canonical line. We prove that $\mathcal L$ is itself an almost abelian Lie algebra; thus the cohomology reduces to the kernel and cokernel of an explicit derivation. Assuming that $A$ is diagonalizable over $\mathbb R$, we obtain a combinatorial formula for the cohomology dimensions and compare with compact quotients. Other comparison results are discussed, noting that the cohomology of invariant forms always injects into that of $Γ\backslash G_A$.

math.DG↗

arXiv2026

A quantum connection approach to rationality

We extract rank and regularized exponents from cyclic generic factors of the quantum connection and show that these data persist under the birational comparisons considered here. Applications include showing non-rationality of cubic threefolds, ordinary Gushel--Mukai threefolds, very general quartic fourfolds, and products of the threefolds with $\PP^1$.

math.AG↗

arXiv2026

On the generalized geometry of almost abelian solvmanifolds

We study left-invariant generalized complex structures on almost abelian Lie groups $G_A$ with Lie algebra $\mathfrak{g}_A=\mathbb{R} e_0\ltimes_A\mathfrak{h}$, where $\mathfrak{h}$ is an abelian ideal of codimension one, and on their compact quotients. First, when $A$ is diagonalizable over $\mathbb{R}$, we characterize all admissible types by pairings of its eigenvalues and characterize the structures admitting a closed invariant pure-spinor generator. For general $A$, we obtain Jordan-theoretic type bounds and a construction using a complex quotient and a symplectic ideal. Finally, in dimension six, we establish nonexistence results and give explicit intermediate-type constructions.

math.DG↗

arXiv2026

Fatness and positive curvatures on Riemannian submersions

Guided by rather counterintuitive properties of \emph{dual holonomy fields}, we introduce different notions of \emph{fatness} for Riemannian submersions. We introduce \emph{integral fatness} as an intrinsic condition implied by positive sectional curvature; \emph{sub-fatness} as a rigidity constraint that naturally arises in \emph{canonical variation} limits; and then develop the concept of \emph{weakly nonnegative curvature} (WNN), an intrinsic condition depending only on the horizontal distribution and the base metric. Positive sectional curvature implies fatness in this class, verifying Petersen--Wilhelm's Conjecture for homogeneous submersions and beyond. Finally, we introduce the notion of Riemannian submersion which are \emph{$A$-Clifford}: we show that them, like fatness and WNN, is intrinsic to the pair $(\mathcal H,b)$; that it singles out a canonical vertical gauge; and that in rank three it characterizes $3$-Sasakian geometry, yielding rigidity on $\mathrm{S}^3,\mathrm{SO}(3)$-principal bundles.

math.DG↗

arXiv2026

A Geometric Realization of Spherical T-Duality via $\star$-Diagrams

We relate spherical T-duality for oriented linear $\mathrm{S}^3$-bundles over $\mathrm{S}^4$ (the Milnor bundles $M_{m,n}$, which are $\mathrm{S}^3$-principal exactly when $m=0$ or $n=0$, and whose total spaces are homotopy $7$-spheres exactly when $m+n=\pm1$) to $\star$-diagrams and to a higher-dimensional generalization of the logarithmic transformations of $4$-manifold topology. For an $\mathrm{S}^3$-principal pair $(P,H)$, $(\widehat P,\widehat H)$ over $\mathrm{S}^4$, we show that the T-duality correspondence space $P\times_{\mathrm{S}^4}\widehat P$ is itself a $\star$-diagram of a distinguished type, which we call \emph{bifree}, and that bifree $\star$-diagrams are precisely the fiber products of principal bundles; spherical T-duality of the decorated pair is then a condition on the fluxes carried by that diagram. For bundles of equal Euler class $ku$, principal or not, we show that the two bundles are spherical T-dual with the diagonal fluxes $[k]$, and that they occur as the two base manifolds of an explicit $\star$-diagram, obtained by pulling back a principal Milnor bundle; this diagram is never bifree. We then introduce product-preserving generalized logarithmic transformations on products $Σ\times\mathrm{S}^1$ of homotopy spheres with the circle, and prove that, after stabilization by $\mathrm{S}^1$, the spherical T-dualities between homotopy $7$-spheres are realized by such transformations. In particular, $Σ^7_{GM}\times\mathrm{S}^1$ is obtained from $\mathrm{S}^7\times\mathrm{S}^1$ by one of them, where $Σ^7_{GM}$ denotes the Gromoll--Meyer exotic sphere: spherical T-duality relates distinct smooth structures on the topological $7$-sphere, and the relation is implemented by an explicit cut-and-paste operation.

math.DG↗

arXiv2026

The $G$-equivariant Kazdan--Warner problem

We establish an equivariant analogue of the Kazdan--Warner trichotomy for admissible scalar curvature functions. Let $M$ be a closed connected manifold of dimension $n \ge 3$ equipped with an effective isometric action of a compact connected Lie group $G$ of cohomogeneity at least one and with no zero-dimensional orbits. All metrics and prescribed functions are taken to be $G$-invariant. We prove that, for such pairs $(M, G)$, the classical trichotomy does not extend verbatim. A distinct class emerges, consisting of \emph{totally $G$-positive} pairs, for which every $G$-invariant metric exhibits positive total scalar curvature. Each such pair admits a $G$-invariant metric of positive constant scalar curvature, but admits no metric of zero or negative constant scalar curvature. Every remaining pair falls into exactly one of three classes mirroring the original trichotomy.

math.DG↗

arXiv2026

The curvature of convex sum of metrics and applications

In this note, we derive explicit formulae for the curvature of a convex sum of Riemannian metrics, $g_t = (1-t)g_0 + t g_1$. We study whether such a deformation can increase the \emph{average} of the Riemann curvature component $R_t(X,Y,Y,X)$ along an immersed, totally geodesic flat torus. Because a first-order increase is prohibited, we obtain necessary and sufficient conditions for $g_t$ to have a positive average variation of order $r \geq 2$. These conditions are applied to paths joining $g_0$ to classical metric deformations, including conformal changes, vertical warpings, and Cheeger deformations.

math.DG↗

arXiv2026

The Cohomology of Solvmanifold SYZ Mirrors

This paper investigates the geometric and cohomological properties of non-Kähler SYZ mirror symmetry for dual torus fibrations over solvmanifolds in the sense of Lau, Tseng and Yau. We are mainly concerned with three questions: \textbf{(a)} How the Lau-Tseng-Yau notion of non-Kähler SYZ is related to the mapping of supersymmetric branes between symplectic and complex sides; \textbf{(b)} Finding explicit non-Kähler SYZ mirror pairs determined purely by Lie-theoretic data; \textbf{(c)} better understand the cohomological correspondence in the Lau-Tseng-Yau framework (given by a Fourier-Mukai transform), especially concerning the role of Tseng-Yau cohomology. We prove that the Fourier-Mukai transform introduced by Lau-Tseng-Yau exchanges type-A supersymmetric cycles, which are given by special Lagrangian sections equipped with flat $\mathrm{U}(1)$ connections, with type-B cycles, corresponding to line bundles whose connections satisfy the deformed Hermitian-Yang-Mills (dHYM) equation. We provide pure Lie-theoretic criteria for the existence of non-Kähler SYZ mirror pairs whose base manifolds are solvmanifolds. Applying these criteria, we construct new explicit families of mirror pairs from almost abelian and generalized Heisenberg Lie groups, and provide a complete classification of such pairs arising from nilpotent Lie groups. To contextualize the role of the Tseng-Yau cohomology, we link it to noncommutative geometry. We introduce the Tseng-Yau and Bott-Chern mirror bicomplexes. We show that (some of) their enclosed cohomologies reduce to the primitive Tseng-Yau and Bott-Chern cohomologies and that for basic forms they are isomorphic under the Fourier-Mukai transform. As a last contribution, we discuss how to explicitly compute the Tseng-Yau and the Bott-Chern cohomology for the non-Kähler SYZ mirror pairs constructed here.

hep-th↗

arXiv2026

Atoms meet symbols

This paper introduces a novel framework for constructing invariants in $G$-equivariant birational geometry by unifying two recent approaches: the theory of atoms recently developed by Katzarkov, Kontsevich, Pantev, and Yu, and the theory of modular symbols due to Kontsevich, Tschinkel, and Pestun. We initiate the theory of Chen-Ruan atoms. Assuming the blowup formula for the quantum Chen-Ruan cohomology, we outline how to extend the theory of atoms to global quotient orbifolds and present some striking applications. In addition, we develop a separate class of purely geometric invariants for $\mathbb{Z}/2$- and $\mathbb{Z}/3$-actions on surfaces and threefolds. We provide many examples of non-$G$-linearizable $G$-actions on projective varieties treated with these new techniques.

math.AG↗

arXiv2026

Singular Riemannian Foliations, variational problems and Principles of Symmetric Criticalities

A singular foliation $\mathcal{F}$ on a complete Riemannian manifold $M$ is called Singular Riemannian foliation (SRF for short) if its leaves are locally equidistant, e.g., the partition of $M$ into the orbits of a Lie group action by isometries. In this paper, we investigate variational problems in compact Riemannian manifolds equipped with SRFs with special properties, which we name as AVP. Examples of such SRFs being considered include isoparametric foliations, SRFs on Euclidean fiber bundles, and the partition of $M$ into the orbits of a Lie group acting by isometries. More precisely, we prove an analog to Palais' Principle of Symmetric Criticality for $\mathcal{F}$-symmetric integral operators on the Banach spaces $W^{1,p}(M)$. This result together with a version of the Rellich--Kondrachov--Hebey--Vaugon Embedding Theorem for $\mathcal{F}$-basic Sobolev functions allows us to circumvent difficulties with Sobolev's critical exponents when considering applications of techniques from Calculus of Variations to find solutions to PDEs. To exemplify this, we prove the existence of countably infinite many weak solutions to a class of variational problems, which includes $p$-Kirchhoff problems for manifolds equipped with AVP.

math.DG↗

arXiv2025

A Gromov-Witten approach to $G$-equivariant birational invariants

In arXiv:2404.19088, we initiated a program linking birational invariants with smooth ones and offering new interpretations of classical invariants, such as the Kervaire-Milnor invariants. Here, we rely on the profound geometric reasoning provided by Lupercio and Uribe in the early 00s to establish a connection between Chen-Ruan cohomology and several $G$-birational invariants introduced in the pioneering works Kontsevich, Kresch, Pestun, Tschinkel, along with presenting applications. Combined with the theory of atoms by Katzarkov, Kontsevich, Pantev, and Yu, the proposal in this paper program will lead to a theory of equivariant atoms.

math.AG↗

arXiv2025

A curvature approach to fatness

This paper delves into the concept of ``fat bundles'' within Riemannian submersions. One explores the structural implications of fat Riemannian submersions, particularly focusing on those with non-negative sectional curvature. The main results include the classification of fibers as symmetric spaces, the isometric correspondence of fat foliations with coset foliations on Lie groups, and the rigidity of dual foliations associated with fat Riemannian submersions.

math.DG↗

arXiv2024

Hearing Exotic Smooth Structures

This paper explores the existence and properties of \emph{basic} eigenvalues and eigenfunctions associated with the Riemannian Laplacian on closed, connected Riemannian manifolds featuring an effective isometric action by a compact Lie group. Our primary focus is on investigating the potential existence of homeomorphic yet not diffeomorphic smooth manifolds that can accommodate invariant metrics sharing common basic spectra. We establish the occurrence of such scenarios for specific homotopy spheres and connected sums. Moreover, the developed theory demonstrates that the ring of invariant admissible scalar curvature functions fails to recover the smooth structure in many examples. We show the existence of homotopy spheres with identical rings of invariant scalar curvature functions, irrespective of the underlying smooth structure.

math.DG↗

arXiv2024

Symmetric spaces as adjoint orbits and their geometries

We realize specific classical symmetric spaces, like the semi-Kähler symmetric spaces discovered by Berger, as cotangent bundles of symmetric flag manifolds. These realizations enable us to describe these cotangent bundles' geodesics and Lagrangian submanifolds. As a final application, we present the first examples of vector bundles over simply connected manifolds with nonnegative curvature that cannot accommodate metrics with nonnegative sectional curvature, even though their associated unit sphere bundles can indeed accommodate such metrics. Our examples are derived from explicit bundle constructions over symmetric flag spaces.

math.DG↗

arXiv2024

Gromoll--Meyer's actions and the geometry of (exotic) spacetimes

Since the advent of new pairwise non-diffeomorphic structures on smooth manifolds, it has been questioned whether two topologically identical manifolds could admit different geometries. Not surprisingly, physicists have wondered whether a smooth structure assumption different from some classical known models could produce different physical meanings. In this paper, we inaugurate a very computational manner to produce physical models on classical and exotic spheres that can be built equivariantly, such as the classical Gromoll--Meyer exotic spheres. As first applications, we produce Lorentzian metrics on homeomorphic but not diffeomorphic manifolds that enjoy the same physical properties, such as geodesic completeness, positive Ricci curvature, and compatible time orientation. These constructions can be pulled back to higher models, such as exotic ten spheres bounding spin manifolds, to be approached in forthcoming papers.

math.DG↗

arXiv2024

A note on the vertizontal curvature of fat bundles

In his unpublished notes on fat bundles, W. Ziller poses a compelling question: given a fat principal $G$-bundle $(P, g) \rightarrow (B, h)$ with $\dim G = 3$, and $g$ representing a Riemannian submersion metric ensuring that the $G$-orbits are totally geodesic, can one modify $h$ to render all vertical curvatures equal to $1$? In this note, we establish a rigidity result for fat Riemannian foliations with bounded holonomy and a specific curvature constraint. Our result addresses Ziller's question for fat fiber bundles with compact structure groups, considering connected compact total spaces under a curvature constraint that holds on various examples, such as locally symmetric spaces. Additionally, we assume that all vertizontal curvatures coincide at a point.

math.DG↗

arXiv2023

On the dynamics of positively curved metrics on $\mathrm{SU}(3)/\mathrm{T}^2$ under the homogeneous Ricci flow

In this note, we show that the classical Wallach manifold $\mathrm{SU}(3)/\mathrm{T}^2$-admits metrics of positive intermediate Ricci curvature $(\mathrm{Ric}_d >0)$ for $d = 1, 2, 3, 4, 5$ that lose these properties under the homogeneous Ricci flow for $d=1, 2, 3, 5$. We make the same analyses to the family of Riemannian flag manifolds $\mathrm{SU}(m+2p)/\mathrm{S}(\mathrm{U}(m)\times\mathrm{U}(p)\times \mathrm{U}(p))$, concluding similar results. These explicitly verify some claims expected to be true among experts (see \cite{BW}) for positive Ricci curvature and intermediate positive Ricci curvature. Our technique is only possible due to the global behavior understanding of the homogeneous Ricci flow for invariant metrics on these manifolds.

math.DG↗