Ugo Bruzzo’s research while affiliated with Federal University of Paraíba and other places

The present paper, which is partially a review, but also contains several completely new results, aims at presenting, in a unified mathematical framework, a complex and articulated lore regarding non-compact symmetric spaces, with negative curvature, whose isometry group is a non-compact, real simple Lie group. All such manifolds are Riemannian normal manifolds, according to Alekseevsky's definition, in the sense that they are metrically equivalent to a solvable Lie group manifold. This identification provides a vision in which, on one side one can derive quite explicit and challenging formulae for the unique distance function between points of the manifold, on the other one, one can organize the entire set of the available manifolds in universality classes distinguished by their common Tits Satake submanifold and, correspondingly, by their non-compact rank. The members of the class are distinguished by their different Paint Groups, the latter notion having been introduced by two of the present authors in an earlier collaboration. In relation to the construction of neural networks, these mathematical structures offer unique possibilities of replacing ad hoc activation functions with the naturally defined non-linear operations that relate Lie algebras to Lie Groups and vice-versa. The Paint Group invariants offer new tokens both to construct algorithms and inspect (hopefully to control) their working. A conspicuous part of the paper is devoted to the study and systematic construction of parabolic/elliptic discrete subgroups of the Lie groups SO(r,r+q), in view of discretization and/or tessellations of the space to which data are to be mapped. Furthermore, it is shown how the ingredients of Special K\"ahler Geometry and the c-map, well known in the supergravity literature, provide a unified classification scheme of the relevant Tits Satake universality classes with non-compact rank r<5.

We provide a splitting criterion for supervector bundles over the projective superspaces $\mathbb{P}^{n|m}$. More precisely, we prove that a rank $p|q$ supervector bundle on $\mathbb{P}^{n|m}$ with vanishing intermediate cohomology is isomorphic to the direct sum of even and odd line bundles, provided that $n \geq 2$. For n=1 we provide an example of a supervector bundle that cannot be written as a sum of line bundles.

We obtain explicit formulae for the Donagi-Markman (Bryant-Griffiths, Yukawa) cubic for Hitchin systems of type $A_2$, $B_2$ and $G_2$. This is achieved by evaluating the quadratic residues in the Balduzzi-Pantev formula, using a previous result of ours. For $G_2$ we also recover earlier results of Hitchin.

This paper is a first step in the study of nonstandard graded algebras having Poincar\'e duality and their Lefschetz properties. We prove the equivalence between the toric setup and the G-graded one, generalize Macaulay-Matlis duality, introduce Lefschetz properties and prove a Hessian criteria in the G-graded setup. We prove a special case of the Codimension One Conjecture of Cattani-Cox-Dickenstein.

In this note we describe explicitly, in terms of Lie theory and cameral data, the covariant (Gauss–Manin) derivative of the Seiberg–Witten differential defined on the weight-one variation of Hodge structures that exists on a Zariski open subset of the base of the Hitchin fibration.

We continue our study of the Noether–Lefschetz loci in toric varieties and investigate deformation of pairs (V, X) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a simplicial projective toric variety PΣ2k+1$\mathbb {P}_{\Sigma }^{2k+1}$, with V⊂X$V\subset X$. The hypersurface X is supposed to be of Macaulay type, which means that its toric Jacobian ideal is Cox–Gorenstein, a property that generalizes the notion of Gorenstein ideal in the standard polynomial ring. Under some assumptions, we prove that the class λV∈Hk,k(X)$\lambda _V\in H^{k,k}(X)$ deforms to an algebraic class if and only if it remains of type (k, k). Actually we prove that locally the Noether–Lefschetz locus is an irreducible component of a suitable Hilbert scheme. This generalizes Theorem 4.2 in our previous work (Bruzzo and Montoya 15(2):682–694, 2021) and the main theorem proved by Dan (in: Analytic and Algebraic Geometry. Hindustan Book Agency, New Delhi, pp 107–115, 2017).

We briefly review an open conjecture about Higgs bundles that are semistable after pulling back to any curve, and prove it in the rank 2 case. We also prove some results in higher rank under suitable additional assumptions. Moreover, we establish a set of inequalities holding for H-nef Higgs bundles that generalize some of the Fulton–Lazarsfeld inequalities for numerically effective vector bundles.

Working in the category of smooth projective varieties over an algebraically closed field of characteristic 0, we review notions of ampleness and numerical nefness for Higgs bundles which "feel" the Higgs field and formulate criteria of the Barton-Kleiman type for these notions. We give an application to minimal surfaces of general type that saturate the Miyaoka-Yau inequality, showing that their cotangent bundle is ample. This will use results by Langer that imply that also for varieties over algebraically closed field of characteristic zero the so-called Simpson system is stable.

D3 brane solutions of type IIB supergravity can be obtained by means of a classical Ansatz involving a harmonic warp factor, H(y,y¯)$H(\textbf{y},\bar{\textbf{y}})$ multiplying at power -1/2$-1/2$ the first summand, i.e., the Minkowski metric of the D3 brane world-sheet, and at power 1/2 the second summand, i.e., the Ricci-flat metric on a six-dimensional transverse space M6$\mathcal {M}_6$, whose complex coordinates y are the arguments of the warp factor. Of particular interest is the case where M6=tot[KMB$\mathcal {M}_6={\text {tot}}[ K\left[ \left( \mathcal {M}_B\right) \right]$ is the total space of the canonical bundle over a complex Kähler surface MB$\mathcal {M}_B$. This situation emerges in many cases while considering the resolution à la Kronheimer of singular manifolds of type M6=C3/Γ$\mathcal {M}_6=\mathbb {C}^3/\Gamma$, where Γ⊂SU(3)$\Gamma \subset \mathrm {SU(3)}$ is a discrete subgroup. When Γ=Z4$\Gamma = \mathbb {Z}_4$, the surface MB$\mathcal {M}_B$ is the second Hirzebruch surface endowed with a Kähler metric having SU(2)×U(1)$\mathrm {SU(2)\times U(1)}$ isometry. There is an entire class Met(FV)${\text {Met}}(\mathcal{F}\mathcal{V})$ of such cohomogeneity one Kähler metrics parameterized by a single function FK(v)$\mathcal{F}\mathcal{K}(\mathfrak {v})$ that are best described in the Abreu–Martelli–Sparks–Yau (AMSY) symplectic formalism. We study in detail a two-parameter subclass Met(FV)KE⊂Met(FV)${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}\subset {\text {Met}}(\mathcal{F}\mathcal{V})$ of Kähler–Einstein metrics of the aforementioned class, defined on manifolds that are homeomorphic to S2×S2$S^2\times S^2$, but are singular as complex manifolds. Actually, Met(FV)KE⊂Met(FV)ext⊂Met(FV)${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}\subset {\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{ext}}\subset {\text {Met}}(\mathcal{F}\mathcal{V})$ is a subset of a four parameter subclass Met(FV)ext${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{ext}}$ of cohomogeneity one extremal Kähler metrics originally introduced by Calabi in 1983 and translated by Abreu into the AMSY action-angle formalism.Met(FV)ext${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{ext}}$ contains also a two-parameter subclass Met(FV)extF2${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{ext}\mathbb {F}_2}$ disjoint from Met(FV)KE${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}$ of extremal smooth metrics on the second Hirzebruch surface that we rederive using constraints on period integrals of the Ricci 2-form. The Kähler–Einstein nature of the metrics in Met(FV)KE${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}$ allows the construction of the Ricci-flat metric on their canonical bundle via the Calabi Ansatz, which we recast in the AMSY formalism deriving some new elegant formulae. The metrics in Met(FV)KE${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}$ are defined on the base manifolds of U(1) fibrations supporting the family of Sasaki–Einstein metrics SEmet5$\textrm{SEmet}_5$ introduced by Gauntlett et al. (Adv Theor Math Phys 8:711–734, 2004), and already appeared in Gibbons and Pope (Commun Math Phys 66:267–290, 1979). However, as we show in detail using Weyl tensor polynomial invariants, the six-dimensional Ricci-flat metric on the metric cone of M5∈Met(SE)5${\mathcal M}_5 \in {\text {Met}}(\textrm{SE})_5$ is different from the Ricci-flat metric on tot[KMKE${\text {tot}}[ K\left[ \left( \mathcal {M}_{\textrm{KE}}\right) \right]$ constructed via Calabi Ansatz. This opens new research perspectives. We also show the full integrability of the differential system of geodesics equations on MB$\mathcal {M}_B$ thanks to a certain conserved quantity which is similar to the Carter constant in the case of the Kerr metric.