GeCo GeDi Papershttps://gecogedi.dimai.unifi.it/papers/en-usSun, 03 Mar 2024 18:42:14 +0000Pseudo-Kähler and hypersymplectic structures on semidirect productshttps://gecogedi.dimai.unifi.it/paper/572/D. Conti, A. Gil-García.<p>We study left-invariant pseudo-K\"ahler and hypersymplectic structures onsemidirect products $G\rtimes H$; we work at the level of the Lie algebra$\mathfrak{g}\rtimes\mathfrak{h}$. In particular we consider the structuresinduced on $\mathfrak{g}\rtimes\mathfrak{h}$ by existing pseudo-K\"ahlerstructures on $\mathfrak{g}$ and $\mathfrak{h}$; we classify all semidirectproducts of this type with $\mathfrak{g}$ of dimension $4$ and$\mathfrak{h}=\mathbb{R}^2$. In the hypersymplectic setting, we consider a moregeneral construction on semidirect products. We construct new $2$-stepnilpotent hypersymplectic Lie algebras; to our knowledge, these are the firstsuch examples whose underlying complex structure is not abelian</p>https://gecogedi.dimai.unifi.it/paper/572/The $*$-exponential as a covering maphttps://gecogedi.dimai.unifi.it/paper/571/A. Altavilla, S. Mongodi.<p>We employ tools from complex analysis to construct the $*$-logarithm of aquaternionic slice regular function. Our approach enables us to achieve threemain objectives: we compute the monodromy associated with the $*$-exponential;we establish sufficient conditions for the $*$-product of two $*$-exponentialsto also be a $*$-exponential; we calculate the slice derivative of the$*$-exponential of a regular function.</p>https://gecogedi.dimai.unifi.it/paper/571/Holomorphic foliations of degree two and arbitrary dimensionhttps://gecogedi.dimai.unifi.it/paper/570/M. Corrêa, A. Muniz.<p>Let $\mathscr{F}$ be a holomorphic foliation of degree $2$ on $\mathbb{P}^n$with dimension $k\geq 2$. We prove that either $\mathscr{F}$ is algebraicallyintegrable, or $\mathscr{F}$ is the linear pull-back of a degree two foliationby curves on $\mathbb{P}^{n-k+1}$, or $\mathscr{F}$ has tangent sheaf$T\mathscr{F}\simeq \mathcal{O}_{\mathbb{P}^n}(1)^{k-2}\oplus(\mathfrak{g}\otimes \mathcal{O}_{\mathbb{P}^n})$, where $\mathfrak{g}\subset\mathfrak{sl}(n+1,\mathbb{C})$ and either $\mathfrak{g}$ is an abelian Liealgebra of dimension 2 or $\mathfrak{g}\simeq \mathfrak{aff}(\mathbb{C})$, or$\mathscr{F}$ is the pull-back by a dominant rational map $\rho: \mathbb{P}^n\dashrightarrow \mathbb{P}(1^{(n-k+1)},2)$ of a non-algebraic foliation bycurves induced by a global vector field on $ \mathbb{P}(1^{(n-k+1)},2)$. Inparticular, the space of foliations of degree 2 and dimension $k\geq 2$ hasexactly 4 distinct irreducible components parameterizing non-algebraicallyintegrable foliations. As a byproduct, we describe the geometry of Poissonstructures on $\mathbb{P}^n$ with generic rank two.</p>https://gecogedi.dimai.unifi.it/paper/570/An algebraic criterion for the vanishing of bounded cohomologyhttps://gecogedi.dimai.unifi.it/paper/569/C. Campagnolo, F. Fournier-Facio, Y. Lodha, M. Moraschini.<p> We prove the vanishing of bounded cohomology with separable dual coefficientsfor many groups of interest in geometry, dynamics, and algebra. These includecompactly supported structure-preserving diffeomorphism groups of certainmanifolds; the group of interval exchange transformations of the half line;piecewise linear and piecewise projective groups of the line, giving stronganswers to questions of Calegari and Navas; direct limit linear groups ofrelevance in algebraic K-theory, thereby answering a question by Kastenholz andSroka and a question of two of the authors and L\"oh; and certain subgroups ofbig mapping class groups, such as the stable braid group and the stable mappingclass group, proving a conjecture of Bowden. Moreover, we prove that in therecently introduced framework of enumerated groups, the generic group hasvanishing bounded cohomology with separable dual coefficients. At the heart ofour approach is an elementary algebraic criterion called the commuting cyclicconjugates condition that is readily verifiable for the aforementioned largeclasses of groups.</p>https://gecogedi.dimai.unifi.it/paper/569/Displacement techniques in bounded cohomologyhttps://gecogedi.dimai.unifi.it/paper/568/C. Campagnolo, F. Fournier-Facio, Y. Lodha, M. Moraschini.<p> Several algebraic criteria, reflecting displacement properties oftransformation groups, have been used in the past years to prove vanishing ofbounded cohomology and stable commutator length. Recently, the authorsintroduced the property of commuting cyclic conjugates, a new displacementtechnique that is widely applicable and provides vanishing of the boundedcohomology in all positive degrees and all dual separable coefficients. In thisnote we consider the most recent along with the by now classical displacementtechniques and we study implications among them as well as counterexamples.</p>https://gecogedi.dimai.unifi.it/paper/568/The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomologyhttps://gecogedi.dimai.unifi.it/paper/567/L. Caputi, C. Collari, E. Ramos.<p> Building upon previous works of Proudfoot and Ramos, and using thecategorical framework of Sam and Snowden, we extend the weak categorical minortheorem from undirected graphs to quivers. As case of study, we investigate theconsequences on the homology of multipath complexes; eg. on its torsion.Further, we prove a comparison result: we show that, when restricted todirected graphs without oriented cycles, multipath complexes and matchingcomplexes yield functors which commute up to a blow-up operation on directedgraphs. We use this fact to compute the homotopy type of matching complexes fora certain class of bipartite graphs also known as half-graphs or ladders. Wecomplement the work with a study of the (representation) category of cones, andwith analysing related consequences on magnitude cohomology of quivers.</p>https://gecogedi.dimai.unifi.it/paper/567/Pseudo-Kähler geometry of properly convex projective structures on the torushttps://gecogedi.dimai.unifi.it/paper/566/N. Rungi, A. Tamburelli.<p> In this paper we prove the existence of a pseudo-K\"ahler structure on thedeformation space $\mathcal{B}_0(T^2)$ of properly convex $\mathbb R\mathbbP^2$-structures over the torus. In particular, the pseudo-Riemannian metric andthe symplectic form are compatible with the complex structure inherited fromthe identification of $\mathcal{B}_0(T^2)$ with the complement of the zerosection of the total space of the bundle of cubic holomorphic differentialsover the Teichm\"uller space. We show that the $S^1$-action on$\mathcal{B}_0(T^2)$, given by rotation of the fibers, is Hamiltonian and itpreserves both the metric and the symplectic form. Finally, we prove theexistence of a moment map for the $\mathrm{SL}(2,\mathbb R)$-action over$\mathcal{B}_0(T^2)$.</p>https://gecogedi.dimai.unifi.it/paper/566/Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of $\mathbb{R}\mathbb{P}^2$-structures in genus onehttps://gecogedi.dimai.unifi.it/paper/565/N. Rungi, A. Tamburelli.<p> In this paper we study a broad class of complete Hamiltonian integrablesystems, namely the ones whose associated Lagrangian fibration is complete andhas non compact fibres. By studying the associated complete Lagrangianfibration, we show that, under suitable assumptions, the integrals of motioncan be taken as action coordinates for the Hamiltonian system. As anapplication we find global Darboux coordinates for a new family of symplecticforms $\boldsymbol{\omega}_f$, parametrized by smooth functions$f:[0,+\infty)\to(-\infty,0]$, defined on the deformation space of properlyconvex $\mathbb{R}\mathbb{P}^2$-structures on the torus. Such a symplectic formis part of a family of pseudo-K\"ahler metrics$(\mathbf{g}_f,\mathbf{I},\boldsymbol{\omega}_f)$ defined on$\mathcal{B}_0(T^2)$ and introduced by the authors. In the last part of thepaper, by choosing $f(t)=-kt, k>0$ we deduce the expression for an arbitraryisometry of the space.</p>https://gecogedi.dimai.unifi.it/paper/565/The $\mathbb{P}\mathrm{S}\mathrm{L}(3,\mathbb{R})$-Hitchin component as an infinite-dimensional pseudo-Kähler reductionhttps://gecogedi.dimai.unifi.it/paper/564/N. Rungi, A. Tamburelli.<p> The aim of this paper is to show the existence and give an explicitdescription of a pseudo-Riemannian metric and a symplectic form on the$\mathbb{P}\mathrm{S}\mathrm{L}(3,\mathbb{R})$-Hitchin component, bothcompatible with Labourie and Loftin's complex structure. In particular, theygive rise to a mapping class group invariant pseudo-K\"ahler structure on aneighborhood of the Fuchsian locus, which restricts to a multiple of theWeil-Petersson metric on Teichm\"uller space. Finally, generalizing a previousresult in the case of the torus, we prove the existence of an Hamiltonian$S^1$-action.</p>https://gecogedi.dimai.unifi.it/paper/564/Riemannian geometry of maximal surface group representations acting on pseudo-hyperbolic spacehttps://gecogedi.dimai.unifi.it/paper/563/N. Rungi.<p> For any maximal surface group representation into $\mathrm{SO}_0(2,n+1)$, weintroduce a non-degenerate scalar product on the the first cohomology group ofthe surface with values in the associated flat bundle. In particular, it givesrise to a non-degenerate Riemannian metric on the smooth locus of the subsetconsisting of maximal representations inside the character variety. In the case$n=2$, we carefully study the properties of the Riemannian metric on themaximal connected components, proving that it is compatible with the orbifoldstructure and finding some totally geodesic sub-varieties. Then, in the generalcase, we explain when a representation with Zariski closure contained in$\mathrm{SO}_0(2,3)$ represents a smooth or orbifold point in the maximal$\mathrm{SO}_0(2,n+1)$-character variety and we show that the associated spaceis totally geodesic for any $n\ge 3$.</p>https://gecogedi.dimai.unifi.it/paper/563/The moduli space of flat maximal space-like embeddings in pseudo-hyperbolic spacehttps://gecogedi.dimai.unifi.it/paper/562/N. Rungi, A. Tamburelli.<p> We study the moduli space of flat maximal space-like embeddings in$\mathbb{H}^{2,2}$ from various aspects. We first describe the associatedCodazzi tensors to the embedding in the general setting, and then, we introducea family of pseudo-K\"ahler metrics on the moduli space. We show the existenceof two Hamiltonian actions with associated moment maps and use them to find ageometric global Darboux frame for any symplectic form in the above family.</p>https://gecogedi.dimai.unifi.it/paper/562/A connection between cut locus, Thom space and Morse-Bott functionshttps://gecogedi.dimai.unifi.it/paper/561/S. Basu, S. Prasad.<p> Associated to every closed, embedded submanifold $N$ in a connectedRiemannian manifold $M$, there is the distance function $d_N$ which measuresthe distance of a point in $M$ from $N$. We analyze the square of this functionand show that it is Morse-Bott on the complement of the cut locus$\mathrm{Cu}(N)$ of $N$, provided $M$ is complete. Moreover, the gradient flowlines provide a deformation retraction of $M-\mathrm{Cu}(N)$ to $N$. If $M$ isa closed manifold, then we prove that the Thom space of the normal bundle of$N$ is homeomorphic to $M/\mathrm{Cu}(N)$. We also discuss several interestingresults which are either applications of these or related observationsregarding the theory of cut locus. These results include, but are not limitedto, a computation of the local homology of singular matrices, a classificationof the homotopy type of the cut locus of a homology sphere inside a sphere, adeformation of the indefinite unitary group $U(p,q)$ to $U(p)\times U(q)$ and ageometric deformation of $GL(n,\mathbb{R})$ to $O(n,\mathbb{R})$ which isdifferent from the Gram-Schmidt retraction.</p>https://gecogedi.dimai.unifi.it/paper/561/Counterexample to a conjecture about dihedral quandlehttps://gecogedi.dimai.unifi.it/paper/560/S. Panja, S. Prasad.<p> It was conjectured that the augmentation ideal of a dihedral quandle of evenorder $n>2$ satisfies $<br>\Delta^k(\text{R}_n)/\Delta^{k+1}(\text{R}_{n})<br>=n$ forall $k\geq 2$. In this article we provide a counterexample against thisconjecture.</p>https://gecogedi.dimai.unifi.it/paper/560/The image of polynomials and Waring type problems on upper triangular matrix algebrashttps://gecogedi.dimai.unifi.it/paper/559/S. Panja, S. Prasad.<p> Let $p$ be a polynomial in non-commutative variables $x_1,x_2,\ldots,x_n$with constant term zero over an algebraically closed field $K$. The object ofstudy in this paper is the image of this kind of polynomial over the algebra ofupper triangular matrices $T_m(K)$. We introduce a family of polynomials calledmulti-index $p$-inductive polynomials for a given polynomial $p$. Using thisfamily we will show that, if $p$ is a polynomial identity of $T_t(K)$ but notof $T_{t+1}(K)$, then $p \left(T_m(K)\right)\subseteq T_m(K)^{(t-1)}$. Equalityis achieved in the case $t=1,~m-1$ and an example has been provided to showthat equality does not hold in general. We further prove existence of $d$ suchthat each element of $T_m(K)^{(t-1)}$ can be written as sum of $d$ manyelements of $p\left( T_m(K) \right)$. It has also been shown that the image of$T_m(K)^\times$ under a word map is Zariski dense in $T_m(K)^\times$.</p>https://gecogedi.dimai.unifi.it/paper/559/Independence complexes of wedge of graphshttps://gecogedi.dimai.unifi.it/paper/558/N. Daundkar, S. Panja, S. Prasad.<p> In this article, we introduce the notion of a wedge of graphs and providedetailed computations for the independence complex of a wedge of path and cyclegraphs. In particular, we show that these complexes are either contractible orwedges of spheres.</p>https://gecogedi.dimai.unifi.it/paper/558/Cut Locus of Submanifolds: A Geometric and Topological Viewpointhttps://gecogedi.dimai.unifi.it/paper/557/S. Prasad.<p> Associated to every closed, embedded submanifold $N$ of a connectedRiemannian manifold $M$, there is the distance function $d_N$ which measuresthe distance of a point in $M$ from $N$. We analyze the square of this functionand show that it is Morse-Bott on the complement of the cut locus$\mathrm{Cu}(N)$ of $N$, provided $M$ is complete. Moreover, the gradient flowlines provide a deformation retraction of $M-\mathrm{Cu}(N)$ to $N$. If $M$ isa closed manifold, then we prove that the Thom space of the normal bundle of$N$ is homeomorphic to $M/\mathrm{Cu}(N)$. We also discuss several interestingresults which are either applications of these or related observationsregarding the theory of cut locus. These results include, but are not limitedto, a computation of the local homology of singular matrices, a classificationof the homotopy type of the cut locus of a homology sphere inside a sphere, adeformation of the indefinite unitary group $U(p,q)$ to $U(p)\times U(q)$ and ageometric deformation of $GL(n,\mathbb{R} )$ to $O(n,\mathbb{R} )$ which isdifferent from the Gram-Schmidt retraction. \bigskip \noindent If a compact Lie group $G$ acts on a Riemannian manifold$M$ freely then $M/G$ is a manifold. In addition, if the action is isometric,then the metric of $M$ induces a metric on $M/G$. We show that if $N$ is a$G$-invariant submanifold of $M$, then the cut locus $\mathrm{Cu}(N)$ is$G$-invariant, and $\mathrm{Cu}(N)/G = \mathrm{Cu}\left( N/G \right) $ in$M/G$. An application of this result to complex projective hypersurfaces hasbeen provided.</p>https://gecogedi.dimai.unifi.it/paper/557/On the Cut Locus of Submanifolds of a Finsler Manifoldhttps://gecogedi.dimai.unifi.it/paper/556/A. Bhowmick, S. Prasad.<p> In this paper, we investigate the cut locus of submanifolds in a Finslermanifolds, a natural generalization of Riemannian manifolds. We explore thedeformation and characterization of the cut locus, extending the results of\cite{BaPr21}. We also obtain a generalization of Klingenberg's lemma forclosed geodesic (\cite{Kli59}) in the Finsler setting.</p>https://gecogedi.dimai.unifi.it/paper/556/Strongly Invertible Legendrian Linkshttps://gecogedi.dimai.unifi.it/paper/555/C. Collari, P. Lisca.<p> We introduce and study strongly invertible Legendrian links in the standardcontact three-dimensional space. We establish the equivariant analogs of basicresults separately well-known for strongly invertible and Legendrian links,i.e. the existence of transvergent front diagrams, an equivariant LegendrianReidemeister theorem, and an equivariant stabilization theorem \`a laFuch-Tabachnikov. We also introduce a maximal equivariant Thurston-Bennequinnumber for strongly invertible links and we exhibit infinitely many such linksfor which the invariant coincides with the usual maximal Thurston-Bennequinnumber. We conjecture that such a coincidence does not hold general and thatthere exist strongly invertible knots having Legendrian representativesisotopic to their reversed Legendrian mirrors but not isotopic to any stronglyinvertible Legendrian knot.</p>https://gecogedi.dimai.unifi.it/paper/555/Dolbeault and $J$-invariant cohomologies on almost complex manifoldshttps://gecogedi.dimai.unifi.it/paper/554/L. Sillari, A. Tomassini.<p> In this paper we relate the cohomology of $J$-invariant forms to theDolbeault cohomology of an almost complex manifold. We find necessary andsufficient condition for the inclusion of the former into the latter to be trueup to isomorphism. We also extend some results obtained by J. Cirici and S.O.Wilson about the computation of the left-invariant cohomology of nilmanifoldsto the setting of solvmanifolds. Several examples are given.</p>https://gecogedi.dimai.unifi.it/paper/554/On the minimal number of solutions of the equation $ φ(n+k)= M \, φ(n) $, $ M=1$, $2$https://gecogedi.dimai.unifi.it/paper/553/M. Ferrari, L. Sillari.<p> We fix a positive integer $k$ and look for solutions of the equations$\phi(n+k) = \phi(n)$ and $\phi(n + k) = 2\phi(n)$. We prove that Fermat primescan be used to build five solutions for the first equation when $k$ is even andfive for the second one when $k$ is odd. These results hold for $k \le 2 \cdot10^{100}$. We also show that for the second equation with even $k$ there are atleast three solutions for $k \le 4 \cdot 10^{58}$. Our work increases theprevious minimal number of known solutions for both equations.</p>https://gecogedi.dimai.unifi.it/paper/553/