arXiv subjects

Jorge Duque Franco

Publications and source records attributed to Jorge Duque Franco.

7 recordsLinked to original sources

arXiv2026

Data-Driven Diffusion Processes on Differential Forms via the Projected Ambient Connection Laplacian

We develop a data-driven approximation of the projected ambient connection Laplacian acting on differential forms over smooth Riemannian manifolds sampled by point clouds. The proposed construction extends the classical framework of diffusion maps and Vector Diffusion Maps from scalar functions and tangent vector fields to differential forms of arbitrary degree. Our approach is based on a novel representation of differential forms as alternating differential arrays obtained through an extension of the classical musical isomorphism. This representation enables the construction of a matrix-valued diffusion operator that approximates the projected ambient connection Laplacian directly from point cloud data without requiring a mesh or simplicial complex. The proposed discretization admits the asymptotically optimal kernel bandwidth scaling inherited from diffusion maps, leading to sharper convergence guarantees than previous data-driven approximations of the Hodge Laplacian. Building upon this operator, we derive a fully data-driven explicit Euler scheme for the heat equation on differential forms and validate the proposed methodology through numerical experiments on the unit sphere. The experiments confirm the predicted decay of the analytical solution and demonstrate the effectiveness of the proposed discretization. The proposed framework provides a natural generalization of Vector Diffusion Maps to differential forms of arbitrary degree and establishes a practical foundation for the numerical approximation of geometric partial differential equations directly from point cloud data.

math.NA↗

arXiv2026

Automorphism groups of rigid complete intersections

We study the automorphism groups of complete intersections of hypersurfaces of strictly increasing degrees in projective space. Under a combinatorial rigidity condition on the tuple of defining polynomials, we show that every automorphism of the complete intersection extends to an automorphism of each defining hypersurface, so that its automorphism group is the intersection of the automorphism groups of the defining hypersurfaces. We apply this principle to two natural families of complete intersections of two hypersurfaces of different degrees. For complete intersections of two Fermat hypersurfaces, we determine the automorphism group in every smooth case. For complete intersections of a Klein hypersurface with the reverse-order Klein hypersurface, we describe the automorphism group under an explicit arithmetic condition relating the two degrees, with Klein hypersurfaces of Wagstaff type as a natural source of examples.

math.AG↗

arXiv2025

On the Picard number and the extension degree of period matrices of complex tori

The rank $ρ$ of the Néron-Severi group of a complex torus $X$ of dimension $g$ satisfies $0\leqρ\leq g^2=h^{1,1}.$ The degree $\mathfrak{d}$ of the extension field generated over $\mathbb{Q}$ by the entries of a period matrix of $X$ imposes constraints on its Picard number $ρ$ and, consequently, on the structure of $X$. In this paper, we show that when $\mathfrak{d}$ is $2$, $3$, or $4$, the Picard number $ρ$ is necessarily large. Moreover, for an abelian variety $X$ of dimension $g$ with $\mathfrak{d}=3,$ we establish a structure-type result: $X$ must be isogenous to $E^g$, where $E$ is an elliptic curve without complex multiplication. In this case, the Picard number satisfies $ρ(X)=\frac{g(g+1)}{2}.$ As a byproduct, we obtain that if $\mathfrak{d}$ is odd, then $ρ(X)\leq\frac{g(g+1)}{2}.$

math.AG↗

arXiv2025

Periods of join algebraic cycles

We show, for all $n\ge 2$ even and $d\ge 2+\frac{4}{n}$, that the moduli of smooth degree $d$ hypersurfaces of $\mathbb{P}^{n+1}$ contains infinitely many different Hodge loci whose Zariski tangent space has the same codimension as the Hodge locus of linear cycles. We construct the Hodge cycles determining those Hodge loci as joins of $0$-dimensional cycles inside hypersurfaces of $\mathbb{P}^1$ with all their closed points defined over $\mathbb{Q}$. In order to analyze the cycle classes of these algebraic cycles, we establish a general formula for the cycle class of the join of any two algebraic cycles inside smooth hypersurfaces, expressed in terms of their periods. Furthermore, we prove that an algebraic cycle is a join of algebraic cycles, if and only if, its associated Artin Gorenstein algebra is the tensor product of the Artin Gorenstein algebras associated to each generating cycle.

math.AG↗

arXiv2025

Hodge Laplacians and Hodge Diffusion Maps

We introduce Hodge Diffusion Maps, a novel manifold learning algorithm designed to analyze and extract topological information from high-dimensional data-sets. This method approximates the exterior derivative acting on differential forms, thereby providing an approximation of the Hodge Laplacian operator. Hodge Diffusion Maps extend existing non-linear dimensionality reduction techniques, including vector diffusion maps, as well as the theories behind diffusion maps and Laplacian Eigenmaps. Our approach captures higher-order topological features of the data-set by projecting it into lower-dimensional Euclidean spaces using the Hodge Laplacian. We develop a theoretical framework to estimate the approximation error of the exterior derivative, based on sample points distributed over a real manifold. Numerical experiments support and validate the proposed methodology.

cs.LG↗

arXiv2022

On fake linear cycles inside Fermat varieties

We introduce a new class of Hodge cycles with non-reduced associated Hodge loci, we call them fake linear cycles. We characterize them for all Fermat varieties and show that they exist only for degrees $d=3,4,6$, where there are infinitely many in the space of Hodge cycles. These cycles are pathological in the sense that the Zariski tangent space of their associated Hodge locus is of maximal dimension, contrary to a conjecture of Movasati. Moreover, they provide examples of algebraic cycles not generated by their periods in the sense of Movasati-Sertöz. To study them we compute their Galois action in cohomology and their second-order invariant of the IVHS. We conclude that for any degree $d\ge 2+\frac{6}{n}$, the minimal codimension component of the Hodge locus passing through the Fermat variety is the one parametrizing hypersurfaces containing linear subvarieties of dimension $\frac{n}{2}$, extending results of Green, Voisin, Otwinowska and the second author.

math.AG↗

arXiv2021

Periods of Hodge cycles and special values of the Gauss' hypergeometric function

We compute periods of perturbations of a Fermat variety. This allows us to consider a subspace of the Hodge cycles defined by "simple" arithmetic conditions. We explore some examples and give an upper bound for the dimension of this subspace. As an application, we find explicit expressions involving some Gauss' hypergeometric functions which are algebraic over the field of rational functions in one variable.

math.AG↗