arXiv subjects

Badre Mounda

Publications and source records attributed to Badre Mounda.

4 recordsLinked to original sources

arXiv2026

Picard groups of completed period images and the Deng-Robles problem

A basic problem in the geometry of degenerating period maps is to determine whether their completed images admit an intrinsic algebraic description. For polarized variations of Hodge structure over smooth quasi-projective surfaces, Deng and Robles formulated such a problem in terms of the Kato-Nakayama-Usui completion of the period image and a conjectural Proj description involving the augmented Hodge line bundle and the boundary divisor on a smooth compactification of the base. We show that the essential obstruction to this description is divisor-theoretic: it may be expressed as a Picard-generation statement on the completed mixed period image. We prove this statement when the pure period image is one-dimensional, and consequently obtain the Deng-Robles Proj description in this case.

math.AG↗

arXiv2025

Constructive Proof of the Hodge Conjecture for K3 Surfaces via Nodal Degenerations

We give a constructive proof of the Hodge conjecture for complex $K3$ surfaces that does not rely on Torelli-type results. Starting with an arbitrary rational $(1,1)$-class $\alpha\in H^{1,1}(X,\mathbb{Q})$, we algorithmically build a one-parameter family of quartic $K3$'s acquiring at most ten $A_1$-nodes. On the central fibre $\widetilde{X}_0$, the class $\alpha$ specializes to a $\mathbb{Q}$-linear combination of the hyperplane class and the exceptional $(-2)$-curves coming from the blow-ups of the nodes. Using the Clemens--Schmid sequence together with Picard--Lefschetz theory, we identify $\Gr^W_2 H^2_{\lim}\cong H^2(\widetilde{X}_0)$ and transport this combination back to the original smooth surface as an algebraic divisor. This yields an explicit, finite-step procedure that realizes any rational $(1,1)$-class by an algebraic cycle. We also formulate an equivariant extension for $(2,2)$-classes on Calabi--Yau threefolds, indicating how the same strategy might apply in higher dimension.

math.AG↗

arXiv2025

Weak Fan Structures for Two-Parameter Degenerations of K3 Surfaces

In this note, we provide an explicit computation of the weak fan associated with a two-parameter degeneration of K3 surfaces. This example serves as a concrete illustration of the general framework developed by Robles and Deng (2023) for the compactification of period maps via nilpotent orbits in the non-Hermitian case. We describe the associated nilpotent cones, examine their compatibility conditions, and construct the weak fan governing the degeneration behavior. This computation contributes to the broader understanding of boundary components of period domains and their relation to limiting mixed Hodge structures.

math.AG↗

arXiv2025

Constructive Degenerations and the Algebraicity of Limiting Hodge

We propose a novel constructive framework for approaching the Hodge Conjecture via explicit degenerations. Building on limiting mixed Hodge structures (LMHS), we formulate a criterion under which a rational class of type (p, p) on a smooth projective variety becomes algebraic in the limit of a semi-stable degeneration. We provide examples where vanishing cycles and monodromy explicitly generate new algebraic classes, and propose a general principle: every rational (p, p) class arises as the limit of algebraic cycles under controlled geometric degenerations. This viewpoint opens a new path toward an effective formulation of the Hodge conjecture.

math.AG↗