arXiv subjects

Rodolfo Aguilar

Publications and source records attributed to Rodolfo Aguilar.

8 recordsLinked to original sources

arXiv2026

Log Conifold Transitions

We define log conifold transitions for Fano threefold pairs of index two and study their deformation theory. Relying on the recent solution to the relative Clemens conjectures in this setting, we construct rational curves with normal bundle $\OO(-1)\oplus \OO(-1)$ by blowing up anchored points on the boundary divisor. Contracting these curves yields a singular space with ordinary double points. We prove that local smoothings of the nodes can be lifted to global first-order deformations, and that the global deformation theory of both the log resolution space and the singular log pair is unconditionally unobstructed. Crucially, the geometry of the boundary del Pezzo surface guarantees this unobstructedness. Furthermore, unlike the classical Calabi-Yau case, the underlying Fano geometry forces the vanishing of global topological balancing conditions, allowing local first-order smoothings of the nodes to be lifted independently. As applications, we construct new non-K\"ahler threefolds via smoothings, we analyze the effective geometry of the smoothed threefolds by determining their Picard groups and proving the persistence of free curves. Finally, we study the Hodge theory of these non-K\"ahler threefolds.

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arXiv2026

A Real Shafarevich Conjecture for Universal Covers

The classical Shafarevich conjecture predicts that the universal cover of a complex smooth projective variety $X$ is holomorphically convex. In this paper, we propose a refinement of this conjecture for varieties defined over the reals. In order to do this, we introduce the notions of real holomorphic convexity and transverse holomorphic convexity to capture the geometric differences dictated by the real locus $X(\mathbb{R})$ of $X$. Specifically, we conjecture that the universal cover is real holomorphically convex when $X(\mathbb{R}) \neq \emptyset$, and dianalytic holomorphically convex when $X(\mathbb{R}) = \emptyset$. We prove this refined conjecture in two main cases: when $X$ is a curve, and when the fundamental group of $X$ is nilpotent.

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arXiv2026

Infinitesimal invariants of mixed Hodge structures II: Log Clemens conjecture and log connectivity

Following previous work, we continue the study of infinitesimal methods in mixed Hodge theory. In the first part, inspired by the deformation theory of curves on Calabi-Yau threefolds, we study deformations of smooth $\mathbb{Q}$-log Calabi-Yau pairs $(X,Y)$. We prove unobstructedness results for these pairs under Fano hypotheses. We define families of infinitesimal Abel-Jacobi maps associated with these deformation problems and show that they control the first-order deformations of smooth curves embedded in the pair. Crucially, for the $\frac{1}{2}$-log Calabi-Yau case, we establish an exact duality between deformations and obstructions, recovering the symmetry found in the absolute Calabi-Yau setting. We apply this framework to the cubic threefold, proposing a relative generalization of the Clemens conjecture regarding the injectivity of the infinitesimal Abel-Jacobi map, and establishing a criterion for its non-vanishing. In the second part, we define infinitesimal invariants for normal functions using extension classes and the log-Leray filtration. Relying on the theory of generalized Jacobian rings developed by Asakura and Saito, we prove a logarithmic Nori connectivity theorem for the universal family of open hypersurfaces, we also deduce a sharp algebraic criterion for the properness of the Hodge loci for open hypersurfaces, generalizing the proof of Carlson-Green-Griffiths-Harris.

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arXiv2026

The relative Clemens Conjectures for $\frac{1}{2}$-log Calabi-Yau threefolds

We formulate a relative analogue of the Clemens conjectures for 1/2-log Calabi-Yau threefold pairs (X,Y) (where K_X+2Y is isomorphic to O_X). This framework rests on the restoration of a perfect deformation/obstruction duality specific to the 1/2-log CY threefold setting. Based on this duality, we conjecture that for a generic intersection configuration on the boundary divisor Y, the number of rational curves anchored to these points is finite, and every such curve possesses the balanced relative normal bundle N_{C/X}(-Y) isomorphic to O_C(-1) + O_C(-1). In a joint appendix with Adrian Zahariuc, we verify this framework for prime Fano threefolds of index two. Using specialization techniques, we demonstrate that the usual virtual complications of relative Gromov-Witten theory are naturally suppressed in this setting. This trivialization of the relative moduli space cleanly reduces the virtual invariants to honest, classical enumerative counts, thereby rigorously proving the geometric conjecture.

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arXiv2025

Lagrangian interpretation of Abel-Jacobi mappings associated to Fano threefolds

Using the general framework due to Donagi-Markman \cite{DM} and Markushevich \cite{M} we shall derive an expression for the differential of Abel-Jacobi mappings on Fano threefolds. This formula involves information normal to the Lagrangian submanifolds constructed in \cite{DM} and \cite{M}. It may be applied to give new proofs of a number of classical results about these varieties.

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arXiv2024

Infinitesimal invariants of mixed Hodge structures

We introduce the notion of infinitesimal variations of mixed Hodge structures and invariants associated to them. We describe these invariants in the case of a pair $(X,Y)$ with $X$ a Fano 3-fold and $Y$ a smooth anticanonical K3 surface and in more detail in the case when $X$ is a cubic threefold. In this last setting, we obtain a generic global Torelli theorem for pairs.

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arXiv2021

The fundamental group of quotients of products of some topological spaces by a finite group -- A generalization of a Theorem of Bauer-Catanese-Grunewald-Pignatelli

We provide a description of the fundamental group of the quotient of a product of topological spaces $X_i$, each admitting a universal cover, by a finite group $G$, provided that there is only a finite number of path-connected components in $X_i^g$ for every $g\in G$. This generalizes previous work of Bauer-Catanese-Grunewald-Pignatelli and Dedieu-Perroni.

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arXiv2019

The fundamental group of partial compactifications of the complement of a real line arrangement

Let $\mathscr{A}$ be a real projective line arrangement and $M(\mathscr{A})$ its complement in $\mathbb{CP}^2$. We obtain an explicit expression in terms of Randell's generators of the meridians around the exceptional divisors in the blow-up $\bar{X}$ of $\mathbb{CP}^2$ in the singular points of $\mathscr{A}$. We use this to investigate the partial compactifications of $M(\mathscr{A})$ contained in $\bar{X}$ and give a counterexample to a statement suggested by A. Dimca and P. Eyssidieux to the effect that the fundamental group of such an algebraic variety is finite whenever its abelianization is.

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