arXiv subjects

André Oliveira

Publications and source records attributed to André Oliveira.

At least 19 recordsLinked to original sources

arXiv2026

Zero velocity Lagrangians in the Hitchin system

Lagrangians of the moduli space of Higgs bundles are a fundamental piece of its study under the lens of mirror symmetry. We construct a new class of Lagrangians defined via a zero 'velocity condition' imposed on the natural action of the non-zero complex numbers on the moduli space, motivated by a certain 'infinitesimal conormal principle'. These are Lagrangian subvarieties analogous to the upward flows considered by [CW19] and [HH22], but which fiber over the $\mathbb{C}^*$-fixed point subvarieties. We then prove a formula for the number of intersection points of this Lagrangian with the generic fiber of the Hitchin map, and build a vector bundle over the dual moduli space which we conjecture to correspond to its mirror hyperholomorphic bundle.

math.AG↗

arXiv2026

On the Influence of Refactoring Types on Merge Effort

Modern software development involves parallel work and concurrent changes, requiring code merging. Prior studies report that 10% to 20% of merge attempts result in conflicts, often requiring manual intervention. The literature explores factors that generate conflicts, including refactorings, but does not analyze how individual refactoring types influence the manual effort required to resolve them. We analyzed 64 open-source Java projects and applied association rule mining to measure the strength of associations between specific refactoring types and merge effort. Our results show that refactoring types relate to merge effort with varying strength. In particular, Rename Attribute, Move Class, Extract Variable, Change Return Type, and Split Parameter exhibit some of the strongest associations, especially when a higher number of such refactorings is present in the merge branches. We also find that both the number of refactorings and their diversity independently increase merge effort, both in terms of occurrence and intensity. Additionally, the co-occurrence of refactorings across parallel branches is associated with higher merge effort, particularly when combining structural transformations with changes to method signatures and data-structure representations, whereas more localized changes are less frequent in the most impactful combinations.

cs.SE↗

arXiv2026

Fourier-Mukai transforms and normalisation of nodal curves

We study Arinkin's Poincaré sheaf $\mathcal{P}_C$ on the singular locus of $\overline{\mathsf{Jac}}_C$, the compactified Jacobian of rank one torsion-free sheaves on an integral nodal projective curve $C$. Each stratum of the singular locus $\mathsf{Sing}(\overline{\mathsf{Jac}}_C)$ is indexed by a partial normalisation $Σ\to C$. We prove that the Poincaré sheaf $\mathcal{P}_C$ restricted to each stratum can be expressed through the Poincaré sheaf $\mathcal{P}_Σ$, obtaining a relation between Fourier-Mukai transforms associated to $\mathcal{P}_C$ and $\mathcal{P}_Σ$. Our approach uses an intermediate geometry: the moduli space of parabolic modules of Bhosle and Cook, to intertwine sheaf data over the two curves. In a sequel, our formulae are used to study mirror symmetry in singular loci of Hitchin systems.

math.AG↗

arXiv2024

A Comprehensive Review and Taxonomy of Audio-Visual Synchronization Techniques for Realistic Speech Animation

In many applications, synchronizing audio with visuals is crucial, such as in creating graphic animations for films or games, translating movie audio into different languages, and developing metaverse applications. This review explores various methodologies for achieving realistic facial animations from audio inputs, highlighting generative and adaptive models. Addressing challenges like model training costs, dataset availability, and silent moment distributions in audio data, it presents innovative solutions to enhance performance and realism. The research also introduces a new taxonomy to categorize audio-visual synchronization methods based on logistical aspects, advancing the capabilities of virtual assistants, gaming, and interactive digital media.

eess.AS↗

arXiv2024

Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces

Let $\mathcal{L}=(L,[\cdot\,,\cdot],δ)$ be an algebraic Lie algebroid over a smooth projective curve $X$ of genus $g\geq 2$ such that $L$ is a line bundle whose degree is less than $2-2g$. Let $r$ and $d$ be coprime numbers. We prove that the motivic class of the moduli space of $\mathcal{L}$-connections of rank $r$ and degree $d$ over $X$ does not depend on the Lie algebroid structure $[\cdot\,,\cdot]$ and $δ$ of $\mathcal{L}$ and neither on the line bundle $L$ itself, but only on the degree of $L$ (and of course on $r$, $d$ and $X$). In particular it is equal to the motivic class of the moduli space of $K_X(D)$-twisted Higgs bundles of rank $r$ and degree $d$, for $D$ any effective divisor with the appropriate degree. As a consequence, similar results (actually slightly stronger) are obtained for the corresponding $E$-polynomials. Some applications of these results are then deduced.

math.AG↗

arXiv2024

A general Cayley correspondence and higher Teichmüller spaces

We introduce a new class of $\mathfrak{sl}_2$-triples in a complex simple Lie algebra $\mathfrak{g}$, which we call magical. Such an $\mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $\mathfrak{g}$. Using this decomposition data, we explicitly parameterize special connected components of the moduli space of Higgs bundles on a compact Riemann surface $X$ for an associated real Lie group, hence also of the corresponding character variety of representations of $π_1X$ in the associated real Lie group. This recovers known components when the real group is split, Hermitian of tube type, or $\mathrm{SO}_{p,q}$ with $1<p\leq q$, and also constructs previously unknown components for the quaternionic real forms of $\mathrm{E}_6$, $\mathrm{E}_7$, $\mathrm{E}_8$ and $\mathrm{F}_4$. The classification of magical $\mathfrak{sl}_2$-triples is shown to be in bijection with the set of $Θ$-positive structures in the sense of Guichard--Wienhard, thus the mentioned parameterization conjecturally detects all examples of higher Teichmüller spaces. Indeed, we discuss properties of the surface group representations obtained from these Higgs bundle components and their relation to $Θ$-positive Anosov representations, which indicate that this conjecture holds.

math.AG↗

arXiv2021

Principal bundles on 2-dimensional CW-complexes with disconnected structure group

Given any topological group $G$, the topological classification of principal $G$-bundles over a finite CW-complex $X$ is long-known to be given by the set of free homotopy classes of maps from $X$ to the corresponding classifying space $BG$. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when $X$ has dimension $2$, it seems there is a case in which such explicit classification has not been explicitly considered. This is the case where $G$ is a Lie group, whose group of components acts non-trivially on its fundamental group $π_1G$. In this note we deal with this case by obtaining the classification, in terms of characteristic classes, of principal $G$-bundles over a finite CW-complex of dimension $2$, with $G$ is a Lie group such that $π_0G$ is abelian.

math.AT↗

arXiv2020

Unramified covers and branes on the Hitchin system

We study the locus of the moduli space of Higgs bundles on a curve given by those Higgs bundles obtained by pushforward under an unramified cover. We equip these loci with a hyperholomorphic bundle so that they can be viewed as BBB-branes, and we introduce corresponding BAA-branes which can be described via Hecke modifications. We then show how these branes are naturally dual via explicit Fourier--Mukai transform, where we recall that the structure group $\mathrm{GL}(n,\mathbb{C})$ is Langlands self dual. It is noteworthy that these branes lie over the singular locus of the Hitchin fibration. As a particular case, our construction describes the behaviour under mirror symmetry of the fixed loci for the action of tensorization by a line bundle of order $n$. These loci play a key role in the work of Hausel and Thaddeus on topological mirror symmetry for Higgs moduli spaces.

math.AG↗

arXiv2020

Complex Lagrangians in a hyperKaehler manifold and the relative Albanese

Let $M$ be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold $X$, and let $\varpi : \widehat{\mathcal{A}} \rightarrow M$ be the relative Albanese over $M$. We prove that $\widehat{\mathcal{A}}$ has a natural holomorphic symplectic structure. The projection $\varpi$ defines a completely integrable structure on the symplectic manifold $\widehat{\mathcal{A}}$. In particular, the fibers of $\varpi$ are complex Lagrangians with respect to the symplectic form on $\widehat{\mathcal{A}}$. We also prove analogous results for the relative Picard over $M$.

math.DG↗

arXiv2020

From Byzantine Replication to Blockchain: Consensus is only the Beginning

The popularization of blockchains leads to a resurgence of interest in Byzantine Fault-Tolerant (BFT) state machine replication protocols. However, much of the work on this topic focuses on the underlying consensus protocols, with emphasis on their lack of scalability, leaving other subtle limitations unaddressed. These limitations are related to the effects of maintaining a durable blockchain instead of a write-ahead log and the requirement for reconfiguring the set of replicas in a decentralized way. We demonstrate these limitations using a digital coin blockchain application and BFT-SMaRt, a popular BFT replication library. We show how they can be addressed both at a conceptual level, in a protocol-agnostic way, and by implementing SMaRtChain, a blockchain platform based on BFT-SMaRt. SMaRtChain improves the performance of our digital coin application by a factor of eight when compared with a naive implementation on top of BFT-SMaRt. Moreover, SMaRtChain achieves a throughput $8\times$ and $33\times$ better than Tendermint and Hyperledger Fabric, respectively, when ensuring strong durability on its blockchain.

cs.DC↗

arXiv2019

Representations of surface groups in the projective general linear group

Given a closed, oriented surface X of genus g>1, and a semisimple Lie group G, let R_G be the moduli space of reductive representations of the fundamental group of X in G. We determine the number of connected components of R_PGL(n,R), for n>=4 even. In order to have a first division of connected components, we first classify real projective bundles over such a surface. Then we achieve our goal, using holomorphic methods through the theory of Higgs bundles over compact Riemann surfaces. We also show that the complement of the Hitchin component in R_SL(3,R) is homotopically equivalent to R_SO(3).

math.AG↗

arXiv2019

SO(p,q)-Higgs bundles and higher Teichmüller components

Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we describe new examples of such `exotic' components in moduli spaces of SO(p,q)-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into the Lie group SO(p,q). Furthermore, we discuss how these exotic components are related to the notion of positive Anosov representations recently developed by Guichard and Wienhard. We also provide a complete count of the connected components of these moduli spaces (except for SO(2,q), with q> 3).

math.AG↗

arXiv2018

Exotic components of $\mathrm{SO}(p,q)$ surface group representations, and their Higgs bundle avatars

For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups $\mathrm{SO}(p,q)$ with $2<p<q$. These groups lie outside formerly know classes of groups associated with exotic components.

math.AG↗

arXiv2017

Maximal Higgs bundles for adjoint forms via Cayley correspondence

For a fixed compact Riemann surface X, of genus at least 2, we count the number of connected components of the moduli space of maximal Higgs bundles over X for the hermitian groups $PSp(2n,R)$, $PSO^*(2n)$, $PSO_0(2,n)$ and $E_6^{-14}$. Hence the same result follows for the number of connected components of the moduli space of maximal representations of $π_1X$ in these groups. We use the Cayley correspondence proved in [3] as our main tool.

math.DG↗

arXiv2016

Quadric bundles and their moduli spaces

Quadric bundles on a compact Riemann surface X generalise orthogonal bundles and arise naturally in the study of the moduli space of representations of $π_1(X)$ in Sp(2n,R). We prove some basic results on the moduli spaces of quadric bundles over X of arbitrary rank and then survey deeper results in the rank 2 case.

math.AG↗

arXiv2016

Quadric bundles applied to non-maximal Higgs bundles

We present a survey on the moduli spaces of rank 2 quadric bundles over a compact Riemann surface X. These are objects which generalise orthogonal bundles and which naturally occur through the study of the connected components of the moduli spaces of Higgs bundles over X for the real symplectic group Sp(4,R), with non-maximal Toledo invariant. Hence they are also related with the moduli space of representations of $π_1(X)$ in Sp(4,R). We explain this motivation in some detail.

math.AG↗

arXiv2016

Connectedness of Higgs bundle moduli for complex reductive Lie groups

We carry an intrinsic approach to the study of the connectedness of the moduli space $\mathcal{M}_G$ of $G$-Higgs bundles, over a compact Riemann surface, when $G$ is a complex reductive (not necessarily connected) Lie group. We prove that the number of connected components of $\mathcal{M}_G$ is indexed by the corresponding topological invariants. In particular, this gives an alternative proof of the counting by J. Li of the number of connected components of the moduli space of flat $G$-connections in the case in which $G$ is connected and semisimple.

math.AG↗

arXiv2014

Homotopy type of free group character varieties

Let G be a real reductive algebraic group with maximal compact subgroup K, and let F be a rank r free group. Here, we summarize the construction of a natural strong deformation retraction from the space of closed orbits in Hom(F,G)/G to the orbit space Hom(F,K)/K. In particular, these spaces have the same homotopy type.

math.AT↗