arXiv subjects

Mauricio Corrêa

Publications and source records attributed to Mauricio Corrêa.

10 recordsLinked to original sources

arXiv2026

Chern-Ricci flow on Kato surfaces

Let $S$ be a Kato surface and $D$ its maximal reduced divisor of rational curves. On $S\setminus D$ we construct Hermitian metrics which are flat along the leaves of the canonical foliation and study their evolution under the Chern-Ricci flow. For Enoki surfaces, we construct an immortal normalised solution which, on every compact sublevel of the natural exhaustion, converges in the Gromov-Hausdorff sense to the elliptic base endowed with an explicit flat metric. For Kato surfaces of intermediate type, the affine Green model and its finite-index extension give, under the assumption $0<μ<2$, an explicit normalised solution which, on every compact Green block, collapses in the Gromov-Hausdorff sense to a circle. In the Enoki case the limiting area is $2πb_2(S)$; in the intermediate case the length of the limiting circle is determined by the Green and leafwise monodromies.

math.DG↗

arXiv2026

Shifted Poisson unfoldings and quantum anomalies

Families of classical field theories often depend on geometric parameters. A basic question is whether the corresponding classical observables can be compared by flat parallel transport, and whether this comparison survives quantization. We study this problem for families of shifted Poisson structures. To such a family we attach a Poisson transverse controller $\mathbb U_π$, a homotopy stabilizer encoding transverse symmetries which preserve the Poisson Maurer--Cartan element up to coherent homotopy. Its flat splittings are precisely transversal shifted Poisson unfoldings, and they act on vertical symmetries, Poisson cohomology and local deformation theory. We then formulate the $\hslash$-adic lifting problem for quantized objects; its degree-two obstruction classes are the transport anomalies. The construction is realized for star-products, BV observables, factorization algebras and AKSZ theories. For the Poisson sigma model, anomaly-free quantization of the flat transport makes the Cattaneo--Felder/Kontsevich boundary product horizontal over the parameter space.

math.AG↗

arXiv2026

A categorical and algebro-geometric theory of localization

We develop a categorical and algebro-geometric theory of localisation for cohomological theories with open--closed recollement. A class whose restriction to the open complement vanishes need not determine a preferred class on the closed stratum; the localisation triangle associates with it instead a torsor of supported refinements, whose secondary indeterminacy is governed by the connecting morphism from the open complement. We prove compatibility with excision, base change, proper pushforward, external products and local indices, and show that compatible supported constructions factor through this torsor. Under explicit Gysin hypotheses, injectivity of Euler multiplication gives a pre-Euler canonicity criterion, making the supported refinement unique before any coefficient localisation. Purity, concentration and Euler rigidification recover the usual Euler-denominator formulae. We also relate the secondary boundary group to link transgression, treat equivariant algebraic K-theory as a multiplicative analogue, and introduce Milnor localisation torsors for characteristic-class defects of singularities.

math.AG↗

arXiv2026

Compactified supermoduli space is almost never projected

We settle the projectedness problem for the compactified unpunctured supermoduli stack in every genus at least two. In genus two, the odd component is split, whereas the even component is non-projected. In every genus $g\geq 3$, both compactified parity components are non-projected.

math.AG↗

arXiv2026

Formal moduli and the splitting theory of supermanifolds

We develop a formal moduli theory for the splitting problem of complex supermanifolds. Starting from Green's obstruction tower, we construct a finite-step filtered dg Lie algebra which controls splittings by filtered Maurer-Cartan theory. We prove that the classical successive obstruction classes are recovered as the leading terms of adapted Maurer-Cartan representatives, and we transfer the theory to a minimal filtered $L_\infty$-model whose higher brackets give the intrinsic Kuranishi relations among the obstruction coordinates. We also prove that, in a precise filtered sense, the affine Atiyah class contains the entire Green obstruction tower: the Donagi-Witten component gives the primary obstruction, while the higher obstructions arise as successive projected defects and residual classes of the same Atiyah cocycle. We then pass to families, by constructing the fixed-retract formal moduli problem with prescribed split model and by identifying the formal neighbourhood of the split section with the fibrewise deformation theory of the split model; under standard finiteness, base-change, and descent hypotheses this yields relative tangent-obstruction and Kuranishi presentations. Finally, we work out explicit examples of non-split supergeometries, including cases with residual higher obstructions and a first nonlinear Kuranishi relation. These examples illustrate the interaction between Green obstructions, Atiyah classes, and higher $L_\infty$-operations.

math.AG↗

arXiv2026

The logarithmic leaf complex and foliated d-semistability

We study holomorphic foliations on normal crossings varieties arising as semistable degenerations. We do so by we exploring the notion of foliated d-semistability using the language of logarithmic structures in the sense of Fontaine-Illusie. First, we identify both local and global obstructions to d-semistability. In order to analyze the existence of smoothings, we develop a logarithmic deformation theory of foliations and show that the corresponding moduli functor admits a versal hull.

math.AG↗

arXiv2026

A Čech--Stokes Pushout Groupoid: a Log/Kummer Betti Presenter for Stokes Torsors

We give an explicit Betti presentation of the Stokes torsors attached to meromorphic flat connections of prescribed irregular type along a simple normal crossings divisor, at fixed Kummer level. Our construction is strictly 1-categorical and cover-based: on a punctured logarithmic collar of the divisor, we define a small Cech--Stokes groupoid and prove that the boundary Stokes moduli is described by sections of a natural forgetful functor, rather than by representations of the decorated boundary groupoid itself. By gluing this boundary model to the ordinary Cech presenter of the complement through an explicit pushout construction, we obtain a groupoid presentation that computes the global Stokes objects. The resulting presentation is canonical up to Morita equivalence, compatible with Kummer descent along all normal crossings strata, and admits an explicit finite local description near corners in terms of nonabelian cocycles and relations.

math.AG↗

arXiv2023

Enumerative geometry of Legendrian foliations: a Tale of Contact

A contact distribution on projective three-space is defined by the 1-form $x_2dx_1-x_1dx_2+x_4dx_3-x_3dx_4$, up to a change of projective coordinates. The family of contact distributions is parameterized by the complement of the Pfaff-Plücker quadric in the projective 5-space of antisymmetric $4\times4$ matrices. A foliation of dimension 1 and degree $d$ is specified by a polynomial vector field $\sum{}p_i\partial_{x_i}, p_i$ homogeneous of degree $d$. The foliation is called Legendrian if tangent to some distribution of contact. Our goal is to give formulas for the dimensions and degrees of the varieties of Legendrian foliations, and of the varieties of foliations tangent to a pencil of planes.

math.AG↗

arXiv2018

On holomorphic distributions on Fano threefolds

This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds threedimensional, with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on smooth weighted projective complete intersection Fano manifolds threefold. We also prove that a codimension one locally free distribution with trivial canonical bundle on any Fano threefold, with Picard number equal to one, has a tangent sheaf which either splits or it is stable.

math.AG↗

arXiv2018

Brunella-Khanedani-Suwa variational residues for invariant currents

In this work we prove a Brunella-Khanedani-Suwa variational type residue theorem for currents invariant by holomorphic foliations. As a consequence, we give conditions for the leaves of a singular holomorphic foliation to accumulate in the intersection of the singular set of the foliation with the support of an invariant current.

math.CV↗