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Pedro Antonio Muniz Martins

Publications and source records attributed to Pedro Antonio Muniz Martins.

7 recordsLinked to original sources
arXiv2026

On the atomic decomposition of complete intersection in flag varieties

We develop a root-theoretic localization formalism for genus-zero Gromov--Witten invariants of flag varieties and of smooth zero loci of globally generated homogeneous vector bundles. The resulting invariants are expressed as finite sums over decorated trees whose contributions are determined by the root system of the ambient flag variety and by the torus weights of the defining bundle. We connect these computations with the Theory of Hodge Atoms of Katzarkov--Kontsevich--Pantev--Yu via the matrix of small quantum multiplication by first Chern class. Let $X$ be a Fano fourfold with $h^{3,1}(X)=1$ arising as a hyperplane section of a suitable Fano fivefold. Its cohomology decomposes into a monodromy-fixed part and the middle vanishing cohomology. This decomposition is preserved by quantum multiplication by the first Chern class. Moreover, it acts by a scalar on the vanishing summand. We show, for the monodromy-fixed block, that (a) if every eigenvalue has algebraic multiplicity at most two and $X$ is Hodge general, then $X$ is irrational; (b) if every eigenvalue has Jordan defect at most one and $X$ is rational, then every weak factorization contains a smooth surface center whose minimal model is a projective K3 surface. Many applications are presented. Using the recently introduced quantum connection approach to irrationality, we obtain from the computations here presented the irrationality of the very general smooth complete intersection of a hyperplane and a quadric in $\operatorname{LG}(3,6)$; the very general smooth quadric section in $G_2/P_2$; and the very general smooth complete intersection of five hyperplanes and a quadric in the spinor 10-fold.

math.AG↗
arXiv2026

On the generalized cohomologies of almost abelian solvmanifolds

Generalized complex structures give rise to generalized Dolbeault, Bott--Chern and Aeppli cohomologies. We study these cohomologies for invariant generalized complex structures on almost abelian Lie algebras $\mathfrak g_A=\mathbb R e_0\ltimes_A\mathfrak h$, where $\mathfrak h$ is an abelian ideal of codimension one, and on almost abelian solvmanifolds $Γ\backslash G_A$. We first show that the invariant generalized Dolbeault cohomology can be identified with the Chevalley--Eilenberg cohomology of the complex Lie algebra $\mathcal L$, the $+{\mathbf i}$-eigenspace of the structure, with coefficients in a character determined by the canonical line. We prove that $\mathcal L$ is itself an almost abelian Lie algebra; thus the cohomology reduces to the kernel and cokernel of an explicit derivation. Assuming that $A$ is diagonalizable over $\mathbb R$, we obtain a combinatorial formula for the cohomology dimensions and compare with compact quotients. Other comparison results are discussed, noting that the cohomology of invariant forms always injects into that of $Γ\backslash G_A$.

math.DG↗
arXiv2026

A quantum connection approach to rationality

We extract rank and regularized exponents from cyclic generic factors of the quantum connection and show that these data persist under the birational comparisons considered here. Applications include showing non-rationality of cubic threefolds, ordinary Gushel--Mukai threefolds, very general quartic fourfolds, and products of the threefolds with $\PP^1$.

math.AG↗
arXiv2026

On the generalized geometry of almost abelian solvmanifolds

We study left-invariant generalized complex structures on almost abelian Lie groups $G_A$ with Lie algebra $\mathfrak{g}_A=\mathbb{R} e_0\ltimes_A\mathfrak{h}$, where $\mathfrak{h}$ is an abelian ideal of codimension one, and on their compact quotients. First, when $A$ is diagonalizable over $\mathbb{R}$, we characterize all admissible types by pairings of its eigenvalues and characterize the structures admitting a closed invariant pure-spinor generator. For general $A$, we obtain Jordan-theoretic type bounds and a construction using a complex quotient and a symplectic ideal. Finally, in dimension six, we establish nonexistence results and give explicit intermediate-type constructions.

math.DG↗
arXiv2026

The Cohomology of Solvmanifold SYZ Mirrors

This paper investigates the geometric and cohomological properties of non-Kähler SYZ mirror symmetry for dual torus fibrations over solvmanifolds in the sense of Lau, Tseng and Yau. We are mainly concerned with three questions: \textbf{(a)} How the Lau-Tseng-Yau notion of non-Kähler SYZ is related to the mapping of supersymmetric branes between symplectic and complex sides; \textbf{(b)} Finding explicit non-Kähler SYZ mirror pairs determined purely by Lie-theoretic data; \textbf{(c)} better understand the cohomological correspondence in the Lau-Tseng-Yau framework (given by a Fourier-Mukai transform), especially concerning the role of Tseng-Yau cohomology. We prove that the Fourier-Mukai transform introduced by Lau-Tseng-Yau exchanges type-A supersymmetric cycles, which are given by special Lagrangian sections equipped with flat $\mathrm{U}(1)$ connections, with type-B cycles, corresponding to line bundles whose connections satisfy the deformed Hermitian-Yang-Mills (dHYM) equation. We provide pure Lie-theoretic criteria for the existence of non-Kähler SYZ mirror pairs whose base manifolds are solvmanifolds. Applying these criteria, we construct new explicit families of mirror pairs from almost abelian and generalized Heisenberg Lie groups, and provide a complete classification of such pairs arising from nilpotent Lie groups. To contextualize the role of the Tseng-Yau cohomology, we link it to noncommutative geometry. We introduce the Tseng-Yau and Bott-Chern mirror bicomplexes. We show that (some of) their enclosed cohomologies reduce to the primitive Tseng-Yau and Bott-Chern cohomologies and that for basic forms they are isomorphic under the Fourier-Mukai transform. As a last contribution, we discuss how to explicitly compute the Tseng-Yau and the Bott-Chern cohomology for the non-Kähler SYZ mirror pairs constructed here.

hep-th↗
arXiv2025

Separating symmetric polynomials over finite fields

The set $S(n)$ of all elementary symmetric polynomials in $n$ variables is a minimal generating set for the algebra of symmetric polynomials in $n$ variables, but over a finite field ${\mathbb F}_q$ the set $S(n)$ is not a minimal separating set for symmetric polynomials in general. We determined when $S(n)$ is a minimal separating set for the algebra of symmetric polynomials having the least possible number of elements.

math.AC↗