arXiv subjects

Álvaro Muñiz-Brea

Publications and source records attributed to Álvaro Muñiz-Brea.

3 recordsLinked to original sources

arXiv2026

Fourier transforms and a filtration on the Lagrangian cobordism group of tori

Given a polarized tropical affine torus, we show that the fibered Lagrangian cobordism group of the corresponding symplectic manifold admits a natural geometric filtration of finite length. This contrasts with results of Sheridan-Smith in dimension four and the present author in higher dimensions, who showed that such group is infinite-dimensional. In the second half of this paper, we construct a Fourier transform between Fukaya categories of dual symplectic tori. We show that, under homological mirror symmetry, it corresponds to the Fourier transform between derived categories of coherent sheaves of dual abelian varieties due to Mukai. We use this to show how our filtration is mirror to the Bloch filtration on Chow groups of abelian varieties, but the results may be of broader interest.

math.SG↗

arXiv2026

Dimensionality of tropical Chow groups

We show that the existence of non-zero tropical forms of degree at least two implies that the tropical Chow group of points of a tropical affine manifold is infinite-dimensional. This can be seen as a tropical analog of classical results of Mumford and Roitman for Chow groups of smooth (complex) projective algebraic varieties. We also show that the existence of tropical 1-forms on tropical surfaces does not imply infinite dimensionality by considering the case of a tropical Klein bottle.

math.AG↗

arXiv2024

Lagrangian cobordisms and K-theory of symplectic bielliptic surfaces

We consider a family of closed symplectic manifolds 4-manifolds which we call symplectic bielliptic surfaces and study its Lagrangian cobordism group of weakly-exact Lagrangian G-branes (that is, Lagrangians equipped with a grading, a Pin structure and a G-local system); relations come from Lagrangian cobordisms satisfying a tautologically unobstructedness-type condition, also equipped with G-brane structures. Our first theorem computes its subgroup generated by tropical Lagrangians. When G is the unitary group of the Novikov field, we use homological mirror symmetry to compute the Grothendieck group of the Fukaya category and show it agrees with our computation for the cobordism group. This leads us to conjecture that tropical Lagrangians generate the whole cobordism group.

math.SG↗