arXiv subjects

Lino Amorim

Publications and source records attributed to Lino Amorim.

11 recordsLinked to original sources

arXiv2025

On the Morita invariance of Categorical Enumerative Invariants

Categorical Enumerative Invariants (CEI) are invariants associated with a unital, cyclic, smooth $A_\infty$-category and a splitting of its non-commutative Hodge filtration. In this paper, we extend the definition of CEI to Calabi-Yau $A_\infty$-categories with a splitting. Moreover, we formulate and prove the Morita invariance of CEI. As part of our proof, we develop tools to construct unital and cyclic models for Calabi-Yau categories. In particular, we prove a unital version of Kontsevich-Soibelman's Darboux theorem. As an application, we compute CEI in some new examples. Also, when applied to derived categories of coherent sheaves, our results yield new invariants of smooth, proper Calabi-Yau 3-folds.

math.AG↗

arXiv2024

Big Quantum cohomology of orbifold spheres

We construct a Kodaira-Spencer map from the big quantum cohomology of a sphere with three orbifold points to the Jacobian ring of the mirror Landau-Ginzburg potential function. This is constructed via the Lagrangian Floer theory of the Seidel Lagrangian and we show that Kodaira-Spencer map is a ring isomorphism.

math.SG↗

arXiv2022

Perversely categorified Lagrangian correspondences

In this article, we construct a $2$-category of Lagrangians in a fixed shifted symplectic derived stack S. The objects and morphisms are all given by Lagrangians living on various fiber products. A special case of this gives a $2$-category of $n$-shifted symplectic derived stacks $Symp^n$. This is a $2$-category version of Weinstein's symplectic category in the setting of derived symplectic geometry. We introduce another $2$-category $Symp^{or}$ of $0$-shifted symplectic derived stacks where the objects and morphisms in $Symp^0$ are enhanced with orientation data. Using this, we define a partially linearized $2$-category $LSymp$. Joyce and his collaborators defined a certain perverse sheaf on any oriented $(-1)$-shifted symplectic derived stack. In $LSymp$, the $2$-morphisms in $Symp^{or}$ are replaced by the hypercohomology of the perverse sheaf assigned to the $(-1)$-shifted symplectic derived Lagrangian intersections. To define the compositions in $LSymp$ we use a conjecture by Joyce, that Lagrangians in $(-1)$-shifted symplectic stacks define canonical elements in the hypercohomology of the perverse sheaf over the Lagrangian. We refine and expand his conjecture and use it to construct $LSymp$ and a $2$-functor from $Symp^{or}$ to $LSymp$. We prove Joyce's conjecture in the most general local model. Finally, we define a $2$-category of $d$-oriented derived stacks and fillings. Taking mapping stacks into a $n$-shifted symplectic stack defines a $2$-functor from this category to $Symp^{n-d}$.

math.SG↗

arXiv2022

The inverse function theorem for curved L-infinity spaces

In this paper we prove an inverse function theorem in derived differential geometry. More concretely, we show that a morphism of curved $L_\infty$ spaces which is a quasi-isomorphism at a point has a local homotopy inverse. This theorem simultaneously generalizes the inverse function theorem for smooth manifolds and the Whitehead theorem for $L_\infty$ algebras. The main ingredients are the obstruction theory for $L_\infty$ homomorphisms (in the curved setting) and the homotopy transfer theorem for curved $L_\infty$ algebras. Both techniques work in the $A_\infty$ case as well.

math.DG↗

arXiv2022

Ungraded matrix factorizations as mirrors of non-orientable Lagrangians

We introduce the notion of ungraded matrix factorization as a mirror of non-orientable Lagrangian submanifolds. An ungraded matrix factorization of a polynomial $W$, with coefficients in a field of characteristic 2, is a square matrix $Q$ of polynomial entries satisfying $Q^2 = W \cdot \mathrm{Id}$. We then show that non-orientable Lagrangians correspond to ungraded matrix factorizations under the localized mirror functor and illustrate this construction in a few instances. Our main example is the Lagrangian submanifold $\mathbb{R}P^2 \subset \mathbb{C}P^2$ and its mirror ungraded matrix factorization, which we construct and study. In particular, we prove a version of Homological Mirror Symmetry in this setting.

math.SG↗

arXiv2022

Categorical primitive forms of Calabi-Yau $A_\infty$-categories with semi-simple cohomology

We study categorical primitive forms for Calabi--Yau $A_\infty$ categories with semi-simple Hochschild cohomology. We classify these primitive forms in terms of certain grading operators on the Hochschild homology. We use this result to prove that, if the Fukaya category ${\sf Fuk}(M)$ of a symplectic manifold $M$ has semi-simple Hochschild cohomology, then its genus zero Gromov--Witten invariants may be recovered from the $A_\infty$-category ${\sf Fuk}(M)$ together with the closed-open map. An immediate corollary of this is that in the semi-simple case, homological mirror symmetry implies enumerative mirror symmetry.

math.SG↗

arXiv2017

Exact Lagrangian submanifolds, Lagrangian spectral invariants and Aubry-Mather theory

We construct graph selectors for compact exact Lagrangians in the cotangent bundle of an orientable, closed manifold. The construction combines Lagrangian spectral invariants developed by Oh and results by Abouzaid about the Fukaya category of a cotangent bundle. We also introduce the notion of Lipschitz-exact Lagrangians and prove that these admit an appropriate generalization of graph selector. We then, following Bernard-Oliveira dos Santos, use these results to give a new characterization of the Aubry and Mane sets of a Tonelli Hamiltonian and to generalize a result of Arnaud on Lagrangians invariant under the flow of such Hamiltonians.

math.SG↗

arXiv2017

The Künneth theorem for the Fukaya algebra of a product of Lagrangians

Given a compact Lagrangian submanifold $L$ of a symplectic manifold $(M,ω)$, Fukaya, Oh, Ohta and Ono construct a filtered $A_\infty$-algebra $\mathcal{F}(L)$, on the cohomology of $L$, which we call the Fukaya algebra of $L$. In this paper we describe the Fukaya algebra of a product of two Lagrangians submanifolds $L_1\times L_2$. Namely, we show that $\mathcal{F}(L_1\times L_2)$ is quasi-isomorphic to $\mathcal{F}(L_1)\otimes_\infty \mathcal{F}(L_2)$, where $\otimes_\infty$ is the tensor product of filtered $A_\infty$-algebras defined in arXiv:1404.7184. As a corollary of this quasi-isomorphism we obtain a description of the bounding cochains on $\mathcal{F}(L_1\times L_2)$ and of the Floer cohomology of $L_1\times L_2$.

math.SG↗

arXiv2016

Tensor product of filtered $A_\infty$-algebras

We define the tensor product of filtered $A_\infty$-algebras. establish some of its properties and give a partial description of the space of bounding cochains in the tensor product. Furthermore we show that in the case of classical $A_\infty$-algebras our definition recovers the one given by Markl and Shnider. We also give a criterion that implies that a given $A_\infty$-algebra is quasi-isomorphic to the tensor product of two subalgebras. This will be used in a sequel to prove a Künneth Theorem for the Fukaya algebra of a product of Lagrangian submanifolds.

math.SG↗

arXiv2015

Tensor product of cyclic A$_\infty$-algebras and their Kontsevich classes

Given two cyclic A$_\infty$-algebras $A$ and $B$, we prove that there exists a cyclic A$_\infty$-algebra structure on their tensor product $A\otimes B$ which is unique up to a cyclic A$_\infty$-quasi-isomorphism. Furthermore, the Kontsevich class of $A\otimes B$ is equal to the cup product of the Kontsevich classes of $A$ and $B$ on the moduli space of curves.

math.QA↗

arXiv2011

Floer cohomology of torus fibers and real lagrangians in Fano toric manifolds

In this article, we consider the Floer cohomology (with $\Z_2$ coefficients) between torus fibers and the real Lagrangian in Fano toric manifolds. We first investigate the conditions under which the Floer cohomology is defined, and then develop a combinatorial description of the Floer complex based on the polytope of the toric manifold. We show that if the Floer cohomology is defined, and the Floer cohomology of the torus fiber is non-zero, then the Floer cohomology of the pair is non-zero. We use this result to develop some applications to non-displaceability and the minimum number of intersection points under Hamiltonian isotopy.

math.SG↗