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Jeff Hicks

Publications and source records attributed to Jeff Hicks.

15 recordsLinked to original sources
arXiv2026

Monodromy action of mirror stops for toric Calabi-Yau surfaces

Mirror symmetry predicts an action by the fundamental group of a conjectural stringy K\"ahler moduli space on the derived category of an algebraic variety. For a toric variety, a model for this space is understood, but constructing the action is still an open problem in general. We propose that this action can be studied on the $A$-side via a moduli space of Legendrians isotopic to the FLTZ Legendrian. For the $A_{n-1}$ singularity, we construct an annular braid-group action on the corresponding partially wrapped Fukaya category by exact autoequivalences. The standard braid subgroup recovers the Seidel--Thomas action on the derived category, while the additional annular generator corresponds to tensor product with $\mathcal O(-1)$. We additionally extend the Floer-theoretic approach to homological mirror symmetry for toric varieties to the setting of semiprojective toric Deligne--Mumford stacks over an arbitrary field.

math.SG↗
arXiv2025

Rigidity and Realizability for Tropical Curves in Dimension 3

We present an unobstructedness criterion for Lagrangian threefolds $L\subset X^A$ using the $H_1(L)$-class associated with the boundary of a pseudoholomorphic disk. As an application, let $X^A\to Q$ be a Lagrangian torus fibration whose base $Q$ is a tropical abelian threefold. Given $V\subset Q$ a rigid tropical curve with a pair-of-pants decomposition, we prove that the Lagrangian lift $L_V\subset X^A$ is unobstructed. Provided that an appropriate homological mirror symmetry statement holds, this implies the existence of a realization $Y_V$ in the mirror abelian threefold $X^B\to Q$.

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arXiv2024

Homological mirror symmetry for Batyrev mirror pairs

We prove Kontsevich's homological mirror symmetry conjecture for a large class of mirror pairs of Calabi--Yau hypersurfaces in toric varieties. These mirror pairs were constructed by Batyrev from dual reflexive polytopes. The theorem holds in characteristic zero and in all but finitely many positive characteristics.

math.SG↗
arXiv2023

Integrality of mirror maps and arithmetic homological mirror symmetry for Greene--Plesser mirrors

We prove the `integrality of Taylor coefficients of mirror maps' conjecture for Greene--Plesser mirror pairs as a natural byproduct of an arithmetic refinement of homological mirror symmetry. We also prove homological mirror symmetry for Greene--Plesser mirror pairs in all characteristics such that the B-side family has good reduction, generalizing work of the fifth author and Smith over the complex numbers. A key technical ingredient is a new versality argument which allows us to work throughout over a Novikov-type ring with integer coefficients.

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arXiv2023

A short computation of the Rouquier dimension for a cycle of projective lines

Given a dg category $\mathcal C$, we introduce a new class of objects (weakly product bimodules) in $\mathcal C^{op}\otimes \mathcal C$ generalizing product bimodules. We show that the minimal generation time of the diagonal by weakly product bimodules provides an upper bound for the Rouquier dimension of $\mathcal C$. As an application, we give a purely algebro-geometric proof of a result of Burban and Drozd that the Rouquier dimension of the derived category of coherent sheaves on an $n$-cycle of projective lines is one. Our approach explicitly gives the generator realizing the minimal generation time.

math.AG↗
arXiv2023

Relating categorical dimensions in topology and symplectic geometry

We study several notions of dimension for (pre-)triangulated categories naturally arising from topology and symplectic geometry. We prove new bounds on these dimensions and raise several questions for further investigation. For instance, we relate the Rouquier dimension of the wrapped Fukaya category of either the cotangent bundle of a smooth manifold $M$ or more generally a Weinstein domain $X$ to quantities of geometric interest. These quantities include the minimum number of critical values of a Morse function on $M$, the Lusternik-Schnirelmann category of $M$, the number of distinct action values of a Hamiltonian diffeomorphism of $X$, and the smallest $n$ such that $X$ admits a Weinstein embedding into $\mathbb{R}^{2n+1}$. Along the way, we introduce a notion of the Lusternik-Schnirelmann category for dg-categories and construct exact Lagrangian cobordisms for restriction to a Liouville subdomain.

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arXiv2023

Reverse isoperimetric inequalities for Lagrangian intersection Floer theory

We extend Groman and Solomon's reverse isoperimetric inequality to pseudoholomorphic curves with punctures at the boundary and whose boundary components lie in a collection of Lagrangian submanifolds with intersections locally modelled on $\mathbb{R}^n\cap (\mathbb{R}^{k}\times \sqrt{-1}\mathbb{R}^{n-k})$ inside $\mathbb{C}^n$. Our construction closely follows the methods used by Duval (2016) and Abouzaid (2021) and corrects an error appearing in the latter approach.

math.CV↗
arXiv2023

Resolutions of toric subvarieties by line bundles and applications

Given any toric subvariety $Y$ of a smooth toric variety $X$ of codimension $k$, we construct a length $k$ resolution of $\mathcal O_Y$ by line bundles on $X$. Furthermore, these line bundles can all be chosen to be direct summands of the pushforward of $\mathcal O_X$ under the map of toric Frobenius. The resolutions are built from a stratification of a real torus that was introduced by Bondal and plays a role in homological mirror symmetry. As a corollary, we obtain a virtual analogue of Hilbert's syzygy theorem for smooth projective toric varieties conjectured by Berkesch, Erman, and Smith. Additionally, we prove that the Rouquier dimension of the bounded derived category of coherent sheaves on a toric variety is equal to the dimension of the variety, settling a conjecture of Orlov for these examples. We also prove Bondal's claim that the pushforward of the structure sheaf under toric Frobenius generates the derived category of a smooth toric variety and formulate a refinement of Uehara's conjecture that this remains true for arbitrary line bundles.

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arXiv2022

Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds

We say that a tropical subvariety $V\subset \mathbb R^n$ is $B$-realizable if it can be lifted to an analytic subset of $(\Lambda^*)^n$. When $V$ is a smooth curve or hypersurface, there always exists a Lagrangian submanifold lift $L_V\subset (\mathbb C^*)^n$. We prove that whenever $L_V$ has well-defined Floer cohomology, we can find for each point of $V$ a Lagrangian torus brane whose Lagrangian intersection Floer cohomology with $L_V$ is non-vanishing. Assuming an appropriate homological mirror symmetry result holds for toric varieties, it follows that whenever $L_V$ is a Lagrangian submanifold that can be made unobstructed by a bounding cochain, the tropical subvariety $V$ is $B$-realizable. As an application, we show that the Lagrangian lift of a genus zero tropical curve is unobstructed, thereby giving a purely symplectic argument for Nishinou and Siebert's proof that genus-zero tropical curves are $B$-realizable. We also prove that tropical curves inside tropical abelian surfaces are $B$-realizable.

math.SG↗
arXiv2021

Lagrangian cobordisms and Lagrangian surgery

Lagrangian $k$-surgery modifies an immersed Lagrangian submanifold by topological $k$-surgery while removing a self-intersection. Associated to a $k$-surgery is a Lagrangian surgery trace cobordism. We prove that every Lagrangian cobordism is exactly homotopic to a concatenation of suspension cobordisms and Lagrangian surgery traces. This exact homotopy can be chosen with as small Hofer norm as desired. Furthermore, we show that each Lagrangian surgery trace bounds a holomorphic teardrop pairing the Morse cochain associated with the handle attachment to the Floer cochain generated by the self-intersection. We give a sample computation for how these decompositions can be used to algorithmically construct bounding cochains for Lagrangian submanifolds. In an appendix, we describe a 2-ended embedded monotone Lagrangian cobordism which is not the suspension of a Hamiltonian isotopy following a suggestion of Abouzaid and Auroux.

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arXiv2020

Aspects of functoriality in homological mirror symmetry for toric varieties

We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety. In particular, we realize tropical Lagrangian sections as objects of a partially wrapped category and construct a Lagrangian correspondence mirror to the inclusion of a toric divisor. As a corollary, we prove that tropical sections generate the Fukaya-Seidel category, completing a Floer-theoretic proof of homological mirror symmetry for projective toric varieties. In the course of the proof, we develop techniques for constructing Lagrangian cobordisms and Lagrangian correspondences in Liouville domains, which may be of independent interest.

math.SG↗
arXiv2020

Tropical Lagrangians in toric del-Pezzo surfaces

We look at how one can construct from the data of a dimer model a Lagrangian submanifold in $(\mathbb{C}^*)^n$ whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori $L_{T^2}$ in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers. An example is worked out for the mirror pair $(\mathbb{CP}^2\setminus E, W), \check X_{9111}$. We find a symplectomorphism of $\mathbb{CP}^2\setminus E$ interchanging $L_{T^2}$ and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier-Mukai transform on $\check X_{9111}$.

math.SG↗
arXiv2019

Wall-crossing from Lagrangian Cobordisms

Biran and Cornea showed that monotone Lagrangian cobordisms give an equivalence of objects in the Fukaya category. However, there are currently no known non-trivial examples of monotone Lagrangian cobordisms with two ends. We look at an extension of their theory to the pearly model of Lagrangian Floer cohomology and unobstructed Lagrangian cobordisms. In particular, we examine the suspension cobordism of a Hamiltonian isotopy and the Haug mutation cobordism between mutant Lagrangian surfaces. In both cases, we show that these Lagrangian cobordisms can be unobstructed by bounding cochain and additionally induce an $A_\infty$ homomorphism between the Floer cohomology of the ends. This gives a first example of a two-ended Lagrangian cobordism giving a non-trivial equivalence of Lagrangian Floer cohomology. A brief computation is also included which shows that the incorporation of bounding cochain from this equivalence accounts for the "instanton-corrections" considered by Auroux, Ekholm, Pascaleff, Rizell, and Tonkonog for the wall-crossing formula between Chekanov and product tori in $(\mathbb C^2)\setminus \{z_1z_2=1\}$. We additionally prove some auxiliary results that may be of independent interest. These include a weakly filtered version of the Whitehead theorem for $A_\infty$ algebras and an extension of Charest-Woodward's stabilizing divisor model of Lagrangian Floer cohomology to Lagrangian cobordisms.

math.SG↗
arXiv2019

Tropical Lagrangian Hypersurfaces are Unobstructed

We produce for each tropical hypersurface $V(\phi)\subset Q=\mathbb{R}^n$ a Lagrangian $L(\phi)\subset (\mathbb{C}^*)^n$ whose moment map projection is a tropical amoeba of $V(\phi)$. When these Lagrangians are admissible in the Fukaya-Seidel category, we show that they are unobstructed objects of the Fukaya category, and mirror to sheaves supported on complex hypersurfaces in a toric mirror.

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