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Nick Sheridan

Publications and source records attributed to Nick Sheridan.

At least 19 recordsLinked to original sources
arXiv2025

The cyclic open--closed map and variations of Hodge structures

We construct the cyclic open--closed map for the big (i.e., bulk-deformed) relative Fukaya category, in the semipositive case, and show that it is a morphism of `polarized variations of semi-infinite Hodge structures'. We also give a natural criterion for the map to be an isomorphism, which is verified for example in the context of Batyrev mirror pairs. We conclude in such Calabi-Yau cases that the rational Gromov--Witten invariants can be extracted from the relative Fukaya category, and hence that enumerative mirror symmetry is a consequence of homological mirror symmetry for Calabi--Yau mirror pairs.

math.AG↗
arXiv2025

Constructing the big relative Fukaya category, and its open--closed maps

This paper continues previous work of the author with Perutz, in which the `small' version of Seidel's relative Fukaya category of a smooth complex projective variety relative to a normal crossings divisor was constructed, under a semipositivity assumption. In the present work, we generalize this to construct the `big' relative Fukaya category in the same setting, as well as its closed--open and open--closed maps, and prove Abouzaid's split-generation criterion in this context. We also establish a general framework for constructing chain-level Floer-theoretic operations in this context, and dealing efficiently with signs and bounding cochains, which will be used in follow-up work with Ganatra to construct the cyclic open--closed map and establish its properties.

math.SG↗
arXiv2024

On the positivity and integrality of coefficients of mirror maps

We present natural conjectural generalizations of the `positivity and integrality of mirror maps' phenomenon, encompassing the mirror maps appearing in the Batyrev--Borisov construction of mirror Calabi--Yau complete intersections in Fano toric varieties as a special case. We find that, given the combinatorial data from which one constructs a mirror pair of Calabi--Yau complete intersections, there are two ways of writing down an associated `mirror map': one which is the `true mirror map', meaning the one which appears in mirror symmetry theorems; and one which is the `naive mirror map'. The two are equal under a certain combinatorial criterion which holds e.g. for the quintic threefold, but not in general. We conjecture (based on substantial computer checks, together with proofs under extra hypotheses) that the naive mirror map always has positive integer coefficients, while the true mirror map always has integer (but not necessarily positive) coefficients. Most previous works on the integrality of mirror maps concern the naive mirror map, and in particular, only apply to the true mirror map under the combinatorial criterion mentioned above.

math.NT↗
arXiv2024

An $L_\infty$ structure on symplectic cohomology

We construct the $L_\infty$ structure on symplectic cohomology of a Liouville domain, together with an enhancement of the closed--open map to an $L_\infty$ homomorphism from symplectic cochains to Hochschild cochains on the wrapped Fukaya category. Features of our construction are that it respects a modified action filtration (in contrast to Pomerleano--Seidel's construction); it uses a compact telescope model (in contrast to Abouzaid--Groman--Varolgunes' construction); and it is adapted to the purposes of our follow-up work where we construct Maurer--Cartan elements in symplectic cochains which are associated to a normal-crossings compactification of the Liouville domain.

math.SG↗
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↗
arXiv2024

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.

math.SG↗
arXiv2023

Constructing the relative Fukaya category

We give a definition of Seidel's `relative Fukaya category', for a smooth complex projective variety, under a semipositivity assumption. We use the Cieliebak--Mohnke approach to transversality via stabilizing divisors. Two features of our construction are noteworthy: that we work relative to a normal crossings divisor that supports an effective ample divisor but need not have ample components; and that our relative Fukaya category is linear over a certain ring of multivariate power series with integer coefficients.

math.SG↗
arXiv2022

The Gamma and Strominger-Yau-Zaslow conjectures: a tropical approach to periods

We propose a new method to compute asymptotics of periods using tropical geometry, in which the Riemann zeta values appear naturally as error terms in tropicalization. Our method suggests how the Gamma class should arise from the Strominger-Yau-Zaslow conjecture. We use it to give a new proof of (a version of) the Gamma Conjecture for Batyrev pairs of mirror Calabi-Yau hypersurfaces.

math.AG↗
arXiv2022

Quantum cohomology as a deformation of symplectic cohomology

We prove that under certain conditions, the quantum cohomology of a positively monotone compact symplectic manifold is a deformation of the symplectic cohomology of the complement of a simple crossings symplectic divisor. We also prove rigidity results for the skeleton of the divisor complement.

math.SG↗
arXiv2020

Homological mirror symmetry for generalized Greene-Plesser mirrors

We prove Kontsevich's homological mirror symmetry conjecture for certain mirror pairs arising from Batyrev-Borisov's `dual reflexive Gorenstein cones' construction. In particular we prove HMS for all Greene-Plesser mirror pairs (i.e., Calabi-Yau hypersurfaces in quotients of weighted projective spaces). We also prove it for certain mirror Calabi-Yau complete intersections arising from Borisov's construction via dual nef partitions, and also for certain Calabi-Yau complete intersections which do not have a Calabi-Yau mirror, but instead are mirror to a Calabi-Yau subcategory of the derived category of a higher-dimensional Fano variety. The latter case encompasses Kuznetsov's `K3 category of a cubic fourfold', which is mirror to an honest K3 surface; and also the analogous category for a quotient of a cubic sevenfold by an order-3 symmetry, which is mirror to a rigid Calabi-Yau threefold.

math.SG↗
arXiv2020

Lagrangian cobordism and tropical curves

We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relation to rational equivalence of 0-cycles on the mirror rigid analytic space.

math.SG↗
arXiv2019

Formulae in noncommutative Hodge theory

We prove that the cyclic homology of a saturated $A_\infty$ category admits the structure of a `polarized variation of Hodge structures', building heavily on the work of many authors: the main point of the paper is to present complete proofs, and also explicit formulae for all of the relevant structures. This forms part of a project of Ganatra, Perutz and the author, to prove that homological mirror symmetry implies enumerative mirror symmetry.

math.KT↗
arXiv2019

Symplectic topology of K3 surfaces via mirror symmetry

We study the symplectic topology of certain K3 surfaces (including the "mirror quartic" and "mirror double plane"), equipped with certain Kähler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated, and derive new constraints on Lagrangian tori. The key input, via homological mirror symmetry, is a result of Bayer and Bridgeland on the autoequivalence group of the derived category of an algebraic K3 surface of Picard rank one.

math.SG↗
arXiv2019

Versality of the relative Fukaya category

Seidel introduced the notion of a Fukaya category `relative to an ample divisor', explained that it is a deformation of the Fukaya category of the affine variety that is the complement of the divisor, and showed how the relevant deformation theory is controlled by the symplectic cohomology of the complement. We elaborate on Seidel's definition of the relative Fukaya category, and give a criterion under which the deformation is versal.

math.SG↗
arXiv2019

Rational equivalence and Lagrangian tori on K3 surfaces

Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L in X to the skyscraper sheaf of a point y of Y. We show there are Lagrangian tori with vanishing Maslov class in X whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about the `Beauville--Voisin subring' in the Chow groups of Y, and fits into a conjectural relationship between Lagrangian cobordism and rational equivalence of algebraic cycles.

math.SG↗
arXiv2018

Versality in mirror symmetry

One of the attractions of homological mirror symmetry is that it not only implies the previous predictions of mirror symmetry (e.g., curve counts on the quintic), but it should in some sense be `less of a coincidence' than they are and therefore easier to prove. In this survey we explain how Seidel's approach to mirror symmetry via versality at the large volume/large complex structure limit makes this idea precise.

math.SG↗
arXiv2016

On the Fukaya category of a Fano hypersurface in projective space

This paper is about the Fukaya category of a Fano hypersurface $X \subset \mathbb{CP}^n$. Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed-open string maps, weak proper Calabi-Yau structure, Abouzaid's split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface $X$: we construct a configuration of monotone Lagrangian spheres in $X$, and compute the associated disc potential. The result coincides with the Hori-Vafa superpotential for the mirror of $X$ (up to a constant shift in the Fano index $1$ case). As a consequence, we give a proof of Kontsevich's homological mirror symmetry conjecture for $X$. We also explain how to extract non-trivial information about Gromov-Witten invariants of $X$ from its Fukaya category.

math.SG↗