arXiv subjects

Umut Varolgunes

Publications and source records attributed to Umut Varolgunes.

14 recordsLinked to original sources

arXiv2026

Homological Mirror Symmetry for Affine Log Calabi-Yau Surfaces

Let $U$ be a smooth complex affine log Calabi--Yau surface, and let $\widehat{U}$ be its complete finite-type Liouville manifold, equipped with its natural grading structure. We give an algorithm that constructs a finite-type quasi-projective $\mathbb{Z}$-scheme $U^\vee$ such that \[ D^\pi\bigl(\mathcal{W}(\widehat{U},\Bbbk)\bigr) \cong D^b\operatorname{Coh}(U^\vee_{\Bbbk}) \] for every field $\Bbbk$. Our main contribution is to construct an almost toric model encoded by an exact eigenray diagram and, using the symplectic Torelli theorem for symplectic log Calabi--Yau pairs, to prove that this model is grading-preserving strongly exact symplectomorphic to $\widehat{U}$. The result then follows from the homological mirror symmetry theorem of Hacking--Keating.

math.SG↗

arXiv2025

Contact big fiber theorems

We prove contact big fiber theorems, analogous to the symplectic big fiber theorem by Entov and Polterovich, using symplectic cohomology with support. Unlike in the symplectic case, the validity of the statements requires conditions on the closed contact manifold. One such condition is to admit a Liouville filling with non-zero symplectic cohomology. In the case of Boothby-Wang contact manifolds, we prove the result under the condition that the Euler class of the circle bundle, which is the negative of an integral lift of the symplectic class, is not an invertible element in the quantum cohomology of the base symplectic manifold. As applications, we obtain new examples of rigidity of intersections in contact manifolds and also of contact non-squeezing.

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arXiv2023

Descent with algebraic structures for symplectic cohomology

We formulate and prove a chain level descent property of symplectic cohomology for involutive covers by compact subsets that take into account the natural algebraic structures that are present. The notion of an involutive cover is reviewed. We indicate the role that the statement plays in mirror symmetry.

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arXiv2023

A characterization of heaviness in terms of relative symplectic cohomology

For a compact subset $K$ of a closed symplectic manifold $(M, \omega)$, we prove that $K$ is heavy if and only if its relative symplectic cohomology over the Novikov field is non-zero. As an application we show that if two compact sets are not heavy and Poisson commuting, then their union is also not heavy. A discussion on superheaviness together with some partial results are also included.

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arXiv2023

On homological mirror symmetry for chain type polynomials

We consider Takahashi's categorical interpretation of the Berglund-Hubsch mirror symmetry conjecture for invertible polynomials in the case of chain polynomials. Our strategy is based on a stronger claim that the relevant categories satisfy a recursion of directed $A_{\infty}$-categories, which may be of independent interest. We give a full proof of this claim on the B-side. On the A-side we give a detailed sketch of an argument, which falls short of a full proof because of certain missing foundational results in Fukaya-Seidel categories, most notably a generation statement.

math.AG↗

arXiv2023

Locality of relative symplectic cohomology for complete embeddings

A complete embedding is a symplectic embedding $ι:Y\to M$ of a geometrically bounded symplectic manifold $Y$ into another geometrically bounded symplectic manifold $M$ of the same dimension. When $Y$ satisfies an additional finiteness hypothesis, we prove that the truncated relative symplectic cohomology of a compact subset $K$ inside $Y$ is naturally isomorphic to that of its image $ι(K)$ inside $M$. Under the assumption that the torsion exponents of $K$ are bounded we deduce the same result for relative symplectic cohomology. We introduce a technique for constructing complete embeddings using what we refer to as integrable anti-surgery. We apply these to study symplectic topology and mirror symmetry of symplectic cluster manifolds and other examples of symplectic manifolds with singular Lagrangian torus fibrations satisfying certain completeness conditions.

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arXiv2022

Closed string mirrors of symplectic cluster manifolds

For the base $B$ of a Maslov $0$ Lagrangian torus fibration with singularities consider the sheaf assigning to each $P\subset B$ the relative symplectic cohomology in degree $0$ of its pre-image. We compute this sheaf for nodal Lagrangian torus fibrations on four dimensional symplectic cluster manifolds. We show that it is the pushforward of the structure sheaf of a certain rigid analytic space under a non-archimedean torus fibration. The rigid analytic space is constructed in a canonical way from the relative SH sheaf and is referred as the \emph{closed string mirror}. The construction relies on computing relative SH for local models by applying general axiomatic properties rather than ad hoc analysis of holomorphic curves. These axiomatic properties include previously established ones such as the Mayer-Vietoris property and locality for complete embeddings; and new ones such as the Hartogs property and the holomorphic volume form preservation property of wall crossing in relative $SH$. We indicate some higher dimensional settings where the same techniques apply.

math.SG↗

arXiv2022

Framed $E_2$ structures in Floer theory

We resolve the long-standing problem of constructing the action of the operad of framed (stable) genus-$0$ curves on Hamiltonian Floer theory; this operad is equivalent to the framed $E_2$ operad. We formulate the construction in the following general context: we associate to each compact subset of a closed symplectic manifold a new chain-level model for symplectic cohomology with support, which we show carries an action of a model for the chains on the moduli space of framed genus $0$ curves. This construction turns out to be strictly functorial with respect to inclusions of subsets, and the action of the symplectomorphism group. In the general context, we appeal to virtual fundamental chain methods to construct the operations over fields of characteristic $0$, and we give a separate account, over arbitrary rings, in the special settings where Floer's classical transversality approach can be applied. We perform all constructions over the Novikov ring, so that the algebraic structures we produce are compatible with the quantitative information that is contained in Floer theory. Over fields of characteristic $0$, our construction can be combined with results in the theory of operads to produce explicit operations encoding the structure of a homotopy $BV$ algebra. In an appendix, we explain how to extend the results of the paper from the class of closed symplectic manifolds to geometrically bounded ones.

math.SG↗

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↗

arXiv2021

On certain quantifications of Gromov's non-squeezing theorem

Let $R>1$ and let $B$ be the Euclidean $4$-ball of radius $R$ with a closed subset ${E}$ removed. Suppose that $B$ embeds symplectically into the unit cylinder $\mathbb{D}^2 \times \mathbb{R}^2$. By Gromov's non-squeezing theorem, ${E}$ must be non-empty. We prove that the Minkowski dimension of ${E}$ is at least $2$, and we exhibit an explicit example showing that this result is optimal at least for $R \leq \sqrt{2}$. In an appendix by Jo\'e Brendel, it is shown that the lower bound is optimal for $R < \sqrt{3}$. We also discuss the minimum volume of ${E}$ in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball.

math.SG↗

arXiv2021

Seifert form of chain type invertible singularities

In this paper, we confirm a conjecture of Orlik-Randell from 1977 on the Seifert form of chain type invertible singularities. We use Lefschetz bifibration techniques as developed by Seidel (inspired by Arnold and Donaldson) and take advantage of the symmetries at hand. We believe that our method will be useful in understanding the homological/categorical version of Berglundt-Hubsch mirror conjecture for invertible singularities.

math.AG↗

arXiv2020

Super-rigidity of certain skeleta using relative symplectic cohomology

This article uses relative symplectic cohomology, recently studied by the second author, to understand rigidity phenomena for compact subsets of symplectic manifolds. As an application, we consider a symplectic crossings divisor in a Calabi-Yau symplectic manifold $M$ whose complement is a Liouville manifold. We show that, for a carefully chosen Liouville structure, the skeleton as a subset of $M$ exhibits strong rigidity properties akin to super-heavy subsets of Entov-Polterovich. Along the way, we expand the toolkit of relative symplectic cohomology by introducing products and units. We also develop what we call the contact Fukaya trick, concerning the behaviour of relative symplectic cohomology of subsets with contact type boundary under adding a Liouville collar.

math.SG↗

arXiv2020

Mayer-Vietoris property for relative symplectic cohomology

In this paper, we construct a Hamiltonian Floer theory based invariant called relative symplectic cohomology, which assigns a module over the Novikov ring to compact subsets of closed symplectic manifolds. We show the existence of restriction maps, and prove some basic properties. Our main contribution is to identify a natural geometric situation in which relative symplectic cohomology of two subsets satisfy the Mayer-Vietoris property. This is tailored to work under certain integrability assumptions, the weakest of which introduces a new geometric object called a barrier - roughly, a one parameter family of rank 2 coisotropic submanifolds. The proof uses a deformation argument in which the topological energy zero (i.e. constant) Floer solutions are the main actors.

math.SG↗

arXiv2017

On the equatorial Dehn twist of a Lagrangian nodal sphere

Let $(M^4,ω)$ be a geometrically bounded symplectic manifold, and $L\subset M$ a Lagrangian nodal sphere such that $ω\mid_{π_2(M,L)}=0$. We show that an equatorial Dehn twist of $L$ does not extend to a Hamiltonian diffeomorphism of $M$. We also confirm a mirror symmetry prediction about the action of a symplectomorphism extending an equatorial Dehn twist on the Floer theory of the nodal sphere. We present analogues of the equatorial Dehn twist for more singular Lagrangians, and make concrete conjectures about them.

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