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

Homological mirror symmetry for hypertoric varieties, II

Homological mirror symmetry for weighted projective spaces and Morse homotopy

Homological mirror symmetry for nodal stacky curves

Abstract We prove that homological mirror symmetry for very affine hypersurfaces respects certain natural symplectic operations (as functors between partially wrapped Fukaya categories), verifying conjectures of Auroux. These conjectures concern compatibility between mirror symmetry for a very affine hypersurface and its complement, itself also a very affine hypersurface. We find that the complement of a very affine hypersurface has, in fact, two natural mirrors, one of which is a derived scheme. These two mirrors are related via a nongeometric equivalence mediated by Knörrer periodicity; Auroux's conjectures require some modification to take this into account. Our proof also introduces new techniques for presenting Liouville manifolds as gluings of Liouville sectors.

In this article, we revisit the classical McKay correspondence via homological mirror symmetry. Specifically, we demonstrate how this correspondence can be articulated as a derived equivalence between the category of vanishing cycles associated with a Kleinian surface singularity and the category of perfect complexes on the corresponding quotient orbifold. We further illustrate how this equivalence allows for the interpretation of the spectrum of a Kleinian surface singularity solely in terms of the representation-theoretic data of the associated binary polyhedral group.

Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves

We prove a homological mirror symmetry result for maximally degenerating families of hypersurfaces in .C / n (B-model) and their mirror toric Landau-Ginzburg A-models. The main technical ingredient of our construction is a "fiberwise wrapped" version of the Fukaya category of a toric Landau-Ginzburg model. With the definition in hand, we construct a fibered admissible Lagrangian submanifold whose fiberwise wrapped Floer cohomology is isomorphic to the ring of regular functions of the hypersurface. It follows that the derived category of coherent sheaves of the hypersurface quasiembeds into the fiberwise wrapped Fukaya category of the mirror. We also discuss an extension to complete intersections.

Given a holomorphic family of Bridgeland stability conditions over a surface, we define a notion of spectral network which is an object in a Fukaya category of the surface with coefficients in a triangulated DG-category. These spectral networks are analogs of special Lagrangian submanifolds, combining a graph with additional algebraic data, and conjecturally correspond to semistable objects of a suitable stability condition on the Fukaya category with coefficients. They are closely related to the spectral networks of Gaiotto–Moore–Neitzke. One novelty of our approach is that we establish a general uniqueness results for spectral network representatives. We also verify the conjecture in the case when the surface is disk with six marked points on the boundary and the coefficients category is the derived category of representations of an A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_2$$\\end{document} quiver. This example is related, via homological mirror symmetry, to the stacky quotient of an elliptic curve by the cyclic group of order six.

On a B-field transform of generalized complex structures over complex tori

We study homological mirror symmetry for Hirzebruch surfaces Fk as complex manifolds by using the Strominger-Yau-Zaslow construction of mirror pair and Morse homotopy. For toric Fano surfaces, Futaki-Kajiura and the author proved homological mirror symmetry by using Morse homotopy in [9,10,16]. In this paper, we extend Futaki-Kajiura's result of the Hirzebruch surface F1 to Fk. We discuss Morse homotopy and show that homological mirror symmetry in the sense above holds true.

We introduce the wrapped Donaldson–Fukaya category of a (generalized) semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical divisor in a toric Calabi–Yau 3-fold. We compute the wrapped Floer cohomology of a Lagrangian section and find that it is the algebra of functions on the Hori–Vafa mirror. The latter result is the key step in proving homological mirror symmetry for this case. The techniques developed here allow the construction in general of the wrapped Fukaya category on an open Calabi–Yau manifold carrying an SYZ fibration with nice behavior at infinity. We discuss the relation of this to the algebraic vs analytic aspects of mirror symmetry.

Following [35] , in this section we exploit the large center of quantizations in characteristic p so as to relate modules over A K with coherent sheaves on M .1/ K . Roughly speaking, upon restriction to fibers of W M .1/ K ! N .1/ K , the quantization becomes the algebra of endomorphisms of a vector bundle, and thus Morita-equivalent to the structure sheaf of the fiber. Theorem 2. 4 [35, Theorems 4.3.1 and 4.3.4] For any 2 t F p , there exists a coherent sheaf A of algebras Azumaya over the structure sheaf on M .1/ K such that .M

Strominger–Yau–Zaslow (SYZ) proposed a way of constructing mirror pairs as pairs of torus fibrations. We apply this SYZ construction to toric Fano surfaces as complex manifolds, and discuss the homological mirror symmetry, where we consider Morse homotopy of the moment polytope instead of the Fukaya category.

We introduce a conjecture on homological mirror symmetry relating the symplectic topology of the complement of a smooth ample divisor in a K3 surface to algebraic geometry of type III degenerations, and prove it when the degree of the divisor is either 2 or 4.

Mirror P=W conjecture and extended Fano/Landau-Ginzburg correspondence

Consider the pairs $(f,G)$ with $f = f(x_1,\dots,x_N)$ being a polynomial defining a quasihomogeneous singularity and $G$ being a subgroup of ${\rm SL}(N,\mathbb{C})$, preserving $f$. In particular, $G$ is not necessary abelian. Assume further that $G$ contains the grading operator $j_f$ and $f$ satisfies the Calabi-Yau condition. We prove that the nonvanishing bigraded pieces of the B-model state space of $(f,G)$ form a diamond. We identify its topmost, bottommost, leftmost and rightmost entries as one-dimensional and show that this diamond enjoys the essential horizontal and vertical isomorphisms.

Abstract 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.

Morse-Novikov cohomology on foliated manifolds

Abstract Motivated by their role in M‐theory, F‐theory, and S‐theory compactifications, all possible complete intersections Calabi‐Yau five‐folds in a product of four or less complex projective spaces are constructed, with up to four constraints. A total of 27 068 spaces are obtained, which are not related by permutations of rows and columns of the configuration matrix, and determine the Euler number for all of them. Excluding the 3909 product manifolds among those, the cohomological data for 12 433 cases are calculated, i.e., 53.7% of the non‐product spaces, obtaining 2375 different Hodge diamonds. The dataset containing all the above information is available here. The distributions of the invariants are presented, and a comparison with the lower‐dimensional analogues is discussed. Supervised machine learning is performed on the cohomological data, via classifier, and regressor (both fully connected and convolutional) neural networks. h1, 1 can be learnt very efficiently, with very high R2 score and an accuracy of 96% is found, i.e., 96% of the predictions exactly match the correct values. For , very high R2 scores are also found, but the accuracy is lower, due to the large ranges of possible values.