Variation of gluing in homological mirror symmetry

This thesis is a collection of seven papers concerned with the relationship between variation of gluing spaces and categories in homological mirror symmetry(HMS). We divide it into three parts according to how we vary gluing what. The first part consists of three papers on algebraic deformations of Calabi–Yau 3-folds(CY3s), where we vary complex structures to glue locally trivial deformations. The second part consists of two papers on cut-and-reglue procedure for relative Jacobians of generic elliptic 3-folds, where we vary Brauer classes to glue smooth elliptic 3-folds with sections. The third part consists of two papers on local-to-global principle for wrapped Fukaya categories of very affine hypersurfaces(VAHs), where we vary Liouville structures to glue pairs of pants. Our main goal of the first two parts is to construct new Fourier– Mukai partners(FMPs), nonbirational derived-equivalent CY3s. While birational CY3s are derived-equivalent, FMPs give highly nontrivial multiple mirrors to the dual manifolds. Our main goal of the third part is to establish HMS for complete intersections of VAHs. Recently, Gammage–Shende established HMS for VAHs under some assumption essential to construct a global skeleton, which allows them to reduce gluing wrapped Fukaya categories to gluing local skeleta. For several reasons we need a different approach to remove their assumption. In the first paper, we prove that the derived equivalence of CY3s extends to their versal de formations over an affine complex variety. This is fundamental for our deformation methods to construct new examples of FMPs. Due to the main theorem of the second paper, the derived category of the generic fiber of a flat proper family can be described as a certain Verdier quo tient. As a consequence, the derived equivalence of the above versal deformations is inherited to their generic fibers. We analyze some good cases where also nonbirationality is inherited, establishing a deformation method to construct new FMPs from known examples. Conversely, the description enables us to prove specialization, i.e., the derived equivalence of the generic fibers extends to general fibers, completing all the relevant inductions of the derived equiva lence of CY3s through deformations. The main theorem of the third paper gives a rigorous explanation of these phenomena. Namely, deformations of a CY3 are equivalent to Morita deformations of its dg category of perfect complexes. We also prove that, analogous to isomor phisms of schemes, the derived equivalence is inherited from effectivizations to their enough close approximations. This is an improvement of the main theorem of the first paper, expected from the equivalence of the two deformation theories. In the fourth paper, we prove that any flat projective family must be what we call an almost coprime twisted power, whenever it is linear derived-equivalent over the base to a generic el liptic CY3. This should be the best possible reconstruction result for generic elliptic CY3s. Combining with the main theorem of the first paper, we obtain a family of pairs of coprime twisted powers whose closed fibers are nonbirational whenever they are nonisomorphic. Un winding our arguments, one sees that generic elliptic CY3s are linear derived-equivalent over the base if and only if their generic fibers are derived-equivalent. This is the key observation for the fifth paper where we give affirmative answers to two of the four conjectures raised by Knapp–Scheidegger–Schimannek. Namely, we prove that each of 12 pairs of elliptic CY3s constructed by them share the relative Jacobian and linear derived-equivalent over the base. Except one self-dual pair, the closed fibers of the family obtained by the above combination are nonisomorphic. Hence we obtain families of new FMPs, establishing another deformation method to construct FMPs. As far as we know, this is the first systematic construction of (fami lies of) FMPs. Moreover, it works for elliptic CY3s with higher multisections, whose examples some string theorists have been looking for. In the sixth paper, we establish HMS for complete intersections of VAHs. The main chal lenge is computing wrapped Fukaya categories of complete intersections. With the aid of equivariantization/de-equivariantization, we reduce it to unimodular case. Proving that locally complete intersections are products of lower dimensional pairs of pants, we reduce it further to hypersurface case without the assumption imposed on the previous result by Gammage– Shende. We extend it by inductive argument following Pascaleff–Sibilla which does not re quire any global skeleton. Besides the invariance of wrapped Fukaya categories under simple Liouville homotopies, one key is to find Weinstein structures on the initial exact symplectic manifold and the additional pair of pants which glue to yield that on the gluing, everytime we proceed the inductive argument. Another is to show that also their wrapped Fukaya categories glue to yield that of the gluing. Our method should work to compute wrapped Fukaya cat egories in other relevant settings. Finally, we glue HMS for pairs of pants along the global combinatorial duality over the tropical hypersurface. The geometry of VAHs is further studied in the seventh paper, where we complete the missing A-side of the SYZ picture over fanifolds. This can be regarded as a generalization of that over tropical hypersurfaces

Biregular and Birational Geometry of Algebraic Varieties

Every area of mathematics is characterized by a guiding problem. In algebraic geometry such problem is the classification of algebraic varieties. In its strongest form it means to classify varieties up to biregular morphisms. However, birationally equivalent varieties share many interesting properties. Therefore for any birational equivalence class it is natural to work out a variety, which is the simplest in a suitable sense, and then study these varieties. This is the aim of birational geometry. In the first part of this thesis we deal with the biregular geometry of moduli spaces of curves, and in particular with their biregular automorphisms. However, in doing this we will consider some aspects of their birational geometry. The second part is devoted to the birational geometry of varieties of sums of powers and to some related problems which will lead us to computational geometry and geometric complexity theory

Geometry of Matroids and Hyperplane Arrangements

There is a trinity relationship between hyperplane arrangements, matroids and convex polytopes. We expand and combinatorialize it as settling the complexity issue expected by Mnev's universality theorem. Based on this theory, we show that for n less than or equal to 9 every matroid tiling in the hypersimplex Delta(3,n) associated to a weighted stable hyperplane arrangement extends to a matroid subdivision of Delta(3,n) and that the bound 9 for n is sharp. As a straightforward application, we completely answer Alexeev's algebro-geometric question.Comment: 46 pages, 24 figures; v3: minor improvement

Geometric and topological recursion and invariants of the moduli space of curves

A thread common to many problems of enumeration of surfaces is the idea that complicated cases can be recovered from simpler ones through a recursive procedure. Solving the problem for the simplest topologies and expressing how to glue them together provides an algorithm to solve the enumerative problem of interest. In this dissertation, we consider three distinct but interconnected topics: integration over the moduli space of curves and its combinatorial model, the enumeration of curves and quadratic differentials, and the enumeration of branched covers of the Riemann sphere. The leitmotif that will connect them all is a recursive procedure known as topological recursion. The moduli space of curves is a key object of study in algebraic geometry. Its combinatorial model has provided powerful tools to compute various invariants of the moduli space, such as the Euler characteristic and Witten's intersection numbers. In this dissertation we further develop the (symplectic) geometry of this combinatorial model, providing a complete parallel with the Weil–Petersson geometry of the hyperbolic model. In particular, we show that certain length and twist coordinates are Darboux, and propose a new geometric approach to Witten's conjecture/Kontsevich's theorem. Namely, it is obtained by integration of a Mirzakhani-type identity on the combinatorial Teichmüller space, which recursively computes the constant function 1 by excision of embedded pairs of pants. The second topic of interest is the enumeration of multicurves with respect to either the hyperbolic or the combinatorial notion of length. Following ideas of Mirzakhani and Andersen–Borot–Orantin, we show that such problems can again be recursively solved by excision of embedded pairs of pants. As a consequence, the average number of multicurves over the corresponding moduli space can be computed by topological recursion. On the other hand, since the work of Mirzakhani, the average number of multicurves is known to be related to the Masur–Veech volumes of the principal stratum of the moduli space of quadratic differentials. Combining these two results, we find a topological recursion formula to compute Masur–Veech volumes. To conclude, we turn our attention to spin Hurwitz theory, that is the enumeration of branched covers of the Riemann sphere with respect to their ramification and parity. Thanks to the connection between the fermion formalism and Hurwitz theory, we are able to formulate a precise conjecture to recursively compute spin Hurwitz numbers from the simplest topologies. We also prove that this recursive formula is equivalent to a description of spin Hurwitz numbers as intersection numbers on the moduli space of curves, that is a spin version of the celebrated ELSV formula

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Tropical varieties, maps and gossip

Tropical geometry is a relatively new field of mathematics that studies the tropicalization map: a map that assigns a certain type of polyhedral complex, called a tropical variety, to an embedded algebraic variety. In a sense, it translates algebraic geometric statements into combinatorial ones. An interesting feature of tropical geometry is that there does not exist a good notion of morphism, or map, between tropical varieties that makes the tropicalization map functorial. The main part of this thesis studies maps between different classes of tropical varieties: tropical linear spaces and tropicalizations of embedded unirational varieties. The first chapter is a concise introduction to tropical geometry. It collects and proves the main theorems. None of these results are new. The second chapter deals with tropicalizations of embedded unirational varieties. We give sufficient conditions on such varieties for there to exist a (not necessarily injective) parametrization whose naive tropicalization is surjective onto the associated tropical variety. The third chapter gives an overview of the algebra related to tropical linear spaces. Where fields and vector spaces are the central objects in linear algebra, so are semifields and modules over semifields central to tropical linear algebra and the study of tropical linear spaces. Most results in this chapter are known in some form, but scattered among the available literature. The main purpose of this chapter is to collect these results and to determine the algebraic conditions that suffice to give linear algebra over the semifield a familiar feel. For example, under which conditions are varieties cut out by linear polynomials closed under addition and scalar multiplication? The fourth chapter comprises the biggest part of the thesis. The techniques used are a combination of tropical linear algebra and matroid theory. Central objects are the valuated matroids introduced by Andreas Dress and Walter Wenzl. Among other things the chapter contains a classification of functions on a tropical linear space whose cycles are tropical linear subspaces, extending an old result on elementary extensions of matroids by Henry Crapo. It uses Mikhalkin’s concept of a tropical modification to define the morphisms in a category whose objects are all tropical linear spaces. Finally, we determine the structure of an open submonoid of the morphisms from affine 2-space to itself as a polyhedral complex. Finally, the fifth and last chapter is only indirectly related to maps. It studies a certain monoid contained in the tropicalization of the orthogonal group: the monoid that is generated by the distance matrices under tropical matrix multiplication (i.e. where addition is replaced by minimum, and multiplication by addition). This monoid generalizes a monoid that underlies the well-known gossip problem, to a setting where information is transmitted only with a certain degree accuracy. We determine this so-called gossip monoid for matrices up to size 4, and prove that in general it is a polyhedral monoid of dimension equal to that of the orthogonal group

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition