The geometric aspects of mirror symmetry are reviewed, with an eye towards future developments. Given a mirror pair (X,Y) of Calabi-Yau threefolds, the best-understood mirror statements relate certain small corners of the moduli spaces of X and of Y. We will indicate how one might go beyond such statements, and relate the moduli spaces more globally. In fact, in the boldest version of mirror symmetry (the Strominger-Yau-Zaslow conjecture), the Calabi-Yau threefolds X and Y should be directly related to each other through a very geometric construction.
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 dev…
We prove that the punctured generalized conifolds and punctured orbifolded conifolds are mirror symmetric under the SYZ program with quantum corrections. This mathematically confirms the gauge-theoretic prediction by Aganagic-Karch-Lüst-Miemiec, and also provides a supportive evidence to Morrison's conjecture that geometric transitions are reversed under mirror symmetry.
We consider compactifications of type IIA superstring theory on mirror-folds obtained as K3 fibrations over two-tori with non-geometric monodromies involving mirror symmetries. At special points in the moduli space these are asymmetric Gepner models. The compactifications are constructed from non-geometric automorphisms that arise from the diagonal action of an automorphism of the K3 surface and of an automorphism of the mirror surface. We identify the corresponding gaugings of N=4 supergravity in four dimensions, and show that the minima of the potential describe the same four-dimensional low…
We present enumerative aspects of the Gross-Siebert program in this introductory survey. After sketching the program's main themes and goals, we review the basic definitions and results of logarithmic and tropical geometry. We give examples and a proof for counting algebraic curves via tropical curves. To illustrate an application of tropical geometry and the Gross-Siebert program to mirror symmetry, we discuss the mirror symmetry of the projective plane.
We study geometric structures on the complement of a toric mirror arrangement associated with a root system. Inspired by those special hypergeometric functions found by Heckman-Opdam, as well as the work of Couwenberg-Heckman-Looijenga on geometric structures on projective arrangement complements, we consider a family of connections on a total space, namely, a $\mathbb{C}^{\times}$-bundle on the complement of a toric mirror arrangement (=finite union of hypertori, determined by a root system). We prove that these connections are torsion free and flat, and hence define a family of affine struct…
在我的预印本[T3]（特殊拉格朗日几何和轻微变形代数几何（spLag和sdAG），沃里克预印本22/1998，alg-geom/9806006，54页）出现后。我收到了Cumrun Vafa的一封电子邮件，他认识到这个主题与他的预印本[V]密切相关（Extending mirror conjecture to Calabi-Yau with bundles，hep-th/9804131，7 pp.）。这条短信一开始是对他信件的电子邮件"回复“。我们提出的所有构造都有众所周知的“谱曲线”原型（例如参见弗里德曼和其他[FMW]，Bershadsky和其他[BJPS]以及其他一些）。粗略地说，我们的构造是谱曲线构造加上[T3]中描述的相位几何。所以这篇文章应该在[T3]之前，作为[T3]中相位图几何发展的动机。
Zhegalkin zebra motives are tilings of the plane by black and white polygons representing certain ${\mathbb F}_2$-valued functions on ${\mathbb R}^2$. They exhibit a rich geometric structure and provide easy to draw insightful visualizations of many topics in the physics and mathematics literature. The present paper gives some pieces of a general theory and a few explicit examples. Many more examples will be shown in the forthcoming article "Zhegalkin zebra motives: algebra and geometry in black and white".
Brane brick models are Type IIA brane configurations that encode the $2d$ $\mathcal{N}=(0,2)$ gauge theories on the worldvolume of D1-branes probing toric Calabi-Yau 4-folds. We use mirror symmetry to improve our understanding of this correspondence and to provide a systematic approach for constructing brane brick models starting from geometry. The mirror configuration consists of D5-branes wrapping 4-spheres and the gauge theory is determined by how they intersect. We also explain how $2d$ $(0,2)$ triality is realized in terms of geometric transitions in the mirror geometry. Mirror symmetry l…
We prove Rietsch's mirror conjecture that the Dubrovin quantum connection for minuscule flag varieties is isomorphic to the character D-module of the Berenstein-Kazhdan geometric crystal. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. We reveal surprising relations with the works of Frenkel-Gross, Heinloth-Ngô-Yun, and Zhu on Kloosterman sheaves. The isomorphism comes from global rigidity results where Hecke eigensheaves are determined by their local ramification. As corollaries we obtain combinatorial identities for counts of rational curve…
We study mirror symmetry (A-side vs B-side) in the framework of quantum differential systems. We focuse on the logarithmic and non-resonant case, which describes the geometric situation. We show that quantum differential systems provide a good framework in order to generalize the construction of the rational structure on the A-side given by Katzarkov, Kontsevitch and Pantev for the projective spaces. As an application, we compute the rational structure obtained in this way on the orbifold cohomology of weighted projective spaces. As an example we also calculate, using quantum differential syst…
This is a report of the author's talk at Kinosaki Algebraic Geometry Symposium 2018. We discuss some recent progress on the geometry of thin exceptional sets in Manin's Conjecture.
In this expository note we discuss some arithmetic aspects of the mirror symmetry for plane cubic curves. We also explain how the Picard-Fuchs equation can be used to reveal part of these arithmetic properties. The application of Picard-Fuchs equations in studying the genus zero Gromov-Witten invariants of more general Calabi-Yau varieties and the Weil-Petersson geometry on their moduli spaces will also be discussed.
This paper studies geometric properties of the Iterated Matrix Multiplication polynomial and the hypersurface that it defines. We focus on geometric aspects that may be relevant for complexity theory such as the symmetry group of the polynomial, the dual variety and the Jacobian loci of the hypersurface, that are computed with the aid of representation theory of quivers.
This is a survey article on Newton-Okounkov bodies in projective geometry focusing on the relationship between positivity of divisors and Newton-Okounkov bodies.
This is a survey article on the recent progress in understanding the Strominger-Yau-Zaslow (SYZ) mirror symmetry conjecture, especially on the effect of quantum corrections, via Witten-Morse theory using the program first depicted by Fukaya to obtain an explicit relation between differential geometric operations, e.g. wedge product of differential forms and Lie-bracket of the Kodaira-Spencer complexes, with combinatorial structures, e.g. Morse $A_\infty$ structures and scattering diagrams.
We state a wall-crossing formula for the virtual classes of epsilon-stable quasimaps to GIT quotients and prove it for complete intersections in projective space, with no positivity restrictions on their first Chern class. As a consequence, the wall-crossing formula relating the genus g descendant Gromov-Witten potential and the genus g epsilon-quasimap descendant potential is established. For the quintic threefold, our results may be interpreted as giving a rigorous and geometric interpretation of the holomorphic limit of the BCOV B-model partition function of the mirror family.
