Ignacio de Gregorio, Etienne Mann
We show that the orbifold Chow ring of a root stack over a well-formed weighted projective space can be naturally seen as the Jacobian algebra of a function on a singular variety given by a partial compactification of its Ginzburg-Landau model.
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Given X a K3 surface, a mirror dual to X can be identified with a component of the moduli space of semistable sheaves on X. We consider fibrations by K3 surfaces over a one dimensional base that are Calabi-Yau and we charaterize the dual fibration that turns to be derived equivalent to the original one relating the problem to mirror symmetry.
Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes
本文确定了由Fano多面体导出的环面$K3$超曲面的Néron-Severi格，通过引入特定椭圆纤维化并证明其生成元，从而证明了Dolgachev猜想。
SYZ mirror conjecture predicts that a Calabi-Yau manifold $X$ consists of a family of tori which are dual to a family of special lagrangian tori on the mirror dual manifold $\hat{X}$. Here we consider a fibration of polarized abelian varieties and we construct a dual one. Moreover we prove that they are equivalent at the level of derived categories.
In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for this class of \cy manifolds in symplectic category.
We investigate a potential relationship between mirror symmetry for Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and Landau-Ginzburg models. More precisely, we show that if a Calabi-Yau admits a so-called Tyurin degeneration to a union of two Fano varieties, then one should be able to construct a mirror to that Calabi-Yau by gluing together the Landau-Ginzburg models of those two Fano varieties. We provide evidence for this correspondence in a number of different settings, including Batyrev-Borisov mirror symmetry for K3 surfaces and Calabi-Yau threefolds, Dolgachev…
We consider the suspension operation on Lefschetz fibrations, which takes p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant, and changes the category of the fibre (or more precisely, the subcategory consisting of a basis of vanishing cycles) in a specific way. As an application, we prove part of Homological Mirror Symmetry for the total spaces of canonical bundles over toric del Pezzo surfaces.
We consider regular Calabi-Yau hypersurfaces in $N$-dimensional smooth toric varieties. On such a hypersurface in the neighborhood of the large complex structure limit point we construct a fibration over a sphere $S^{N-1}$ whose generic fibers are tori $T^{N-1}$. Also for certain one-parameter families of such hypersurfaces we show that the monodromy transformation is induced by a translation of the $T^{N-1}$ fibration by a section. Finally we construct a dual fibration and provide some evidence that it describes the mirror family.
Arithmetic geometry of the moduli stack of Weierstrass fibrations over $\mathbb{P}^1$
本文研究了无参数化射影线上Weierstrass纤维的模堆，构造了其显式形式，并确定了其在Grothendieck环中的动机。
Degenerations and Fibrations of K3 Surfaces: Lattice Polarisations and Mirror Symmetry
本文研究K3曲面Tyurin退化及椭圆纤维化的格极化理论，建立兼容现有定义的格极化概念，并探讨两种设置间的镜像对称对应。
This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related by the existence of dual special Lagrangian torus fibrations. We explore this conjecture without reference to the special Lagrangian condition. In this setting, natural questions include: does there exist a nice class of T^3-fibrations for which the dual fibration can be constructed? Do such fibrations exist on a manifold such as the quintic threefold? If so, is the dual fibration the mirror? We answe…
In this survey paper, we briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various examples are presented, some of them new.
We give finite presentations for the fundamental group of moduli stacks of smooth Weierstrass curves over complex projective space P^n which extend the classical result for elliptic curves to positive dimensional base. We thus get natural generalisations of SL_2(Z) and pave the way to understanding the fundamental group of moduli stacks of elliptic surfaces in general. Our approach exploits the natural involution on Weierstrass curves and the identification of its fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over P^n. The fundamental group of…
We consider fibrations by abelian surfaces and K3 surfaces over a one dimensional base that are Calabi-Yau and we obtain dual fibrations that are derived equivalent to the original fibration. Finally, we relate the problem to mirror symmetry.
We study sextic del Pezzo surface fibrations via root stacks.
In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in toric variety. We will also be able to give precise formulation of SYZ mirror conjecture in general (including singular locus and duality of singular fibres).
We discuss homological mirror symmetry for the conifold from the point of view of the Strominger-Yau-Zaslow conjecture.
Mirror of Orbifold Singularities in the Hitchin Fibration: the case $(\text{SL}_n,\text{PGL}_n)$
研究 Hitchin 纤维化中轨道奇点的镜像，证明 $\ ext{PGL}_n$ 模空间奇点对应 $\ ext{SL}_n$ 可约纤维，并验证 Fourier-Mukai 变换满足分数 Hecke 特征性质。
We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the quintic threefold. We interpret our results as corroboration of the view that in homological mirror symmetry, an anti-symplectic involution is the mirror of duality. In the same setting, we construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle.
Let E be a toric fibration arising from symplectic reduction of a direct sum of line bundles over (almost-) Kähler base B. Then each torus-fixed point of the toric manifold fiber defines a section of the fibration. Let L_a be convex line bundles over B, A_a smooth divisors of B arising as the zero loci of generic sections of L_a, and \a:B\to E a particular fixed-point section of E. Further assume the \{A_a\} to be mutually disjoint. We compute genus-0 Gromov--Witten invariants of the blowup of E along \a(\coprod_a A_a) in terms of genus-0 Gromov--Witten invariants of B and of \{A_a\}, the matr…
We give a precise relation between the mirror transformation and the Seidel elements for weak Fano toric Deligne-Mumford stacks. Our result generalizes the corresponding result for toric varieties proved by González and Iritani in \cite{GI}.
We find that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration. In the simplest examples, the generic CY elliptic fibration over any toric base surface $B$ that supports an elliptic Calabi-Yau threefold has a mirror that is an elliptic fibration over a dual toric base surface $\tilde{B}$ that is related through toric geometry to the line bundle $-6K_B$. The Kreuzer-Skarke database includes all these examples and gives a wide range of other more complicated constructions where mirror symmetry also fact…
We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$ in a toric variety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of $V\times\mathbb{C}$ along $H\times 0$, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to $H$. The main applications concern affine hypersurfaces of general type, for which…
We discuss various topics on degenerations and special Lagrangian torus fibrations of Calabi-Yau manifolds in the context of mirror symmetry. A particular emphasis is on Tyurin degenerations and the Doran-Harder-Thompson conjecture, which builds a bridge between mirror symmetry for Calabi-Yau manifolds and that for quasi-Fano manifolds. The proof of the conjecture is of interest in its own right and leads us to a few other related topics such as SYZ mirror symmetry, theta functions and geometric quantization. Inspired by the conjecture, we also propose a new construction of Landau-Ginzburg mod…
For a toric Calabi-Yau (CY) orbifold $\mathcal{X}$ whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian torus fibration on $\mathcal{X}$, which we call the Gross fibration. We apply the Strominger-Yau-Zaslow (SYZ) recipe to the Gross fibration of $\mathcal{X}$ to construct its mirror with the instanton corrections coming from genus 0 open orbifold Gromov-Witten (GW) invariants, which are virtual counts of holomorphic orbi-disks in $\mathcal{X}$ bounded by fibers of the Gross fibration. We explicitly evaluate all these invariants by first proving an o…
In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau manifold and the derived category of coherent sheaves on the dual Calabi-Yau manifold). Our point of view on the origin of torus fibrations is based on the standard differential-geometric picture of collapsing Riemannian manifolds as well as analogous considerations for Conformal Field Theories. It seems to give a description of mirror manifolds much more transparent than th…
Family Floer program and non-archimedean SYZ mirror construction
本文基于家族 Floer 理论，用全纯圆盘计数、非阿基米德分析和 $A_\infty$ 同调代数构造含刚性解析空间与超势的镜像 Landau-Ginzburg 模型，并将假设减弱到半正 Lagrangian 纤维化。
Given a category fibered in groupoids over schemes with a log structure, one produces a category fibered in groupoids over log schemes. We classify the groupoid fibrations over log schemes that arise in this manner in terms of a categorical notion of "minimal" objects. The classification is actually a purely category-theoretic result about groupoid fibrations over fibered categories, though most of the known applications occur in the setting of log geometry, where our categorical framework encompasses many notions of "minimality" previously extant in the literature.
The integral Chow rings of moduli of Weierstrass fibrations
本文计算了射影直线上极小 Weierstrass 纤维化模空间的整系数 Chow 环，利用商栈表示和等变交叠理论确定了各 Chow 环的完整生成元与关系，并对 $N=1$ 和 $N=2$ 的情况给出了更明确的计算。
Given any smooth cubic curve $E\subseteq \mathbb{P}^2$, we show that the complex affine structure of the special Lagrangian fibration of $\mathbb{P}^2\setminus E$ constructed by Collins--Jacob--Lin arXiv:1904.08363 coincides with the affine structure used in Carl--Pomperla--Siebert for constructing mirror. Moreover, we use the Floer-theoretical gluing method to construct a mirror using immersed Lagrangians, which is shown to agree with the mirror constructed by Carl--Pomperla--Siebert.
We use the mirror theorem for toric Deligne-Mumford stacks, proved recently by the authors and by Cheong-Ciocan-Fontanine-Kim, to compute genus-zero Gromov-Witten invariants of a number of toric orbifolds and gerbes. We prove a mirror theorem for a class of complete intersections in toric Deligne-Mumford stacks, and use this to compute genus-zero Gromov-Witten invariants of an orbifold hypersurface.
Projective moduli of log Calabi--Yau fibrations over curves
研究曲线上对数卡拉比 - 丘纤维化，引入新稳定性条件，证明该条件能产生有射影粗模空间的恰当层，其边界可参数化曲线上的平坦纤维化。
We prove a Givental-style mirror theorem for toric Deligne--Mumford stacks X. This determines the genus-zero Gromov--Witten invariants of X in terms of an explicit hypergeometric function, called the I-function, that takes values in the Chen--Ruan orbifold cohomology of X.
Shifted Contact Structures on Exact Symplectic Fibrations
本文在派生叠上证明移位接触Thurston定理，即若态射具有移位恰当辛纤维化结构且基底有移位接触结构，则源叠也有相容的移位接触结构，并给出余法叠等应用。
We continue the study of the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, this states that if two Calabi-Yau manifolds X and Y are mirror partners, then X and Y have special Lagrangian torus fibrations which are dual to each other. Much work on this conjecture is necessarily of a speculative nature, as in dimension 3 it is still a very difficult problem of how to construct such fibrations. Nevertheless, assuming the existence of such fibrations there are many things one can prove. This paper covers a number of issues. First it applies results from the theory of completely int…
We extend the functor Sing of singular chains to the category of topological stacks and establish its main properties. We prove that Sing respects weak equivalences and takes a morphism of topological stacks that is both a Serre and a Reedy fibration to a Kan fibration of simplicial sets. When restricted to the category of topological spaces Sing coincides with the usual singular functor.
Given a smooth projective variety $X$ with a smooth nef divisor $D$ and a positive integer $r$, we construct an $I$-function, an explicit slice of Givental's Lagrangian cone, for Gromov--Witten theory of the root stack $X_{D,r}$. As an application, we also obtain an $I$-function for relative Gromov--Witten theory following the relation between relative and orbifold Gromov--Witten invariants.
On the cone conjecture for log Calabi-Yau mirrors of Fano 3-folds
该研究针对带K3纤维化的光滑射影3-fold，证明当三阶上同调群为零时伪自同构群在可动锥余维一面上的轨道有限，验证了锥猜想的特例，且该类3-fold可作为Fano 3-folds的对数Calabi-Yau镜像。
Family Floer SYZ mirror algorithm for the Grassmannian $Gr(2,4)$
为格拉斯曼流形\(Gr(2,4)\)的朗道 - 金兹堡镜像给出非阿基米德SYZ构造，从A侧拉格朗日纤维化出发，在B侧朗兰兹对偶格拉斯曼流形的贝科维奇解析化中构造镜像纤维化，证明二者相关性质一致。
Given a smooth projective variety $X$ with a simple normal crossing divisor $D:=D_1+D_2+...+D_n$, where $D_i\subset X$ are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks $X_{D,\vec r}$ by constructing an $I$-function, a slice of Givental's Lagrangian cone for Gromov--Witten theory of multi-root stacks. We provide three applications: (1) We show that some genus zero invariants of $X_{D,\vec r}$ stabilize for sufficiently large $\vec r$. (2) We state a generalized local-log-orbifold principle conjecture and prove a version of it. (3) We show that regularized quantum…
Motivic Mirror Symmetry for Character Stacks
本文提出任意半单群特征标栈的motivic镜像对称版本，推广胞腔分解至连通约化群，用其描述特征标栈自同构并通过Weil配对建立胞腔对偶性，还在秩二特殊线性群上验证了猜想。
We introduce a new version of 3d mirror symmetry for toric stacks, inspired by a 3d $\mathcal{N} = 2$ abelian mirror symmetry construction in physics. Given some toric data, we introduce the $K$-theoretic $I$-function with effective level structure for the associated toric stack. When a particular stability condition is chosen, it restricts to the $I$-function for the particular toric GIT quotient. The mirror of a toric stack is defined by the Gale dual of the original toric data. We then proved the mirror conjecture that the $I$-functions of a mirror pair coincide, under the mirror map, which…
This is an expository article on the Gross--Siebert approach to mirror symmetry and its interactions with the Strominger--Yau--Zaslow conjecture from a topological perspective.
We study the the moduli space of KSBA stable pairs $(X,sS+\sum a_i F_i)$, consisting of a Weierstrass fibration $X$, its section $S$, and some fibers $F_i$. We find a compactification which is a DM stack, and we describe the objects on the boundary. We show that the fibration in the definition of Weierstrass fibration extends to the boundary, and it is equidimensional when $s \ll 1$. We prove that there are wall-crossing morphisms when the weights $s$ and $a_i$ change. When $s=1$, this recovers the work of La Nave (arXiv:math/0205035); and a special case of the work of Ascher-Bejleri (arXiv:17…
Inspired by the work of Gross on topological Mirror Symmetry we construct candidate Lagrangian torus fibration models for the 105 families of smooth Fano threefolds. We prove, in the case the second Betti number is one, that the total space of each fibration is homeomorphic to the expected Fano threefold, and show that the numerical invariants coincide for all 105. Our construction relies on a notion of toric degeneration for affine manifolds with singularities, and the correspondence we obtain between polytopes and Fano manifolds is compatible with that appearing in the work of Coates-Corti-K…
Let π: M \ra B be a Lagrangian torus fibration with singularities such that the fibers are of Maslov index zero, and unobstructed. The paper constructs a rigid analytic space M_0^\chk over the Novikov field which is a deformation of the semi-flat complex structure of the dual torus fibration over the smooth locus B_0 of π. Transition functions of M_0^\chk are obtained via A-\infty homomorphisms which capture the wall-crossing phenomenon of moduli spaces of holomorphic disks.
Intrinsic mirror symmetry and Frobenius structure theorem via Gromov-Witten theory of root stacks
该研究通过根栈Gromov-Witten理论，建立了Gross-Siebert与Tseng-You两类对数Calabi-Yau镜像代数的结构常数关联，给出对数镜像代数结合性新证，完整证明弱Frobenius结构定理，并引入相关扭化不变量研究分歧基变换下的行为。
Shifted Symplectic Fibrations and Derived Thurston Theorem
本文证明了经典辛纤维化中瑟斯顿定理的导出辛类比，即在满足一定条件时，可从基的移位辛结构构造出源的移位辛结构，并发展了移位辛纤维化的仿射模型。
We construct pseudotoric structures (à la Tyurin) on the two-step flag variety $F\ell_{1, n-1; n}$, and explain a general relation between pseudotoric structures and special Lagrangian torus fibrations, the latter of which are important in the study of SYZ mirror symmetry. As an application, we speculate how our constructions can explain the number of terms in the superpotential of Rietsch's Landau-Ginzburg mirror.
We prove homological mirror symmetry for Lefschetz fibrations obtained as disconnected sums of polynomials of types A or D. The proof is based on the behavior of the Fukaya category under the addition of a polynomial of type D.
Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big equivariant quantum D-module of a toric Deligne-Mumford stack is isomorphic to the Saito structure associated to the mirror Landau-Ginzburg potential. We give a GKZ-style presentation of the quantum D-module, and a combinatorial description of quantum cohomology as a quantum Stanley-Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant …
Irregular Hodge numbers of stacky Clarke mirror pairs
本文证明堆叠 Clarke 镜面对扭曲上同调的不规则 Hodge 滤过分次片段对偶，并推广相关猜想与 Hodge 数对偶结果。
Homological mirror symmetry for weighted projective spaces and Morse Homotopy
本文将环面流形上的加权Morse同伦范畴与凝聚层导出范畴的同调镜像对称研究推广至环面轨形，讨论了加权射影空间的对应版本同调镜像对称。
The singular Hitchin fibration, cameral data, and representation theory
本文研究具有任意约化结构群的Higgs丛模栈上的Hitchin纤维化，利用Higgs场的中心化子分析其奇异轨迹，通过阿贝尔化纤维化分解Hitchin映射，并推广Donagi-Gaitsgory的相机数据描述纤维，最后应用于实群并揭示与李代数表示论的联系。
On the mirrors of low-degree del Pezzo surfaces
本文比较了度数 $d \leq 3$ 的 del Pezzo 曲面的不同镜像构造，通过提取 Lefschetz 纤维化并利用类曲面伪格理论与突变方法，证明了这些纤维化定义了范畴镜像，并建立了相关 Fukaya-Seidel 范畴的等价性。
We develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne-Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over $\mathbb{P}^1$, into other …
Gross and Siebert identified a class of singular Lagrangian torus fibrations which arise when smoothing toroidal degenerations, and which come in pairs that are related by mirror symmetry. We identify an immersed Lagrangian in each of these local models which supports a moduli space of objects that is isomorphic to the mirror space, and prove a homological mirror statement along the way.
In this article, we construct the Hitchin fibration for groups following the scheme outlined by Frenkel-Ngo in the case of SL_{2}. This construction uses as a decisive tool the Vinberg's semigroup. The total space of Hitchin is obtained by taking the fiber product of the Hecke stack with the diagonal of the stack of G-bundles $Bun_{G}$; we prove a transversality statement between the intersection complex of the Hecke stack and the diagonal of $Bun_{G}$, over a sufficiently big open subset, in order to get local applications, such that the fundamental lemma for the spherical Hecke algebra. Alon…
We construct a Lagrangian torus fibration on a smooth hypertoric variety and a corresponding SYZ mirror variety using $T$-duality and generating functions of open Gromov-Witten invariants. The variety is singular in general. We construct a resolution using the wall and chamber structure of the SYZ base.
We study fppf descent for enhanced derived categories. We revisit the work of [HS] and [TV08] in a lax context. More precisely, we construct a Cartesian and coCartesian fibration ${}^{\mathrm{op}}\mathscr D^+_S\rightarrow N(\mathop{\mathrm{Sch}}_S)$ whose fibre over an $S$-scheme $T$ is the opposite $\mathscr D^+(T)^{\mathrm{op}}$ of the quasicategory of bounded below complexes of $\mathscr O_T$-modules. We show that this fibration satisfies fppf-descent for schemes.
Mirror Symmetry for Quiver Algebroid Stacks
本文提出箭袋栈概念及其 $A_\infty$ 范畴表示理论，利用全局中间代理粘合不相交 Lagrangian，并在穿孔椭圆曲线中构造出镜像 nc 局部射影平面。
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