Jan Christian Rohde
In this note we provide examples of families of Calabi-Yau 3-manifolds over Shimura varieties, whose mirror families contain subfamilies over Shimura varieties. Therefore these original families and subfamilies on the mirror side contain dense sets of complex multiplication fibers. In view of the work of S. Gukov and C. Vafa this is of special interest in theoretical physics.
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This is a survey article on mirror symmetry and Fourier-Mukai partners of Calabi-Yau threefolds with Picard number one based on recent works by the authors [HoTa1,2,3,4]. For completeness, mirror symmetry and Fourier-Mukai partners of K3 surfaces are also discussed.
We show that complex multiplication on abelian varieties is equivalent to the existence of a constant rational Kähler metric. We give a sufficient condition for a mirror of an abelian variety of CM-type to be of CM-type as well. We also study the relationship between complex multiplication and rationality of a toroidal lattice vertex algebra.
In this note we present pairs of hyperkaehler orbifolds which satisfy two different versions of mirror symmetry. On the one hand, we show that their Hodge numbers (or more precisely, stringy E-polynomials) are equal. On the other hand, we show that they satisfy the prescription of Strominger, Yau, and Zaslow (which in the present case goes back to Bershadsky, Johansen, Sadov and Vafa): that a Calabi-Yau and its mirror should fiber over the same real manifold, with special Lagrangian fibers which are tori dual to each other. Our examples arise as moduli spaces of local systems on a curve with s…
Mirror partner for a Klein quartic polynomial
本文研究了Klein四次多项式与Landau-Ginzburg奇点的镜像关系，证明其镜像奇点为Klein四元群的Z/2Z扩张。
By normalizing the space of commuting pairs of elements in a reductive Lie group G, and the corresponding space for the Langlands dual group, we construct pairs of hyperkahler orbifolds which satisfy the conditions to be mirror partners in the sense of Strominger-Yau-Zaslow. The same holds true for commuting quadruples in a compact Lie group. The Hodge numbers of the mirror partners, or more precisely their orbifold E-polynomials, are shown to agree, as predicted by mirror symmetry. These polynomials are explicitly calculated when G is a quotient of SL(n).
An extension of polar duality of toric varieties and its consequences in Mirror Symmetry
本文将 Fano 环面簇的极对偶性扩展为 framed 对偶性，提出一种统一方法以生成任意 Kodaira 维数下环面簇超曲面与完全交集的镜像伙伴，并探讨了其与 Landau-Ginzburg 模型及同调、内蕴镜像对称的联系。
We first describe a canonical mirror partner (B-model) of the small quantum orbifold cohomology of weighted projective spaces (A-model) in the framework of differential equations: we attach to the A-model (resp. B-model) a D-module on the torus and we show that these two D-modules are isomorphic. This makes the A and B-models mirror partners and yields, in this situation, an explicit and finer version of a recent result of Iritani. Then we study, using the theory of the Kashiwara-Malgrange filtration, their degenerations at the origin and we apply our results to the construction of (classical,…
The moduli spaces of flat $\mathrm{SL}_2$- and $\mathrm{PGL}_2$-connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tamás Hausel in Remark 3.30 of "Global topology of the Hitchin system".
We show that under the hypotheses of Strominger, Yau and Zaslow's paper, a mirror partner of a K3 surface $X$ with a fibration in special Lagrangian tori can be obtained by rotating the complex structure of $X$ within its hyperkähler family of complex structures. The same hypotheses force the B-field to vanish.
We study a conjecture, due to Voisin, on 0-cycles on varieties with $p_g=1$. Using Kimura's finite dimensional motives and recent results of Vial's on the refined (Chow-)Künneth decomposition, we provide a general criterion for Calabi-Yau manifolds of dimension at most $5$ to verify Voisin's conjecture. We then check, using in most cases some cohomological computations on the mirror partners, that the criterion can be successfully applied to various examples in each dimension up to $5$.
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…
Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci in mirror moduli spaces, we prove homological mirror symmetry equivalences between them. Our construction uses the coherent-constructible correspondence and a recent result of Ganatra-Pardon-Shende t…
We consider smooth complete intersection Calabi-Yau 3-folds in minuscule Schubert varieties, and study their mirror symmetry by degenerating the ambient Schubert varieties to Hibi toric varieties. We list all possible Calabi-Yau 3-folds of this type up to deformation equivalences, and find a new example of smooth Calabi-Yau 3-folds of Picard number one; a complete intersection in a locally factorial Schubert variety ${\boldsymbolΣ}$ of the Cayley plane ${\mathbb{OP}}^2$. We calculate topological invariants and BPS numbers of this Calabi-Yau 3-fold and conjecture that it has a non-trivial Fouri…
Hall algebra multiplication for stable envelopes on bow varieties
针对椭圆稳定包络这一量子群表示几何实现的基本组件，给出A型切尔基斯弓型簇上椭圆稳定包络的霍尔代数乘积公式，并结合3维镜像对称性质推导出同行列和01矩阵对的theta函数恒等式。
Families of Calabi-Yau manifolds and mirror symmetry
本文从Calabi-Yau流形族与周期积分视角综述镜像对称，强调GKZ超几何级数的性质，并基于中心荷公式证明自反多胞形相关流形对的镜像对称，展示模空间全局描述带来的双有理几何与Fourier-Mukai配对例子。
We study generalized complex structures on K3 surfaces, in the sense of Hitchin. For each real parameter t between one and infinity we exhibit two families of generalized K3 surfaces, (M,cal{I}_{zeta}) and (M,cal{J}_{zeta}), parametrized by zeta in CP^1, which are Mukai dual for zeta=0 and infinity, amd mirror partners for zeta not equal to 0 and infinity. Moreover, the Fourier-Mukai equivalence D^b(M,cal{I}_0) -> D^b(M,cal{J}_0) induces an isomorphism phi_T between the spaces of first order deformations of (M,cal{I}_0) and (M,cal{J}_0) as generalized complex manifolds, and the deformations (M…
Framed duality and mirror symmetry for toric complete intersections
本文将完全环面簇中的超曲面族$f$-镜像对称系统扩展至完全交子簇族，基于分次对偶发展了相关理论，并探讨了广义Landau-Ginzburg/完全交对应。
Mirror symmetry for singular double cover Calabi--Yau varieties: quantum test
本文研究奇异Calabi--Yau双重覆盖簇的镜像对，证明其B模型与A模型等价，并推广至nef-分割平凡情形，验证了Morrison猜想。
It is frequently possible to produce new Calabi-Yau threefolds from old ones by a process of allowing the complex structure to degenerate to a singular one, and then performing a resolution of singularities. (Some care is needed to ensure that the Calabi-Yau condition be preserved.) There has been speculation that all Calabi-Yau threefolds could be linked in this way, and considerable evidence has been amassed in this direction. We propose here a natural way to relate this construction to the string-theoretic phenomenon known as ``mirror symmetry.'' We formulate a conjecture which in principle…
The Last Picard Rank 1 Double--Mirror Calabi--Yau Pair?
本文证明两个Picard秩1的Calabi-Yau三维簇族（$\mathcal{X}$与$\mathcal{Y}$）之间存在导出等价，通过数学规范线性sigma模型建立联系，解决了Miura关于Fourier-Mukai伙伴的猜想，并指出这可能是最后的双镜像对。
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