K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space III: the case r(M)\geq18

Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, II: the singularity of the invariant (with an Appendix by Ken-Ichi Yoshikawa)

We study the boundary behavior of the invariant of $K3^{[2]}$-type manifolds with antisymplectic involution, which we obtained using equivariant analytic torsion. We show the algebraicity of the singularity of the invariant by using the asymptotic of equivariant Quillen metrics and equivariant $L^2$-metrics. We prove that, in some cases, the invariant coincides with Yoshikawa's invariant for 2-elementary K3 surfaces. Hence, in these cases, our invariant is expressed as the Petersson norm of a Borcherds product and a Siegel modular form.

K3曲面上的对合与导出等价

Involutions on K3 surfaces and derived equivalence

本文研究K3曲面上对合在导出等价共轭下的行为，并将其应用于等变双有理几何。

Faisceaux equivariants sur $\mathbb{P}^1$ et faisceaux automorphes

Inspired by [Pan21], we define and study a functor which associates to equivariant sheaves on $\mathbb{P}^1$ automorphic sheaves on modular curves.

On the equivariant blow-Nash classification of simple invariant Nash germs

We make progress towards the classification of simple Nash germs invariant under the involution changing the sign of the first coordinate, with respect to equivariant blow-Nash equivalence, which is an equivariant Nash version of blow-analytic equivalence, taking advantage of invariants for this relation, the equivariant zeta functions.

Igusa quartic and Borcherds products

By applying Borcherds' theory of automorphic forms on bounded symmetric domains of type IV, we give a 5-dimensional linear system of automorphic forms of weight 6 on Igusa quartic 3-fold which induces an S_6-equivariant rational map of degree 16 from Igusa quartic to Segre cubic.

Associated forms of binary quartics and ternary cubics

Let ${\mathcal Q}_n^d$ be the vector space of forms of degree $d\ge 3$ on ${\mathbb C}^n$, with $n\ge 2$. The object of our study is the map $Φ$, introduced in papers [EI], [AI1], that assigns every nondegenerate form in ${\mathcal Q}_n^d$ the so-called associated form, which is an element of ${\mathcal Q}_n^{n(d-2)*}$. We focus on two cases: those of binary quartics ($n=2$, $d=4$) and ternary cubics ($n=3$, $d=3$). In these situations the map $Φ$ induces a rational equivariant involution on the projectivized space ${\mathbb P}({\mathcal Q}_n^d)$, which is in fact the only nontrivial rational …

仿射二元二面体卡拉比 - 丘三流形的对合等变拓扑递归与镜像对称

Involution-equivariant topological recursion and mirror symmetry for the affine binary dihedral Calabi--Yau threefold

研究仿射二元二面体卡拉比 - 丘三流形的闭弦重塑，通过对合等变拓扑递归等方法，将其等变orbifold量子上同调弗罗贝尼乌斯流形与托达曲线结构等同，证明相关$R$矩阵等式，还建立了递归与格罗莫夫 - 威滕生成函数及自由能的联系。

On an invariant bilinear form on the space of automorphic forms via asymptotics

This article concerns the study of a new invariant bilinear form $\mathcal B$ on the space of automorphic forms of a split reductive group $G$ over a function field. We define $\mathcal B$ using the asymptotics maps from Bezrukavnikov-Kazhdan and Sakellaridis-Venkatesh, which involve the geometry of the wonderful compactification of $G$. We show that $\mathcal B$ is naturally related to miraculous duality in the geometric Langlands program through the functions-sheaves dictionary. In the proof, we highlight the connection between the classical non-Archimedean Gindikin-Karpelevich formula and c…

Elliptic canonical bases for toric hyper-Kahler manifolds

Lusztig defined certain involutions on the equivariant K-theory of Slodowy varieties and gave a characterization of certain bases called canonical bases. In this paper, we give a conjectural generalization of these involutions and K-theoretic canonical bases to conical symplectic resolutions which have good Hamiltonian torus actions and state several conjectures related to them which we check for toric hyper-Kahler manifolds. We also propose an elliptic analogue of these bar involutions. As a verification of our proposal, we explicitly construct elliptic lifts of K-theoretic canonical bases an…

通过等变库兹涅佐夫分量研究非埃卡特型三次三重体的范畴托雷利定理的三种途径

Three approaches to a categorical Torelli theorem for cubic threefolds of non-Eckardt type via the equivariant Kuznetsov components

本文针对非埃卡特型三次三重体，通过非交换霍奇理论、布里奇兰模空间理论和周环理论三种途径，证明了库兹涅佐夫分量的等变范畴可决定原三次三重体的同构类，并进一步建立了等变无穷小范畴托雷利定理。

经典群在ALE空间上的瞬子、箭图簇上的对合与量子对称对

Instantons on ALE spaces for classical groups, involutions on quiver varieties, and quantum symmetric pairs

本文研究经典群在ALE空间上的瞬子模空间（即箭图簇上对合的不动点集），通过稳定包络构造扭曲Yangian表示，计算K-矩阵实例并明确良定义条件。

On Hermite's invariant for binary quintics

The Hermite invariant H is the defining equation for the hypersurface of binary quintics in involution. This paper analyses the geometry and invariant theory of H. We determine the singular locus of this hypersurface and show that it is a complete intersection of a linear covariant of quintics. The projective dual of this hypersurface can be identified with itself via an involution. It is shown that the Jacobian ideal of H is perfect of height two, and we describe its SL_2-equivariant minimal resolution. The last section develops a general formalism for evectants of covariants of binary forms,…

On the canonical real structure on wonderful varieties

We study equivariant real structures on spherical varieties. We call such a structure canonical if it is equivariant with respect to the involution defining the split real form of the acting reductive group G. We prove the existence and uniqueness of a canonical structure for homogeneous spherical varieties G/H with H self-normalizing and for their wonderful embeddings. For a strict wonderful variety we give an estimate of the number of real form orbits on the set of real points.

仿射格拉斯曼流形上可构造层的E₃融合-卷积乘积模型

A model for the E3 fusion-convolution product of constructible sheaves on the affine Grassmannian

本文将复简约群球面Hecke范畴的卷积乘积提升为∞范畴中的左t正合E₃-幺半结构，利用Beilinson–Drinfeld格拉斯曼流形等工具完成内蕴自守侧的构造。

K3-surfaces with special symmetry

In this dissertation classification problems for K3-surfaces with finite group actions are considered. Special emphasis is put on K3-surfaces with antisymplectic involutions and compatible actions of symplectic transformations. Given a finite group $G$ it is investigated if it can act in a compatible fashion on a K3-surfaces $X$ with antisymplectic involution and to what extend the action of $G$ determines the geometry of $X$. If $G$ has large order or rich structure a precise description of $X$ as a branched double cover of a Del Pezzo surface is obtained. In particular, this leads to classif…

K3-surfaces with special symmetry: An example of classification by Mori-reduction

The classification problem for K3-surfaces equipped with finite groups $H$ of symplectic symmetry centralized by an antisymplectic involution is considered. An approach via equivariant Mori-reduction is employed. This method, which has proved to be successful even for rather small groups, is exemplified here by giving a complete classification in the case $H = C_3 \ltimes C_7$. The consideration of this particular group is related to the study of K3-surfaces with maximal finite groups of symplectic automorphisms. Applications to the case $L_2(7)$ are given.

关于K3表面上的两个Enriques范畴家族

On two families of Enriques categories over K3 surfaces

本文研究K3表面上的两个Enriques范畴家族的模空间，涉及四次双曲面和特殊Gushel-Mukai三维流形的构造，恢复经典几何构造，并描述模空间的奇异位点和O'Grady超Kähler十维空间的birational自映射。

Notes on genus one real Gromov-Witten invariants

In this paper, we propose a definition of genus one real Gromov-Witten invariants for certain symplectic manifolds with real a structure, including Calabi-Yau threefolds, and use equivariant localization to calculate certain genus one real invariants of the projective space. For this definition, we combine three moduli spaces corresponding to three possible types of involutions on a symplectic torus, by gluing them along common boundaries, to get a moduli space without codimension-one boundary and then study orientation of the total space. Modulo a technical conjectural lemma, we can prove tha…

Topology and geometry of the canonical action of $T^4$ on the complex Grassmannian $G_{4,2}$ and the complex projective space $CP^{5}$

We consider the canonical action of the compact torus $T^4$ on the Grassmann manifold $G_{4,2}$ and prove that the orbit space $G_{4,2}/T^4$ is homeomorphic to the sphere $S^5$. We prove that the induced differentiable structure on $S^5$ is not the smooth one and describe the smooth and the singular points. We also consider the action of $T^4$ on $CP^5$ induced by the composition of the second symmetric power $T^4\subset T^6$ and the standard action of $T^6$ on $CP^5$ and prove that the orbit space $CP^5/T^4$ is homeomorphic to the join $CP^2\ast S^2$. The Plücker embedding $G_{4,2}\subset CP^…