Tautological equations in genus 2 via invariance conjectures

Relations in the Tautological Ring of the Universal Curve

We bound the dimensions of the graded pieces of the tautological ring of the universal curve from below for genus up to 27 and from above for genus up to 9. As a consequence we obtain the precise structure of the tautological ring of the universal curve for genus up to 9. In particular, we see that it is Gorenstein for these genera.

The tautological ring of the space of pointed genus two curves of compact type

We prove that the tautological ring of $M_{2,n}^{ct}$, the moduli space of n-pointed genus two curves of compact type, does not have Poincaré duality for any $n \geq 8$. This result is obtained via a more general study of the cohomology groups of $M_{2,n}^{ct}$. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^k(M_{2,n}^{ct})$ for any $k$ and $n$ considered both as $S_n$-representation and as mixed Hodge structure/$\ell$-ad…

Intersections of Tautological Classes on Blowups of Moduli Spaces of Genus-One Curves

We give two recursions for computing top intersections of tautological classes on blowups of moduli spaces of genus-one curves. One of these recursions is analogous to the well-known string equation. As shown in previous papers, these numbers are useful for computing genus-one enumerative invariants of projective spaces and Gromov-Witten invariants of complete intersections.

Non-tautological Hurwitz cycles

We show that various loci of stable curves of sufficiently large genus admitting degree $d$ covers of positive genus curves define non-tautological algebraic cycles on $\overline{\mathcal{M}}_{g,N}$, assuming the non-vanishing of the $d$-th Fourier coefficient of a certain modular form. Our results build on those of Graber-Pandharipande and van Zelm for degree 2 covers of elliptic curves; the main new ingredient is a method to intersect the cycles in question with boundary strata, as developed recently by Schmitt-van Zelm and the author.

The tautological rings of the moduli spaces of stable maps

We show that the rational cohomology of the genus zero stable map spaces to SL flag varieties is entirely tautological.

The tautological ring of $M_{1,n}^{ct}$

We describe the tautological ring of the moduli space of $n$-pointed curves of genus one of compact type. It is proven that it is a Gorenstein algebra.

On the tautological ring of $M_{g,n}$

We state and check the analogue of Faber's conjectures for the tautological ring of the moduli spaces $M_{g,n}$ of $n$-pointed smooth curves of genus $g$.

Tautological classes with twisted coefficients

Let $M_g$ be the moduli space of smooth genus $g$ curves. We define a notion of Chow groups of $M_g$ with coefficients in a representation of $Sp(2g)$, and we define a subgroup of tautological classes in these Chow groups with twisted coefficients. Studying the tautological groups of $M_g$ with twisted coefficients is equivalent to studying the tautological rings of all fibered powers $C_g^n$ of the universal curve $C_g \to M_g$ simultaneously. By taking the direct sum over all irreducible representations of the symplectic group in fixed genus, one obtains the structure of a twisted commutativ…

Witten's conjecture, Virasoro conjecture, and invariance of tautological equations

The main goal of this paper is to prove the following two conjectures for genus up to two: 1. Witten's conjecture on the relations between higher spin curves and Gelfand--Dickey hierarchy. 2. Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main technique used in the proof is the invariance of tautological equations under loop group action.

On the tautological ring of Humbert curves

We exhibit a 2-dimensional family of non-hyperelliptic curves of genus 5, called Humbert curves, for which the tautological ring injects into cohomology. In particular, Humbert curves have a multiplicative Chow-Künneth decomposition (in the sense of Shen-Vial), and their Ceresa cycle is torsion.

The Chow ring of the moduli space of curves of genus 6

We determine the Chow ring (with Q-coefficients) of M_6 by showing that all Chow classes are tautological. In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus 6 case relies on the particularly beautiful Brill-Noether theory in this case, and in particular on a rank 5 vector bundle "relativizing" a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the …

Tautological Cycles on Jacobian Varieties

Let C be a complex curve of genus g, let J(C) be its Jacobian and let R(C) be its tautological ring, that is, the group of algebraic cycles modulo algebraic equivalence. We study the algebraic structure of R(C). In particular, we give a detailed description of all the possibilities that may occur for g<9: we construct convenient basis and we determine the matrices representing the Fourier transform and both intersection and Pontryagin products explicitly. In particular, we estimate the dimension of R(C).

Intersection Numbers on the Moduli Spaces of Stable Maps in Genus 0

Let $V$ be a smooth, projective, convex variety. We define tautological $ψ$ and $κ$ classes on the moduli space of stable maps $\M_{0,n}(V)$, give a (graphical) presentation for these classes in terms of boundary strata, derive differential equations for the generating functions of the Gromov-Witten invariants of $V$ twisted by these tautological classes, and prove that these intersection numbers are completely determined by the Gromov-Witten invariants of $V$. This results in families of Frobenius manifold structures on the cohomology ring of $V$ which includes the quantum cohomology as a spe…

$\mathcal{S}_5^-$ 的 Chow 环是重言的

The Chow ring of $\mathcal{S}_5^-$ is tautological

本文证明了亏格5的奇自旋曲线模空间$\mathcal{S}_5^-$的Chow环是重言的，通过分析典范亏格5曲线和全切超平面的几何性质，并证明了$\mathcal{M}_{5,4}$中支配$\mathcal{S}_5^-$的微分层的理性。

A conjectural connection between R^*(C_g^n) and R^*(M_{g,n}^rt)

Let M_{g,n}^rt be the moduli space of stable n-pointed curves of genus g>1 with rational tails. We also consider the space C_g^n classifying smooth curves of genus g with not necessarily distinct n ordered points. There is a natural proper map from M_{g,n}^rt to C_g^n which contracts all rational components. Tautological classes on these spaces are natural algebraic cycles reflecting the geometry of curves. In this short note we study the connection between tautological classes on M_{g,n}^rt and C_g^n. We show that there is a natural filtration on the tautological ring of M_{g,n}^rt consisting…

Tautological stable pair invariants of Calabi-Yau 4-folds

Let $X$ be a Calabi-Yau 4-fold and $D$ a smooth divisor on it. We consider tautological complex associated with $L=\mathcal{O}_X(D)$ on the moduli space of Le Potier stable pairs and define its counting invariant by integrating the Euler class against the virtual class. We conjecture a formula for their generating series expressed using genus zero Gopakumar-Vafa invariants of $D$ and genus one Gopakumar-Vafa type invariants of $X$, which we verify in several examples. When $X$ is the local resolved conifold, our conjecture reproduces a conjectural formula of Cao-Kool-Monavari in the PT chamber…

$\overline{\mathcal{M}}_{g,n}$ 的环式环很少是 Gorenstein 环

The tautological ring of $\overline{\mathcal{M}}_{g,n}$ is rarely Gorenstein

本文证明模空间 $\overline{\mathcal{M}}_{g,n}$ 的环式环在 $g\geq 2$ 且 $2g+n\geq 24$ 时非 Gorenstein，推广了 Petersen 和 Tommasi 的结果，并猜想其逆命题成立，且证明了若干新情形。

二重水平的代数拓扑模形式

Tautological modular forms of level two and degree two

通过构造特定的向量值模形式，研究二重水平的Siegel模形式，并揭示其与双曲曲线模空间的紧密联系。

The kappa ring of the moduli of curves of compact type: II

The subalgebra of the tautological ring of the moduli of curves of compact type generated by the kappa classes is studied. Relations, constructed via the virtual geometry of the moduli of stable maps, are used to prove universality results relating the kappa rings in genus 0 to higher genus. Predictions for kappa classes of the Gorenstein conjecture are proven.

族上阿贝尔簇的格罗莫夫-维滕理论与模形式

Gromov-Witten theory of abelian varieties in families and modular forms

本文为系列研究首篇，将椭圆曲线格罗莫夫-维滕理论拟模性推广至高维，经重言投影后证明亏格2情形的相关猜想，引入商格罗莫夫-维滕不变量并给出其显式公式。

二阶向量丛在双曲曲线上的 universal moduli 堆叠中的循环

Cycles in the universal moduli stack of bundles of rank two over genus two curves

本文提出并证明了双曲曲线上的 universal moduli 堆叠的 Chow 环猜想，确定了生成元和关系，并计算了 genus 二曲线的 universal Jacobians 乘积的 Chow 环。

Tautological relations and the r-spin Witten conjecture

In a series of two preprints, Y.-P. Lee studied relations satisfied by all formal Gromov-Witten potentials, as defined by A. Givental. He called them "universal relations" and studied their connection with tautological relations in the cohomology ring of moduli spaces of stable curves. Building on Y.-P. Lee's work, we give a simple proof of the fact that every tautological relation gives rise to a universal relation (which was also proved by Y.-P. Lee modulo certain results announced by C. Teleman). In particular, this implies that in any semi-simple Gromov-Witten theory where arbitrary correl…

DR/DZ equivalence conjecture and tautological relations

In this paper we present a family of conjectural relations in the tautological ring of the moduli spaces of stable curves which implies the strong double ramification/Dubrovin-Zhang equivalence conjecture. Our tautological relations have the form of an equality between two different families of tautological classes, only one of which involves the double ramification cycle. We prove that both families behave the same way upon pullback and pushforward with respect to forgetting a marked point. We also prove that our conjectural relations are true in genus $0$ and $1$ and also when first pushed f…

关于 genus 11 K3 表面的 O'Grady 广义 Franchetta 猜想

On O'Grady's generalized Franchetta conjecture for genus 11 K3 surfaces

本文通过 Mukai 程序和 tautological 生成结果，证实了 genus 11 K3 表面的广义 Franchetta 猜想。

亏格 2 曲线上半稳定秩 2 丛的泛模空间的有理 Chow 环

Rational Chow ring of the universal moduli space of semistable rank two bundles over genus two curves

本文确定亏格 2 曲线上秩 2 半稳定丛泛模空间的有理 Chow 环，证明其由重言类生成，并顺带得到相关泛 Jacobian 的 Chow 环。

椭圆亏格在墙交叉下的不变性

Invariance of elliptic genus under wall-crossing

该研究提出一个拓扑判据，证明利用主空间得到的各类椭圆亏格墙交叉公式具有平凡性，并将其应用于GIT商、层模空间、Donaldson-Thomas及Vafa-Witten理论等场景。

The kappa ring of the moduli of curves of compact type: I

The subalgebra of the tautological ring of the moduli of curves of compact type generated by the kappa classes is studied in all genera. Relations, constructed via the virtual geometry of the moduli of stable quotients, are used to obtain minimal sets of generators. Bases and Betti numbers of the kappa rings are computed. A universality property relating the higher genus kappa rings to the genus 0 rings is stated and proved in a sequel. The lambda_g formula for Hodge integrals arises as the simplest consequence.

亏格二稳定映射、局部方程与模消解

Genus Two Stable Maps, Local Equations and Modular Resolutions

本研究通过对参数化亏格二预稳定对的阿廷叠进行模胀开构造，局部对角化相关稳定映射模空间的导出对象，消解主分支奇点并使全空间仅含正规交叉奇点，方法有望推广至高亏格。

$\u001e{mathcal{M}}_{g,n}$中 tautological 闭链的 Schur-Weyl 对偶与高阶 Abel-Jacobi 不变量

Schur-Weyl Duality and Higher Abel-Jacobi Invariants for Tautological Cycles in $\mathcal{M}_{g,n}$

本文研究带标记点的亏格g曲线模空间的Hodge理论，构建表示论框架建立辛李代数的Schur-Weyl对偶与高阶Abel-Jacobi不变量的联系，引入高阶Faber-Pandharipande闭链并证明其非挠性与滤过位置，首次搭建两类不变量的系统关联。

关于Buryak-Shadrin猜想的一个 tautological 关系

On A Tautological Relation Conjectured By Buryak-Shadrin

本文研究Buryak-Shadrin提出的曲线模空间tautological关系猜想，证明只需验证m=2且|d|=2g+1的情形即可推得猜想成立，将证明简化为对每个亏格g验证有限个情形，并证明了g=1时猜想成立。

Topological Recursion Relations on $\bar{\cal M}_{3,2}$

In this paper, we give some new genus-3 universal equations for Gromov-Witten invariants of compact symplectic manifolds. These equations were obtained by studying new relations in the tautological ring of the moduli space of 2-pointed genus-3 stable curves. A byproduct of our search for genus-3 equations is a new genus-2 universal equation for Gromov-Witten invariants.

Hitchin fibrations, abelian surfaces, and the P=W conjecture

We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus $2$ curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a…

重言相交数与顺序连续划分序列

Tautological Intersection Numbers and Order-Consecutive Partition Sequences

本文证明在特定参数下稳定曲线模空间重言相交数对应多项式的 $f^*$-向量与 $h^*$-向量具有顺序连续划分序列等枚举解释，并提出并验证了 $f^*$-向量的对数凹性猜想。

On the birational invariance of the arithmetic genus and Euler characteristic

The aim of this note is to use elementary methods to give a large class of examples of projective varieties $ Y \subseteq \mathbb{P}^d_k$ over a field $k$ with the property that $Y$ is not isomorphic to a hypersurface $H\subseteq \mathbb{P}^N_k$ in projective space $\mathbb{P}^N_k$ with $N:=dim(Y)+1$. We apply this construction to the study of the arithmetic genus $p_a(Y)$ of $Y$ and the problem of determining if $p_a(Y)$ is a birational invariant of $Y$ in general. We give an infinite number of examples of pairs of projective varieties $(Y, Y')$ in any dimension $dim(Y)=dim(Y')\geq 4$ where $…

Curves on K3 surfaces in divisibility two

We prove a conjecture of Maulik, Pandharipande, and Thomas expressing the Gromov--Witten invariants of K3 surfaces for divisibility two curve classes in all genus in terms of weakly holomorphic quasimodular forms of level two. Then, we establish the holomorphic anomaly equation in divisibility two in all genus. Our approach involves a refined boundary induction, relying on the top tautological group of the moduli space of smooth curves, together with a degeneration formula for reduced virtual fundamental class with imprimitive curve classes. We use the double ramification relations with target…

Invariance property of orbifold elliptic genus for multi-fans

Multi-fan is an analogous notion of fan. As a fan is associated to a toric variety a multi-fan is associated to a torus orbifold. Orbifold elliptic class and orbifold elliptic genus are defined for a triple of a multi-fan, a set of generating integral vectors of one dimensional cones and a $\mathbb{Q}$ divisor. They are shown to behave functorially with respect to birational morphisms between these triples. The result may be considered as a combinatorial or topological analogue of the main result of Borisov and Libgober, McKay correspondence for elliptic genera, Ann. of Math., 161 (2005).