Y. -P. Lee
The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the framework gives an efficient algorithm to calculate all tautological equations using only finite dimensional linear algebra. Other applications are also indicated.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We verify the Invariance Conjectures of tautological equations in genus two. In particular, a uniform derivation of all known genus two equations is given.
A new tautological equation of $\Mbar_{3,1}$ in codimension 3 is derived and proved, using the invariance condition explained in earlier works.
The aim of Part II is to explore the technique of invariance of tautological equations in the realm of Gromov--Witten theory. The main result is a proof of Invariance Theorem (Invariance Conjecture~1 in [14]), via the techniques from Gromov--Witten theory. It establishes some general inductive structure of the tautological rings, and provides a new tool to the study of this area.
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.
Let $N$ be the smallest integer such that there is a non-tautological cohomology class of even degree on $\mathcal{\bar M}_{2,N}$. We remark that there is such a non-tautological class on $\mathcal{\bar M}_{2,20}$, by work of Graber and Pandharipande. We show that $\mathcal{\bar M}_{2,N}$ has non-tautological cohomology only in one degree, which is not the middle degree. In particular, it follows that the tautological ring of $\mathcal{\bar M}_{2,N}$ is not Gorenstein. We present some evidence suggesting that N=20 holds.
We propose a conjecture on the generating series of Chern numbers of tautological bundles on Hilbert schemes of points on curves and establish the rank 1 and rank -1 case of this conjecture. Thus we compute explicitly the generating series of integrals of the Segre classes of tautological bundles of line bundles on curves, which has a similar structure as Lehn's conjecture for surfaces.
The purpose of this paper is to formulate a number of conjectures giving a rather complete description of the tautological ring of M_g and to discuss the evidence for these conjectures.
In this paper, we discuss the cycle theory on moduli spaces $\cF_h$ of $h$-polarized hyperkähler manifolds. Firstly, we construct the tautological ring on $\cF_h$ following the work of Marian, Oprea and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces. We study the tautological classes in cohomology groups and prove that most of them are linear combinations of Noether-Lefschetz cycle classes. In particular, we prove the cohomological version of the tautological conjecture on moduli space of K3$^{[n]}$-type hyperkähler manifolds with $n\leq 2$. Secondly, we prove the…
We discuss analogs of Faber's conjecture for two nested sequences of partial compactifications of the moduli space of smooth curves. We show that their tautological rings are one-dimensional in top degree but do not satisfy Poincare duality.
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…
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 $g$.
We prove that the dimension 0 part of the tautological ring of the moduli space of stable pointed curves is one-dimensional. This provides the first genus-free evidence for a conjecture of Faber and Pandharipande that the tautological ring of the moduli space is Gorenstein, answering in the affirmative a question of Hain and Looijenga.
An inclusion-exclusion principle for tautological sheaves on Hilbert schemes of points
本研究证明双点退化纤维上点希尔伯特概型tautological层的欧拉特征满足容斥型等式，推导出光滑真代数空间上对应欧拉特征的通用多项式存在性，并将Zhou的相关猜想约简至射影空间乘积情形，主要工具为优态退化与代数配边。
Using a simple geometric argument, we obtain an infinite family of nontrivial relations in the tautological ring of $M_g$ (and in fact that of $M_{g,2}$). One immediate consequence of these relations is that the classes $κ_1,...,κ_{[g/3]}$ generate the tautological ring of $M_g$, which has been conjectured by Faber, and recently proven at the level of {\em cohomology} by Morita.
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。
We develop a new method to study intersection theory of the main component of the Hilbert scheme of points on complex manifolds. The main result is an iterated residue formula for tautological integrals. We formulate a Chern-Segre-type positivity conjecture for these integrals, and we introduce higher dimensional Segre and Chern integrals.
The tautological ring of $M_{25}$ is not Gorenstein
本文证明了模空间 $M_{25}$ 的 tautological 环非 Gorenstein，推翻 Faber 猜想，并确认 Faber-Zagier 关系为完整关系列表。
The Gorenstein property and Pixton's conjecture for compact type moduli
该研究证明了紧致型模空间$\u039c_{g,n}^{\mathrm{ct}}$在特定参数下重言环非Gorenstein，并验证了Pixton猜想的多个新案例，相关结果是主极化阿贝尔簇模空间非重言闭链研究的关键基础。
We study the interplay of the moduli of curves and the moduli of K3 surfaces via the virtual class of the moduli spaces of stable maps. Using Getzler's relation in genus 1, we construct a universal decomposition of the diagonal in Chow in the third fiber product of the universal K3 surface. The decomposition has terms supported on Noether-Lefschetz loci which are not visible in the Beauville-Voisin decomposition for a fixed K3 surface. As a result of our universal decomposition, we prove the conjecture of Marian-Oprea-Pandharipande: the full tautological ring of the moduli space of K3 surfaces…
We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on surfaces within the framework of Nakajima's oscillator algebra. This leads to an identification of the cohomology ring of Hilbert schemes of the affine plane with a ring of differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilb^n up to n=7, and give a conjecture for the generating series.
Gromov-Witten theory, degenerations, and the tautological ring
该研究开发新的退化技术，结合对数GW理论的重构定理，证明了几类代数簇的GW闭链的tautological性与退化相关猜想，为GW理论与tautological环的关系提供关键证据。
We give a new method of producing relations in the tautological ring R(M_{g, 1}), using the sl_2-action on the Chow ring of the universal Jacobian. With these relations, we prove that R(M_{g, 1}) is generated by κ_i for i no greater than g/3, together with ψ. Our computation shows that Faber's conjectures for M_{g, 1} are true for g up to 19. Further, by pushing relations forward to M_g, we obtain a new proof of Faber's conjectures (for M_g) for g up to 23. For g = 24, our method recovers all the Faber-Zagier relations. We also give an algebraic proof of an identity of Morita.
Recently, Oh and Thomas constructed algebraic virtual cycles for moduli spaces of sheaves on Calabi-Yau 4-folds. The purpose of this paper is to provide a virtual pullback formula between these Oh-Thomas virtual cycles. We find a natural compatibility condition between 3-term symmetric obstruction theories that induces a virtual pullback formula. There are two types of applications. Firstly, we introduce a Lefschetz principle in Donaldson-Thomas theory, which relates the tautological DT4 invariants of a Calabi-Yau 4-fold with the DT3 invariants of its divisor. As corollaries, we prove the Cao-…
Looijenga recently proved that the tautological ring of M_g vanishes in degree d>g-2 and is at most one-dimensional in degree g-2, generated by the class of the hyperelliptic locus. Here we show that K_{g-2} is non-zero on M_g. The proof uses the Witten conjecture, proven by Kontsevich. With similar methods, we expect to be able to prove some (possibly all) of the identities in degree g-2 in the tautological ring that are part of the author's conjectural explicit description of the ring.
We study tautological classes on the moduli space of stable $n$-pointed hyperelliptic curves of genus $g$ with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invarian…
We connect two notions of tautological ring: one for the moduli space of curves (after Mumford, Faber, etc.), and the other for the Jacobian of a curve (after Beauville, Polishchuk, etc.). The motivic Lefschetz decomposition on the Jacobian side produces relations between tautological classes, leading to results about Faber's Gorenstein conjecture on the curve side. We also relate certain Gorenstein properties on both sides and verify them for small genera. Further, we raise the question whether all tautological relations are motivic, giving a possible explanation why the Gorenstein properties…
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…
Let $g \geq 2$. Let $M_{g,n}^{rt}$ be the moduli space of $n$-pointed genus $g$ curves with rational tails. Let $C_g^n$ be the $n$-fold fibered power of the universal curve over $M_g$. We prove that the tautological ring of $M_{g,n}^{rt}$ has Poincaré duality if and only if the same holds for the tautological ring of $C_g^n$. We also obtain a presentation of the tautological ring of $M_{g,n}^{rt}$ as an algebra over the tautological ring of $C_g^n$. This proves a conjecture of Tavakol. Our results are valid in the more general setting of wonderful compactifications.
We describe a new perspective on the intersection theory of the moduli space of curves involving both Virasoro constraints and Gorenstein conditions. The main result of the paper is the computation of a basic 1-point Hodge integral series occurring in the tautological ring of the moduli space of nonsingular curves. In the appendix by D. Zagier, "Polynomials arising from the tautological ring", a detailed study is made of certain polynomials whose coefficients are intersection numbers on moduli space. The paper and the appendix together provide proofs of all previously conjectured formulas for …
The virtual geometry of the moduli space of stable quotients is used to obtain Chow relations among the kappa classes on the moduli space of nonsingular genus g curves. In a series of steps, the stable quotient relations are rewritten in successively simpler forms. The final result is the proof of the Faber-Zagier relations (conjectured in 2000).
Logarithmic tautological rings of the moduli spaces of curves
本文定义曲线模空间的对数重言环，借助同调分段多项式与锥叠组合完成亏格0计算，并将标准重言环关系及 Pixton 猜想提升至对数情形。
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。
Witten's class on the moduli space of 3-spin curves defines a (non-semisimple) cohomological field theory. After a canonical modification, we construct an associated semisimple CohFT with a non-trivial vanishing property obtained from the homogeneity of Witten's class. Using the classification of semisimple CohFTs by Givental-Teleman, we derive two main results. The first is an explicit formula in the tautological ring of Mbar_{g,n} for Witten's class. The second, using the vanishing property, is the construction of relations in the tautological ring of Mbar_{g,n}. Pixton has previously conjec…
Tautological and non-tautological cycles on the moduli space of abelian varieties
本文通过Torelli拉回研究阿贝尔簇模空间的重言环，证明$[\mathcal{A}_1\times \mathcal{A}_5]$在$\mathcal{A}_6$中非重言，并利用Gorenstein核给出几何解释，同时探讨Noether-Lefschetz轨迹的推广。
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…
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…
Independence of tautological classes and cohomological stability for strata of differentials
本文研究了微分分层的tautological环生成关系及上同调稳定性，证明了在特定条件下不存在低次关系，并推测在大g时tautological环非平凡。
Relations among tautological classes on the moduli space of stable curves are obtained via the study of Witten's r-spin theory for higher r. In order to calculate the quantum product, a new formula relating the r-spin correlators in genus 0 to the representation theory of sl2 is proven. The Givental-Teleman classification of CohFTs is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the R-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten's r-spin class is obtained (along with tautolo…
We show how the "finite Quot scheme method" applied to Le Potier's strange duality on del Pezzo surfaces leads to conjectures (valid for all smooth complex projective surfaces) relating two sets of universal power series on Hilbert schemes of points on surfaces: those for top chern classes of tautological sheaves, and those for Euler characteristics of line bundles. We have verified these predictions computationally for low order. We then give an analysis of these conjectures in small ranks. We also give a combinatorial proof of a formula predicted by our conjectures: the top chern class of th…
We describe a theory of logarithmic Chow rings and tautological subrings for logarithmically smooth algebraic stacks, via a generalisation of the notion of piecewise-polynomial functions. Using this machinery we prove that the double-double ramification cycle lies in the tautological subring of the (classical) Chow ring of the moduli space of curves, and that the logarithmic double ramification cycle is divisorial (as conjectured by Molcho, Pandharipande, and Schmitt).
Hodge-Chern classes and strata-effectivity in tautological rings
本文研究zip周期映射下重言环中类的层效应性，提出Griffiths-Hodge丛的Chern类均为层有效的猜想，推广了Ekedahl-van der Geer关于Hodge向量丛的结果，并证明了若干实例。
After recalling the various tautological algebras of the moduli space of curves and some of its partial compactifications and stating several well-known results and conjectures concerning these algebras, we prove that the natural extension to the case of pointed curves of a 1996 conjecture of Hain and Looijenga is true if and only if two of the stated conjectures are true.
The McKay Correspondence for Dihedral Groups: The Moduli Space and the Tautological Bundles
本文证实了关于二面体群（作为复反射群）的McKay对应猜想，即解消与模空间同构当且仅当被极大解消支配，并推广了McKay对应。
We verify a formula on the solution rank of the tautological system arising from ample complete intersections in a projective homogeneous space of a semisimple group conjectured by Huang--Lian--Yau--Yu arXiv:1801.01194. As an application, we prove the existence of the rank one point for such a system, where mirror symmetry is expected.
We prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert schemes of points of a K3 surface X. We derive relations among the Segre classes via equivariant localization of the virtual fundamental classes of Quot schemes on X. The resulting recursions are then solved explicitly. The formula proves the K-trivial case of a conjecture of M. Lehn from 1999. The relations determining the Segre classes fit into a much wider theory. By localizing the virtual classes of certain relative Quot schemes on surfaces, we obtain new systems of relations among…
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 product identities and the Chow rings of holomorphic symplectic varieties
本文针对K3曲面上稳定层模空间提出推广Beauville-Voisin恒等式的猜想，并证明当模空间为Hilbert概形时猜想成立，同时探讨其对tautological子环结构的影响。
We introduce a new family of tautological relations of the moduli space of stable curves of genus $g$. These relations are obtained by computing the Poincaré-dual class of empty loci in the Hodge bundle. We use these relations to obtain a new expression for the Chern classes of the Hodge bundle. We prove that the $(g-i)$th class can be expressed as a linear combination of tautological classes involving only stable graphs with at most $i$ loops. In particular the top Chern class may be expressed with trees. This property was expected as a consequence of the DR/DZ equivalence conjecture by Burya…
Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification profile over $\infty$, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers. In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first …
Pushforward of Siegel flag varieties in the Chow ring
本文计算了 Siegel 情形下子旗簇在 Chow 环中的类，并指出其对应于阿贝尔簇紧化模空间 tautological 环中支撑在环面边界上的类生成元，猜想该对应关系并非巧合。
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.
The goal of this article is to motivate and describe how Gromov-Witten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from Gromov-Witten theory have led to both conjectures and theorems showing that the tautological part of the cohomology ring has a remarkable and profound structure. As an illustration, we describe a new approach to Faber's intersection number conjecture via branched covers of the projective line (work with I.P. Goulden and D.M. Jackson, based on work with T. Graber). En route we review the work of a large number of m…
The Beauville-Voisin conjecture for a hyperkähler manifold X states that the subring of the Chow ring A^*(X) generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of X. We prove a weak version of this conjecture when X is the Hilbert scheme of points on a K3 surface, for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn's formula and the Li-Qin-Wang W_{1+infinity} algebra action from cohomology to Chow gro…
Torelli loci, product cycles, and the homomorphism conjecture for $\mathcal{A}_g$
本文研究主极化阿贝尔簇模空间Chow环的tautological子代数，通过计算Torelli轨迹与乘积轨迹的交积，表明投影算子taut是Q-代数同态，并构造了tautological环Gorenstein核中的新元素。
The paper is Part III of our ongoing project to study a case of Crepant Transformation Conjecture: K-equivalence Conjecture for ordinary flops. In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain non-split toric bundles over arbitrary smooth base. An essential ingredient in the proof is a quantum splitting principle, which reduces a statement in Gromov--Witten theory on non-split bundles to the case of split bundles.
Tautological relations and integrable systems
本文提出了一族关于带n个标记点的亏格g稳定代数曲线模空间上同调类的猜想关系，并证明这些关系蕴含与F-上同调场论相关的Dubrovin-Zhang和双分歧层次的基本性质，同时证明了n=1任意g及g=0任意n情形下的所有关系。
Rooted trees with level structures, $Ω$-classes and double ramification cycles
本文在曲线模空间重言环中证明了一类涉及带层级结构有根树、霍奇丛顶部陈类、Ω-类及双分歧结构的新关系系统，解决了相关猜想，并与DR/DZ等价性建立联系。
We prove a conjecture of Pixton, namely that his proposed formula for the double ramification cycle on Mbar_{g,n} vanishes in codimension beyond g. This yields a collection of tautological relations in the Chow ring of Mbar_{g,n}. We describe, furthermore, how these relations can be obtained from Pixton's 3-spin relations via localization on the moduli space of stable maps to an orbifold projective line.
We show that any degree at least $g$ polynomial in descendant or tautological classes vanishes on $M_{g,n}$ when $g\ge 2$. This generalizes a result of Looijenga and proves a version of Getzler's conjecture. The method we use is the study of the relative Gromov-Witten invariants of $P^1$ relative 2 points combined with the degeneration formulas of [IP1]. At the end of the paper, we also included a quick proof of a very recent conjecture made by Vakil.
We prove that the Tate conjecture is invariant under Homological Projective Duality (=HPD). As an application, we prove the Tate conjecture in the new cases of linear sections of determinantal varieties, and also in the cases of complete intersections of two quadrics. Furthermore, we extend the Tate conjecture from schemes to stacks and prove it for certain global orbifolds
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。