Tautological algebras of moduli spaces of
                    curves

Faber, Carel

doi:10.1090/pcms/020/07

Cited by 10 publications

(15 citation statements)

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“…Based on empirical observations, Faber [Fab99] formulated an ambitious conjecture giving a precise description of the structure of the tautological ring of M g . Most of these conjectures have by now been proven by work of a large amount of people, see [Fab11] for a survey. But the most elusive part of the conjecture has turned out to be the "Gorenstein" part of the conjecture, which asserts that R • (M g ) is a Gorenstein ring with socle in dimension g − 2 (in other words, that the ring satisfies Poincaré duality).…”

Section: Introductionmentioning

confidence: 98%

The Gorenstein conjecture fails for the tautological ring of $\mathcal{\overline{M}}_{2,n}$

Petersen

Tommasi²

2013

Invent. math.

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Let N be the smallest integer such that there is a non-tautological cohomology class of even degree on M 2,N . We remark that there is such a non-tautological class on M 2,20 , by work of Graber and Pandharipande. We show that 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 R • (M 2,N ) is not Gorenstein. We present some evidence suggesting that N = 20 holds.1.1. Conventions. In Sections 2, 4 and 5, all cohomology means étale cohomology with Q ℓ -coefficients, for some fixed prime ℓ. The only properties we

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Section: Introductionmentioning

confidence: 98%

The Gorenstein conjecture fails for the tautological ring of $\mathcal{\overline{M}}_{2,n}$

Petersen

Tommasi²

2013

Invent. math.

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“…This represents work of a large number of people, and all of the statements now have several different proofs, enlightening in their own way. See the survey [Faber 2013] and the detailed references therein.…”

Section: Introductionmentioning

confidence: 99%

Tautological rings of spaces of pointed genus two curves of compact type

Petersen¹

2016

Compositio Math.

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We prove that the tautological ring of M ct 2,n , the moduli space of n-pointed genus two curves of compact type, does not have Poincaré duality for any n ≥ 8. This result is obtained via a more general study of the cohomology groups of M ct 2,n . 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 ct 2,n ) for any k and n considered both as Snrepresentation and as mixed Hodge structure/ℓ-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of M 2,n is tautological for n < 20, and that the tautological ring of M 2,n fails to have Poincaré duality for all n ≥ 20. This improves and simplifies results of the author and Orsola Tommasi.

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“…The maximal algebraic structure, encoding all natural operations that use only the structure of morphisms of modular operads on the deformation complexes of classical and quantum homotopy CohFTs can be extended further once we use the specifics of the Deligne-Mumford-Knudsen modular operad. Namely, we use that it is a Hopf modular operad and that is contains a distinguished and easy to describe suboperad of tautological classes [Fab13]. So, we extend the algebraic structure that exists of the deformation compelexes of classical and quantum homotopy CohFTs with the action of tautological classes combined with the non-stable parts of the endomorphism modular operad of the target vector space (or, more generally, one can thing of non-stable components of the target modular operad).…”

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confidence: 99%

“…with the algebraic structures of the cohomology algebras (the Poincaré duality, the push-forward in the cohomology, and the intersection product). To this end, there is an intensively studied system of subalgebras of the cohomology rings of the moduli spaces of curves closed under the push-forwards with respect to all natural maps: the subalgebra of tautological classes, see e. g. [Fab13] for a survey. The additive generators of this subalgebra are fully understood, they are represented by the natural strata (which can be considered as the results of the iterative application of the operations ξ ij and • j i ) decorated by ψ-and κ-classes in all possible ways, and we use these classes and push-forwards with respect to the natural maps to enrich the operad acting on g and g .…”

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confidence: 99%

Deformation theory of Cohomological Field Theories

Dotsenko,

Shadrin,

Vaintrob

et al. 2020

Preprint

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A. We develop the deformation theory of cohomological field theories (CohFTs), which is done as a special case of a general deformation theory of morphisms of modular operads. This leads us to introduce two new natural extensions of the notion of a CohFT: homotopical (necessary to structure chain-level Gromov-Witten invariants) and quantum (with examples found in the works of Buryak-Rossi on integrable systems). We introduce a new version of Kontsevich's graph complex, enriched with tautological classes on the moduli spaces of stable curves. We use it to study a new universal deformation group which acts naturally on the moduli spaces of quantum homotopy CohFTs, by methods due to Merkulov-Willwacher. This group is shown to contain both the prounipotent Grothendieck-Teichmüller group and the Givental group. CWe call the latter group the Givental-Grothendieck-Teichmüller group, and both a full description of its structure and the consequences of its action on quantum homotopy CohFTs (for instance, for the Buryak-Rossi theory of quantization of integrable hierarchies of topological type) are very interesting open questions for future research.Layout. The first section recalls the notion of a modular operad together their homological constructions. In the second section, we develop the deformation theory of morphisms of modular operads and study the particular case of the deformation theory of (homotopy) CohFTs. The third section deals with the universal deformation group of morphisms of modular operads, its relationship with the prounipotent Grothendieck-Teichmüller group, and its action of the moduli space of gauge equivalent classes of Maurer-Cartan elements. The last section introduces a new graph operad enriched with tautological classes whose deformation complex is shown to act on quantum homotopy CohFTs and to contain the prounipotent Grothendieck-Teichmüller group and the Givental group.Conventions. We denote by k the fixed ground field of characteristic 0. We mainly work over the underlying category of differential Z-graded vector spaces, with the usual monoidal structure including the Koszul sign rules and conventions. We use the homological degree convention, for which differentials have degree −1. Cochain complexes, that are usually cohomologically graded, are considered with opposite homological degree. We denote by s an element of degree 1 and tensoring with it produces the shift functor. At three points of the paper (Section 2.2, Section 3.2, and Section 4), we switch from the Z-grading to the induced Z/2Z-grading.

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Tautological algebras of moduli spaces of curves

Cited by 10 publications

References 58 publications

The Gorenstein conjecture fails for the tautological ring of $\mathcal{\overline{M}}_{2,n}$

The Gorenstein conjecture fails for the tautological ring of $\mathcal{\overline{M}}_{2,n}$

Tautological rings of spaces of pointed genus two curves of compact type

Deformation theory of Cohomological Field Theories

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