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| Faber–Pandharipande | |
|---|---|
| Name | Faber–Pandharipande |
| Notable works | Intersection theory on moduli spaces, Hodge integrals, Gromov–Witten relations |
| Fields | Algebraic geometry, Enumerative geometry, Symplectic topology |
| Institutions | Harvard University, Institute for Advanced Study, Princeton University |
Faber–Pandharipande
The Faber–Pandharipande framework denotes a collection of conjectures, results, and techniques developed by Carel Faber and Rahul Pandharipande that shaped modern approaches to intersection theory on the moduli space of stable curves, Hodge integrals, and relations with Gromov–Witten theory. The work connects ideas from the Deligne–Mumford compactification, Kontsevich, Witten's conjectures, and the geometry of tautological rings, influencing research at institutions such as the Institute for Advanced Study and Harvard University. It established bridges between enumerative predictions by Gromov–Witten invariants and structural properties conjectured for tautological classes on moduli spaces.
The origin traces to collaborations in the 1990s between Carel Faber and Rahul Pandharipande while interacting with results of Maxim Kontsevich and the proof of Witten's conjecture by Kontsevich relating intersection numbers on moduli spaces to the KdV equation. Early milestones include Faber's conjectures on the structure of the tautological ring of the moduli space of genus-g curves, formulated in dialogue with work from the Moduli of Curves Conference and consolidated through papers appearing in venues influenced by the American Mathematical Society and presentations at the International Congress of Mathematicians. Pandharipande's contributions linked these conjectures to relative Gromov–Witten theory and virtual class techniques developed with collaborators linked to the Mathematical Sciences Research Institute and the Clay Mathematics Institute.
Faber–Pandharipande statements concern tautological rings R^*(\overline{M}_{g,n}) of the Deligne–Mumford stack \overline{M}_{g,n}, specifying predictions about generators, relations, and intersection pairings. Central components include: - The description of socle and vanishing: predictions for R^i(\overline{M}_{g}) vanishing above degree g-2, echoing structures studied by Mumford and Harer. - Relations expressed via Hodge integrals ⟨\lambda_g \psi_1^{a_1}\cdots\psi_n^{a_n}⟩ that tie to formulas from Faber and computations influenced by Getzler and Vakil. - Connections to the virtual fundamental class constructed by Behrend and Fantechi in Gromov–Witten theory, relying on techniques developed in papers by Li and Graber.
These formulations employ tautological generators (ψ-classes, κ-classes, λ-classes) on \overline{M}_{g,n} and conjectural relations conjectured by Faber and extended by Pandharipande, with explicit expressions often involving boundary divisors associated to clutching maps studied by Mumford and Knudsen.
The Faber–Pandharipande program forges explicit links between tautological relations on \overline{M}_{g,n} and enumerative invariants computed via Gromov–Witten theory of target varieties such as \mathbb{P}^1, \mathbb{P}^2, and Calabi–Yau threefolds including the Quintic threefold. Techniques employ degeneration methods of Jun Li and localization formulas of Graber–Pandharipande to express Gromov–Witten invariants in terms of Hodge integrals. This perspective unites hypotheses from Witten about relations satisfied by intersection numbers with the virtual localization framework, enabling translation between relative Gromov–Witten invariants and tautological relations on moduli spaces, informed by work of Ionel and Parker on symplectic field theory.
Concrete computations driven by Faber–Pandharipande techniques include evaluations of Hodge integrals for low genus and small numbers of marked points, explicit intersection numbers on \overline{M}_{g,n} for g≤3 computed following methods of Faber and corroborated using the Virasoro constraints approach of Getzler and Ellingsrud–Göttsche–Lehn style calculations. Examples cover: - Hodge integral identities used to compute genus-one Gromov–Witten invariants of \mathbb{P}^1 and local Calabi–Yau geometries documented alongside computations by Zinger. - Verification of tautological relations via test curves and boundary strata computations influenced by combinatorial descriptions from Keel and Pandharipande. These examples illustrate how virtual localization and degeneration reduce enumerative problems to evaluations of tautological intersections.
Consequences span enumerative predictions for curves in projective varieties, structural clarity about R^*(\overline{M}_{g,n}), and algorithmic tools for computing Gromov–Witten invariants used by researchers at Princeton University, ETH Zurich, and University of California, Berkeley. Applications include proofs and partial proofs of Faber conjectures in specific ranges (thanks to work by Looijenga, Faber–Pandharipande collaborators, and M. Kazarian), formulae for generating functions governing Hurwitz numbers connecting to the ELSV formula developed with input from Ekedahl, and constraints informing mirror symmetry predictions tested against computations for the Quintic threefold by Candelas and successors.
Open problems remain central: proving the full Faber conjectures for all genera, characterizing all relations in tautological rings, and establishing a complete dictionary between tautological relations and identities in Gromov–Witten theory. Conjectural extensions propose links to double ramification cycles studied by Hain and Grushevsky, relations to the Gorenstein property envisioned by Faber, and broader connections to integrable hierarchies inspired by Witten and Kontsevich. Active research groups at institutions including IHES, MPI MiS, and Cambridge University pursue these challenges with techniques from virtual cycle technology, logarithmic Gromov–Witten theory of Abramovich–Chen–Gross–Siebert, and advances in moduli of maps by Ruan and Liang.