Correspondences in enumerative geometry: Hilbert schemes, K3 surfaces and modular forms

Project Information

K3Mod

Grant agreement ID: 101041491

DOI

10.3030/101041491

EC signature date 2 September 2022

Start date 1 February 2023

End date 31 January 2029

Funded under

• European Research Council (ERC)

Total cost

€ 1 429 135,00

EU contribution

€ 1 429 135,00

Coordinated by

KUNGLIGA TEKNISKA HOEGSKOLAN
Sweden

Periodic Reporting for period 1 - K3Mod (Correspondences in enumerative geometry: Hilbert schemes, K3 surfaces and modular forms)

Reporting period: 2023-02-01 to 2025-07-31

Summary of the context and overall objectives of the project

Enumerative geometry is concerned with counting geometric objects on spaces defined by polynomial equations. The subject, which has roots going back to the ancient Greeks, was revolutionized by string theory in the 90s and has since become a fundamental link between algebraic geometry, representation theory, number theory and physics. The project K3Mod aims to establish and make use of a wide range of new correspondences in enumerative geometry. These link together different enumerative theories and open new perspectives to attack long-standing problems concerning the quantum cohomology of the Hilbert scheme of points on surfaces, modular properties of enumerative invariants of K3 geometries, and string partition functions of Calabi-Yau threefolds.

The geometry of the Hilbert scheme of points on a surface will play a central role. A new correspondence between its Gromov-Witten theory and the Donaldson-Thomas theory of certain threefold families will allow for new insight into its quantum cohomology which will give new connection to representation theory and modular forms. Important examples that will be considered are the K3 surfaces and the elliptic surfaces.

The project will consider the enumerative invariants of K3 geometries also in itself. After elliptic curves, which were treated over 20 years ago with much success and rich results, K3 geometries form the simplest compact Calabi-Yau geometries for which a complete understanding of the Gromov-Witten theory is in reach. This will be considered for K3 surfaces as well as related geometries such as the classical Enriques surfaces, which are quotients of K3 surfaces by fixed-point free involutions, or K3-fibrations.

The string-partition functions of smooth projective threefolds will be studied using correspondences between enumerative theories. For elliptic threefolds, the project will relate Donaldson-Thomas invariants with Jacobi forms, using both degeneration techniques and wallcrossing formulae. This will yield new insight and methods in proving modularity of string partition functions. Further important examples that will be studied are the Chaudhuri–Hockney–Lykken Calabi-Yau threefolds and the Enriques Calabi-Yau threefold.

The proposal will lead to new connections between geometry, modular forms and representation theory. The results provide a new understanding of the interplay between Hilbert schemes, K3 surfaces, and modularity in enumerative geometry.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

New results on the quantum cohomology of the Hilbert scheme of points on surfaces have been obtained. In joint work of G. Oberdieck (PI) and A. Pixton, the the quantum multiplication by divisor classes on the Hilbert scheme of points of an elliptic surface was explicitly computed. The structure constants are quasi-Jacobi forms (functions with modular properties) and have interesting representation-theoretic features. For Hilbert schemes of points on K3 surfaces, the structure constants of the reduced quantum cohomology ring are quasi-Jacobi forms (G. Oberdieck). M. Schimpf obtained fully explicit formulas for the full descendent Pandharipande-Thomas theory of local curves in terms of Bethe roots, giving new insight into quantum multiplication on the Hilbert scheme of points on the plane.

Major progress was achieved on the enumerative geometry of K3 geometries. The genus 1 Gromov-Witten theory of the Enriques surface has been completely solved. More generally, the Klemm-Marino formula computing the string-partition function of the Enriques Calabi-Yau threefold (in fiber direction) has been shown by G. Oberdieck, leading to a proof of a 20-year-old conjecture made in the physics literature. In a follow-up work a new K-theoretic refinedment of the Klemm-Marino formula has been conjecturally found. For K3 surface itself explicit formulas for the full stationary descendent were conjectured in terms of quasi-modular forms.

In joint work of the team members G. Oberdieck and M. Schimpf new conjectures and partial results have been obtained for the modularity of the Pandharipande-Thomas theory of elliptic fibered threefolds. This generalizes earlier work of Huang-Katz-Klemm to the non-Calabi-Yau case. Moreover, in this work we find for the first time holomorphic anomaly equations in sheaf-counting theories. For the Chaudhuri–Hockney–Lykken Calabi-Yau threefolds the partition functions of Pandharipande-Thomas invariants were shown to be quasi-Jacobi forms using earlier results on the Hilbert scheme of points of a K3 surface.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The project moved beyond the state-of-the-art in several directions. The quantum cohomology of the Hilbert scheme of points of a projective surface has been computed in explicitly in a first non-trivial case, extending our knowledge from the local cases to global geometries. For the Enriques-Calabi-Yau threefold the string partition function (in fiber direction) was computed, leading to a complete solution of the genus 1 Gromov-Witten theory of the Enriques surface.