Abstract
We give a general overview of the Donaldson-Thomas invariants of elliptic fibrations and their relation to Jacobi forms. We then focus on the specific case of where the fibration is S × E, the product of a K3 surface and an elliptic curve. Oberdieck and Pandharipande conjectured (Oberdieck and Pandharipande, K3 Surfaces and Their Moduli, Progress in Mathematics, vol. 315 (Birkhäuser/Springer, Cham, 2016), pp. 245–278, arXiv:math/1411.1514) that the partition function of the Gromov-Witten/Donaldson-Thomas invariants of S × E is given by minus the reciprocal of the Igusa cusp form of weight 10. For a fixed primitive curve class in S of square 2h − 2, their conjecture predicts that the corresponding partition functions are given by meromorphic Jacobi forms of weight − 10 and index h − 1. We calculate the Donaldson-Thomas partition function for primitive classes of square − 2 and of square 0, proving strong evidence for their conjecture. Our computation uses reduced Donaldson-Thomas invariants which are defined as the (Behrend function weighted) Euler characteristics of the quotient of the Hilbert scheme of curves in S × E by the action of E. Our technique is a mixture of motivic and toric methods (developed with Kool in (Bryan and Kool, Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex (2016), arXiv:math/1608.07369)) which allows us to express the partition functions in terms of the topological vertex and subsequently in terms of Jacobi forms. We compute both versions of the invariants: unweighted and Behrend function weighted Euler characteristics. Our Behrend function weighted computation requires us to assume Conjecture 18 in (Bryan and Kool, Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex (2016), arXiv:math/1608.07369).
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Notes
- 1.
Although, from general considerations, \(\widehat {DT}_{\beta ,n}(X)\) is a constructible function on the deformation space of X and hence it is an invariant of the generic member of each deformation type.
- 2.
The traditional definition is only for projective X. When X is only quasi-projective, the weighted Euler characteristic definition is well defined, but no longer guarenteed to be a deformation invariant.
- 3.
It is not clear what the optimal hypotheses on π : X → S should be. In [9], they primary consider the case of Fano S and where all fibers π −1(pt) are irreducible, but they discuss other cases, including the trivial fibrations studied in this paper.
- 4.
Oberdieck and Pandharipande were the first to formulate a mathematically precise conjecture, but the appearence of the Igusa cusp form in closely related partition functions in physics dates back to the mid-nineties [6].
- 5.
Since this paper was originally written in 2015, Oberdieck and Pixton have given a complete proof of this conjecture using methods (different from ours) from both Donaldson-Thomas theory and Gromov-Witten theory [13].
- 6.
The value of the Behrend function at a closed point of a scheme only depends on the local ring of that point, therefore the Behrend function of a scheme is invariant under any group action.
- 7.
Our insertion of the sign on the p variable in the Donaldson-Thomas partition function is somewhat non-standard. It makes subsequent formulas simpler to state. Geometrically, the extra sign can be interpreted as using the Behrend function coming from the moduli stack of structure sheaves of the curves as oppose to the moduli space of ideal sheaves (see [3, § 3,Thm 3.1]). Since the naïve Donaldson-Thomas invariants do not involve the Behrend function, no sign appears.
- 8.
This follows from fpqc descent since the set U and the sets \(\widehat {X} _{\{y_{i} \}\times E}\) form a fpqc cover. Since C 0 × x 0 is reduced there are no conditions on the overlaps of the cover. Thus the subscheme is uniquely determined by its restriction to the cover.
- 9.
i.e. Identifying the partition α with its Ferrer’s diagram \(\alpha \subset ({\mathbb Z} _{\geq 0})^{2}\), the ideal of Z α is generated by the monomials u i v j where (i, j)∉α.
- 10.
However, see Sect. 7 for the action of a related group.
- 11.
This means that formally locally C red is either smooth, nodal, or the union of the three coordinate axes. That is at p ∈ C red ⊂ Y the ideal \(\widehat {I}_{C_{\mathrm {red}}}\subset \widehat {\mathcal {O}}_{Y,p}\) is given by (x 1, x 2), (x 1, x 2 x 3), or (x 1 x 2, x 2 x 3, x 1 x 3) for some isomorphism \(\widehat {\mathcal {O}}_{Y,p}\cong {\mathbb C} [[x_{1},x_{2},x_{3}]]\).
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Acknowledgements
I’d like to thank George Oberdieck, Rahul Pandharipande, and Yin Qizheng for invaluable discussions. I’ve also benefited with discussions with Tom Coates, Sheldon Katz, Martijn Kool, Davesh Maulik, Tony Pantev, Balazs Szendroi, Andras Szenes, and Richard Thomas. The computational technique employed in this paper was developed in collaboration with Martijn Kool whom I owe a debt of gratitude. I would also like to thank the Institute for Mathematical Research (FIM) at ETH for hosting my visit to Zürich, and for Matematisk Institutt, UiO and Jan Arthur Christophersen for organizing the 2017 Abel Symposium.
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Bryan, J. (2018). The Donaldson-Thomas Theory of K3 × E via the Topological Vertex. In: Christophersen, J., Ranestad, K. (eds) Geometry of Moduli. Abelsymposium 2017. Abel Symposia, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-94881-2_2
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