Abstract
We compute the Donaldson–Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in Bryan et al. [‘Trace identities for the topological vertex’, Selecta Math. (N.S.)24 (2) (2018), 1527–1548, arXiv:math/1603.05271], we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz–Klemm–Vafa formula for primitive curve classes which is independent of the computation of Kawai–Yoshioka.
Highlights
Let p : S → B be a nontrivial elliptic surface over a complex smooth projective curve B
The main results of this paper are closed product formulas for the partition functions DT(X ) and DTfib(X )
Our result provides a new derivation of the KKV formula for primitive classes
Summary
Let p : S → B be a nontrivial elliptic surface over a complex smooth projective curve B. For the Behrend function weighted DT invariants, we require Conjecture 21, see Theorem 3.) The KKV formula was proved in all curve classes in [17]. The appearance of the Jacobi form η24Θ2 in previous proofs of the KKV formula [14, 17] comes from the calculation of Euler characteristics of relative Hilbert schemes of points on curves on K3 by Kawai–Yoshioka [10]. We show that ρ∗(1), the push-forward measure, has nice multiplicative properties that allow us to compute the weighted Euler characteristic over Sym B using a general result about symmetric products (Lemma 32). To compute the push-forward measure ρ∗(1) explicitly, we must compute the Euler characteristics of the fibers of ρ These fibers are strata in the Hilbert scheme parameterizing subschemes whose maximal Cohen–Macaulay subscheme is a fixed partition thickened comb curve C. This gives a geometric bijection QuotnX (IC ) → Hilbn(X, C) from which the lemma follows
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