Curve counting on elliptic Calabi–Yau threefolds via derived categories

Oberdieck, Georg; Shen, Junliang

doi:10.4171/jems/938

Cited by 18 publications

(27 citation statements)

References 39 publications

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“…Relation (4.1) was conjectured in [5] and proved in [11,13]. Corollary 4.2 confirms the coefficient of q via Quot schemes, when Y is the Jacobian of a general curve.…”

Section: The Dt Theory Of An Abel-jacobi Curvesupporting

confidence: 57%

The DT/PT correspondence for smooth curves

Ricolfi

2018

Math. Z.

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“…Relation (4.1) was conjectured in [5] and proved in [11,13]. Corollary 4.2 confirms the coefficient of q via Quot schemes, when Y is the Jacobian of a general curve.…”

Section: The Dt Theory Of An Abel-jacobi Curvesupporting

confidence: 57%

The DT/PT correspondence for smooth curves

Ricolfi

2018

Math. Z.

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“…where ξ(γ) ∈ {±1} is determined by ∆ By Theorem 1 the series Z α also satisfies the elliptic transformation law of Jacobi forms in the variable z. The elliptic transformation law in the genus variable u is conjectured by Huang-Katz-Klemm [17] and corresponds to the expected symmetry of Donaldson-Thomas invariants under the Fourier-Mukai transforms by the Poincaré sheaf of π 2 , see [34]. Hence conjecturally we find that Z α is a meromorphic Jacobi form (of weight and index as in Corollary 1).…”

Section: The Schoen Calabi-yau Threefoldmentioning

confidence: 89%

Gromov–Witten theory of elliptic fibrations : Jacobi forms and holomorphic anomaly equations

Oberdieck

Pixton

2019

Geom. Topol.

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We conjecture that the relative Gromov-Witten potentials of elliptic fibrations are (cycle-valued) lattice quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture for the rational elliptic surface in all genera and curve classes numerically. The generating series are quasi-Jacobi forms for the lattice E8. We also show the compatibility of the conjecture with the degeneration formula. As Corollary we deduce that the Gromov-Witten potentials of the Schoen Calabi-Yau threefold (relative to P 1 ) are E8 × E8 quasi-bi-Jacobi forms and satisfy a holomorphic anomaly equation. This yields a partial verification of the BCOV holomorphic anomaly equation for Calabi-Yau threefolds. For abelian surfaces the holomorphic anomaly equation is proven numerically in primitive classes. The theory of lattice quasi-Jacobi forms is reviewed.In the Appendix the conjectural holomorphic anomaly equation is expressed as a matrix action on the space of (generalized) cohomological field theories. The compatibility of the matrix action with the Jacobi Lie algebra is proven. Holomorphic anomaly equations for K3 fibrations are discussed in an example.Recently it became clear that we should expect properties (i, ii) not only for Calabi-Yau threefolds but also for varieties X (of arbitrary dimension) which are Calabi-Yau relative to a base B, i.e. those which admit a fibration π : X → B whose generic fiber has trivial canonical class. The potential F g (q) is replaced here by a π-relative Gromov-Witten potential which takes values in cycles on M g,n (B, k), the moduli space of stable maps to the base. In this paper we conjecture and develop such a theory for elliptic fibrations with section. Our main theoretical result is a conjectural link between the Gromov-Witten theory of elliptic fibrations and the theory of lattice quasi-Jacobi forms. This framework allows us to conjecture a holomorphic anomaly equation. 4 The elliptic curve (or more generally, trivial elliptic fibrations) is the simplest case of our conjecture and was proven in [33]. In this paper we prove the following new cases (see Section 5.3):(a) The P 1 -relative Gromov-Witten potentials of the rational elliptic surface are E 8 -quasi-Jacobi forms numerically 5 .(b) The holomorphic anomaly equation holds for the rational elliptic surface numerically.In particular, (a) solves the complete descendent Gromov-Witten theory of the rational elliptic surface in terms of E 8 -quasi-Jacobi forms. We also show:(c) The quasi-Jacobi form property and the holomorphic anomaly equation are compatible with the degeneration formula (Section 4.6).These results directly lead to a proof of Theorem 1 and 2 as follows. The Schoen Calabi-Yau X admits a degenerationwhere E i ⊂ R i are smooth elliptic fibers. By the degeneration formula [27] we are reduced to studying the case R i × E j . By the product formula [25] the claim then follows from the holomorphic anomaly equation for the rational elliptic surface and the elliptic curve [33]. For completeness we also prove the follo...

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“…In case h = 0 the invariants N g,h,d were obtained by Maulik in [27]. The cases h ∈ {0, 1} were shown by Bryan [5] and a second time in [36]. The cases d ∈ {1, 2} can be found in [33].…”

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confidence: 90%

“…Both sides yield modular constraints and taken together, they determine the partition function from a single coefficient. The sheaf theory side was developed in [34,36] and yields the elliptic transformation law of Jacobi forms (proven by derived auto-equivalences and wall-crossing in the motivic Hall algebra). On the Gromov-Witten side we apply the product formula [3] and study the theory for the K3 surface and the elliptic curve separately.…”

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confidence: 99%

Holomorphic anomaly equations and the Igusa cusp form conjecture

2018

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Let S be a K3 surface and let E be an elliptic curve. We solve the reduced Gromov-Witten theory of the Calabi-Yau threefold S × E for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture.The proof relies on new results in the Gromov-Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov-Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov-Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for elliptic Calabi-Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov-Witten theory of K3 surfaces in primitive classes.

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Curve counting on elliptic Calabi–Yau threefolds via derived categories

Cited by 18 publications

References 39 publications

The DT/PT correspondence for smooth curves

The DT/PT correspondence for smooth curves

Gromov–Witten theory of elliptic fibrations : Jacobi forms and holomorphic anomaly equations

Holomorphic anomaly equations and the Igusa cusp form conjecture

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