Perverse curves and mirror symmetry

Ruddat, Helge

doi:10.1090/jag/666

Cited by 8 publications

(3 citation statements)

References 21 publications

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“…The SYZ mirror of this configuration was studied by Abouzaid, Auroux, and Katzarkov in [1]. The banana configuration also appears in the Landau-Ginzburg mirror of a genus two curve as studied by Gross, Katzarkov, and Ruddat in [18] and Ruddat in [36]. .…”

mentioning

confidence: 85%

“…Since the arguments of a genus 2 Siegel modular form are coordinates on the moduli space of genus 2 curves (or Abelian surfaces), we expect the complex moduli space of Xban , the mirror of the banana manifold, to contain a subspace isomorphic to the moduli space of genus 2 curves. Indeed, it has already been observed that the mirror of a local banana configuration should be a genus 2 curve [1,18,36].…”

Section: Skoruppa Maassmentioning

confidence: 99%

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The Donaldson-Thomas partition function of the banana manifold

Bryan¹

2019

Preprint

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A banana manifold is a compact Calabi-Yau threefold, fibered by Abelian surfaces, whose singular fibers have a singular locus given by a "banana configuration of curves". A basic example is given by X ban := Bl ∆ (S × P 1 S), the blowup along the diagonal of the fibered product of a generic rational elliptic surface S → P 1 with itself.In this paper we give a closed formula for the Donaldson-Thomas partition function of the banana manifold X ban restricted to the 3-dimensional lattice Γ of curve classes supported in the fibers of X ban → P 1 . It is given by, and the coefficients c(a, k) have a generating function given by an explicit ratio of theta functions. This formula has interesting properties and is closely related to the equivariant elliptic genera of Hilb(C 2 ). In an appendix with S. Pietromonaco, it is shown that the corresponding genus g Gromov-Witten potential F g is a genus 2 Siegel modular form of weight 2g − 2 for g ≥ 2, namely it is the Skoruppa-Maass lift of a multiple of an Eisenstein series, namely 6|B2g | g(2g−2)! E 2g (τ ). JIM BRYAN 3.1. Overview 3.2. Preliminaries on Notation and Euler Characteristics 3.3. Pushing forward the ν-weighted Euler characteristic measure. 3.4. Subschemes supported in an infinitesimal neighborhood of a fiber. 3.5. Mordell-Weil groups and actions on Hilb( F y ). 3.6. Reduction of the computation to the singular fibers. 3.7. Reduction to C * × C * -fixed subschemes. 3.8. Analysis of C * × C * -fixed subschemes. 4. The vertex calculation 4.1. Overview of computation 4.2. Notation and Schur function identities 4.3. The main derivation 5. Geometry of the Banana manifold 5.1. Relation to the Schoen threefold and the Hodge numbers 5.2. Proof of Proposition 24. 6. BPS invariants from a Donaldson-Thomas partition function in product form. Appendix A. The Gromov-Witten potentials are Siegel modular forms (with Stephen Pietromonaco) A.1. Overview. A.2. Modular forms and lifts A.3. The λ expansion of Ell q,y (C 2 , t). A.4. Gromov-Witten potentials via the GW/DT correspondence A.5. Gopakumar-Vafa invariants References

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