Punctured logarithmic maps

Abramovich, Dan; Chen, Qile; Gross, Mark; Siebert, Bernd

doi:10.48550/arxiv.2009.07720

Cited by 19 publications

(90 citation statements)

References 51 publications

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“…Punctured maps. The theory of punctured stable maps [ACGS20] provides an alternative avenue for exploring recursive descriptions of the boundary in the unexpanded setting. In our experience the unexpanded and expanded theories complement each other well, with their differences usually making one or the other better suited to a particular purpose.…”

Section: Future Directionsmentioning

confidence: 99%

Rubber tori in the boundary of expanded stable maps

Carocci,

Nabijou

2021

Preprint

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We investigate torus actions on logarithmic expansions in the context of enumerative geometry. Our main result is an intrinsic and coordinate-free description of the higher-rank rubber torus appearing in the boundary of the space of expanded stable maps. The rubber torus is constructed canonically from the tropical moduli space, and its action on each stratum of the expanded target is encoded in a linear tropical position map. The presence of 2-morphisms in the universal target forces expanded stable maps differing by the rubber action to be identified. This provides the first step towards a recursive description of the boundary of the expanded moduli space, with future applications including localisation and rubber calculus.

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Section: Future Directionsmentioning

confidence: 99%

Rubber tori in the boundary of expanded stable maps

Carocci,

Nabijou

2021

Preprint

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“…Remark 9.1. Another well-known invariant for simple normal crossing pairs is (punctured) logarithmic Gromov-Witten invariants of [AC14], [Che14], [GS13], [ACGS20]. Punctured invariants are essential for the intrinsic mirror symmetry construction in the Gross-Siebert program [GS19].…”

Section: Gromov-witten Invariantsmentioning

confidence: 99%

Degenerations, fibrations and higher rank Landau-Ginzburg models

Doran¹,

Kostiuk²,

You³

2021

Preprint

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We study semi-stable degenerations of quasi-Fano varieties to unions of two pieces. We conjecture that the higher rank Landau-Ginzburg models mirror to these two pieces can be glued together to lower rank Landau-Ginzburg models which are mirror to the original quasi-Fano varieties. We prove this conjecture by relating their Euler characteristics, generalized functional invariants as well as periods. We also use it to conjecture a relation between the degenerations to the normal cones and the fibrewise compactifications of higher rank Landau-Ginzburg models. Furthermore, we use it to iterate the Doran-Harder-Thompson conjecture and obtain higher codimension Calabi-Yau fibrations. 2 CHARLES F. DORAN, JORDAN KOSTIUK, AND FENGLONG YOU 9. Gromov-Witten invariants 36 9.1. Orbifold Gromov-Witten invariants 36 9.2. The formal Gromov-Witten invariants of infinite root stacks 37 9.3. Mirror theorems 38 References 40

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“…Historically this has precedent, the work of [25] came only after the work of [17]. In future work we intend to use the gluing calculations of [9] and [35] and the period calculations of [29] to prove the period gluing formulae of [12] from the perspective of tropical curves, even though we do not know a precise method to prove that fibres are mirror to D.…”

Section: Comments On Restrictionsmentioning

confidence: 99%

Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program

Barrott¹,

Doran²

2021

Preprint

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The Doran-Harder-Thompson "gluing/splitting" conjecture unifies mirror symmetry conjectures for Calabi-Yau and Fano varieties, relating fibration structures on Calabi-Yau varieties to the existence of certain types of degenerations on their mirrors. This was studied for the case of Calabi-Yau complete intersections in toric varieties in [12] for the Hori-Vafa mirror construction. In this paper we prove one direction of the conjecture using a modified version of the Gross-Siebert program. This involves a careful study of the implications within tropical geometry and applying modern deformation theory for singular Calabi-Yau varieties.

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Punctured logarithmic maps

Cited by 19 publications

References 51 publications

Rubber tori in the boundary of expanded stable maps

Rubber tori in the boundary of expanded stable maps

Degenerations, fibrations and higher rank Landau-Ginzburg models

Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program

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