The log product formula

Herr, Leo

doi:10.2140/ant.2023.17.1281

Cited by 4 publications

(1 citation statement)

References 27 publications

(16 reference statements)

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“…The naive spaces provide an alternative perspective for probing the geography and invariants of the multiroot spaces. The iterated blowup construction of [22] gives a method for comparing the logarithmic invariants to the naive/orbifold invariants; see also [25,16] for treatments of related ideas.…”

Section: Comparison With Naive Invariantsmentioning

confidence: 99%

The Local-Orbifold Correspondence for Simple Normal Crossing Pairs

Battistella

Nabijou

Tseng

et al. 2022

J. Inst. Math. Jussieu

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For X a smooth projective variety and $D=D_1+\dotsb +D_n$ a simple normal crossing divisor, we establish a precise cycle-level correspondence between the genus $0$ local Gromov–Witten theory of the bundle $\oplus _{i=1}^n \mathcal {O}_X(-D_i)$ and the maximal contact Gromov–Witten theory of the multiroot stack $X_{D,\vec r}$ . The proof is an implementation of the rank-reduction strategy. We use this point of view to clarify the relationship between logarithmic and orbifold invariants.

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Section: Comparison With Naive Invariantsmentioning

confidence: 99%

The Local-Orbifold Correspondence for Simple Normal Crossing Pairs

Battistella

Nabijou

Tseng

et al. 2022

J. Inst. Math. Jussieu

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Pluricanonical cycles and tropical covers

Cavalieri,

Markwig,

Ranganathan

2024

Trans. Amer. Math. Soc.

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We extract a system of numerical invariants from logarithmic intersection theory on pluricanonical double ramification cycles, and show that these invariants exhibit a number of properties that are enjoyed by double Hurwitz numbers. Among their properties are (i) the numbers can be efficiently calculated by counts of tropical curves with a modified balancing condition, (ii) they are piecewise polynomial in the entries of the ramification vector, and (iii) they are matrix elements of operators on the Fock space. The numbers are extracted from the logarithmic double ramification cycle, which is a lift of the standard double ramification cycle to a blowup of the moduli space of curves. The blowup is determined by tropical geometry. We show that the traditional double Hurwitz numbers are intersections of the refined cycle with the cohomology class of a piecewise polynomial function on the tropical moduli space of curves. This perspective then admits a natural, combinatorially motivated, generalization to the pluricanonical setting. Tropical correspondence results for the new invariants lead immediately to the structural results for these numbers.

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Logarithmic Gromov–Witten theory and double ramification cycles

Ranganathan,

Kumaran

2024

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We examine the logarithmic Gromov–Witten cycles of a toric variety relative to its full toric boundary. The cycles are expressed as products of double ramification cycles and natural tautological classes in the logarithmic Chow ring of the moduli space of curves. We introduce a simple new technique that relates the Gromov–Witten cycles of rigid and rubber geometries; the technique is based on a study of maps to the logarithmic algebraic torus. By combining this with recent work on logarithmic double ramification cycles, we deduce that all logarithmic Gromov–Witten pushforwards, for maps to a toric variety relative to its full toric boundary, lie in the tautological ring of the moduli space of curves. A feature of the approach is that it avoids the as yet undeveloped logarithmic virtual localization formula, instead relying directly on piecewise polynomial functions to capture the structure that would be provided by such a formula. The results give a common generalization of work of Faber and Pandharipande, and more recent work of Holmes and Schwarz as well as Molcho and Ranganathan. The proof passes through general structure results on the space of stable maps to the logarithmic algebraic torus, which may be of independent interest.

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The log product formula

Cited by 4 publications

References 27 publications

The Local-Orbifold Correspondence for Simple Normal Crossing Pairs

The Local-Orbifold Correspondence for Simple Normal Crossing Pairs

Pluricanonical cycles and tropical covers

Logarithmic Gromov–Witten theory and double ramification cycles

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