Gromov–Witten theory with maximal contacts

Nabijou, Navid; Ranganathan, Dhruv

doi:10.1017/fms.2021.78

“…We thank Makoto Miura for a discussion on Hibi varieties. Finally, it is a pleasure to thank Navid Nabijou and Dhruv Ranganathan for discussions on their parallel work [26].…”

Section: A C K N O W L E D G E M E N T Smentioning

confidence: 99%

“…Relation to [26] and [7,8] After this paper was finished, we received the manuscript [26] where the log-local principle is considered for simple normal crossings divisors. The respective strategies have different flavours in the proof and complementary virtues in the outcome: [26] consider the log/local correspondence for 𝑋 smooth and 𝐷 𝑗 a hyperplane section, with a beautiful geometric argument reducing the simple normal crossings case to the case of smooth pairs, and with no restrictions on 𝑋.…”

mentioning

confidence: 99%

“…Relation to [26] and [7,8] After this paper was finished, we received the manuscript [26] where the log-local principle is considered for simple normal crossings divisors. The respective strategies have different flavours in the proof and complementary virtues in the outcome: [26] consider the log/local correspondence for 𝑋 smooth and 𝐷 𝑗 a hyperplane section, with a beautiful geometric argument reducing the simple normal crossings case to the case of smooth pairs, and with no restrictions on 𝑋. The combinatorial pathway we pursued in the toric setting allows on the other hand to relax the hypotheses on the smoothness of 𝑋, the normal crossings nature of 𝐷, and the very ampleness of 𝐷 𝑗 , and it lends itself to a wider application to the case when 𝐷 is not the toric boundary and the refinement to include all-genus invariants.…”

mentioning

confidence: 99%

“…Remark 1. In its most recent version, [26] gives a counter-example in principle to Conjecture 1. It is proven that there is a choice of (unspecified) insertion leading to a counter-example.…”

mentioning

confidence: 99%

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Let 𝑋 be a smooth projective complex variety and let 𝐷 = 𝐷 1 + ⋯ + 𝐷 𝑙 be a reduced normal crossing divisor on 𝑋 with each component 𝐷 𝑗 smooth, irreducible and numerically effective. The log-local principle put forward in van Garrel et al. (Adv. Math. 350 (2019) 860-876) conjectures that the genus 0 log Gromov-Witten theory of maximal tangency of (𝑋, 𝐷) is equivalent to the genus 0 local Gromov-Witten theory of 𝑋 twisted by ⨁ 𝑙 𝑗=1 (−𝐷 𝑗 ). We prove that an extension of the loglocal principle holds for 𝑋 a (not necessarily smooth) ℚ-factorial projective toric variety, 𝐷 the toric boundary, and descendant point insertions. M S C ( 2 0 2 0 )14N35 (primary), 14M25, 14J33, 14T90 (secondary) INTRODUCTIONLet 𝑋 be a smooth projective complex variety of dimension 𝑛 and let 𝐷 = 𝐷 1 + ⋯ + 𝐷 𝑙 be an effective reduced normal crossing divisor with each component 𝐷 𝑗 smooth, irreducible and numerically effective. We can then consider two, a priori very different, geometries associated to the pair (𝑋, 𝐷):-the 𝑛-dimensional log geometry of the pair (𝑋, 𝐷), -the (𝑛 + 𝑙)-dimensional local geometry of the total space Tot( ⨁ 𝑙 𝑗=1  𝑋 (−𝐷 𝑗 )). The genus 0 log Gromov-Witten invariants of (𝑋, 𝐷) virtually count rational curves 𝑓 ∶ ℙ 1 → 𝑋

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“…It is discussed in [CMR17] that M(x) may be obtained as an iterated blow-up of M 0,n . The results in [NR22] suggest that in the case M(x) = M n , this sequence of blow-ups can be given an explicit algorithmic description. Conjecture 4.1 suggests that the complexity of this iterated blow-up increases dramatically with n. 4.4.…”

Section: Further Directionsmentioning

confidence: 95%

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Preprint

0

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We study constructible invariants of the moduli space M(x) of stable maps from genus zero curves to P 1 , relative to 0 and ∞, with ramification profiles specified by x ∈ Z n . These spaces are central to the enumerative geometry of P 1 , and provide a large family of birational models of the Deligne-Mumford-Knudsen moduli space M 0,n . For the sequence of vectors x corresponding to maps which are maximally ramified over 0 and unramified over ∞, we prove that a generating function for the topological Euler characteristics of these spaces satisfies a differential equation which allows for its recursive calculation. We also show that the class [M(x)] ∈ K 0 (Var/C) of the moduli space in the Grothendieck ring of varieties is constant as x varies within a fixed chamber in the resonance decomposition of Z n . We conclude by suggesting several further directions in the study of these spaces, giving conjectures on (1) the asymptotic behavior of the Euler characteristic and (2) a potential chamber structure for the Chern numbers.

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Moduli of relative stable maps to $$\mathbb {P}^1$$: cut-and-paste invariants

Kannan

¹

2023

Sel. Math. New Ser.

1

0

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We study constructible invariants of the moduli space $$\hspace{0.83328pt}\overline{\hspace{-0.83328pt}\mathcal {M}\hspace{-0.83328pt}}\hspace{0.83328pt}(\varvec{x})$$ M ¯ ( x ) of stable maps from genus zero curves to $$\mathbb {P}^1$$ P 1 , relative to 0 and $$\infty $$ ∞ , with ramification profiles specified by $${\varvec{x}\in \mathbb {Z}^n}$$ x ∈ Z n . These spaces are central to the enumerative geometry of $$\mathbb {P}^1$$ P 1 , and provide a large family of birational models of the Deligne–Mumford–Knudsen moduli space $$\hspace{0.83328pt}\overline{\hspace{-0.83328pt}\mathcal {M}\hspace{-0.83328pt}}\hspace{0.83328pt}_{0,n}$$ M ¯ 0 , n . For the sequence of vectors $$\varvec{x}$$ x corresponding to maps which are maximally ramified over 0 and unramified over $$\infty $$ ∞ , we prove that a generating function for the topological Euler characteristics of these spaces satisfies a differential equation which allows for its recursive calculation. We also show that the class $${[\hspace{0.83328pt}\overline{\hspace{-0.83328pt}\mathcal {M}\hspace{-0.83328pt}}\hspace{0.83328pt}(\varvec{x})] \in K_0(\textsf{Var}/\mathbb {C})}$$ [ M ¯ ( x ) ] ∈ K 0 ( Var / C ) of the moduli space in the Grothendieck ring of varieties is constant as $$\varvec{x}$$ x varies within a fixed chamber in the resonance decomposition of $$\mathbb {Z}^n$$ Z n . We conclude by suggesting several further directions in the study of these spaces, giving conjectures on (1) the asymptotic behavior of the Euler characteristic and (2) a potential chamber structure for the Chern numbers.

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Gromov–Witten theory with maximal contacts

Cited by 11 publications

References 24 publications

On the log–local principle for the toric boundary

On the log–local principle for the toric boundary

Moduli of relative stable maps to $\mathbb{P}^1$: cut-and-paste invariants

Moduli of relative stable maps to $$\mathbb {P}^1$$: cut-and-paste invariants

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