Moduli of hybrid curves and variations of canonical measures

Amini, Omid; Nicolussi, Noema

doi:10.48550/arxiv.2007.07130

Cited by 5 publications

(44 citation statements)

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“…Proof. Recall that the well-known local structure of M g around the boundary and universal curves over it, says in particular there is a natural one to one order (closure relation) reversing bijection between the set of strata of M g DM and the strata of ) is nothing but the parametrizes the limit stable curves, while its σ o π -part parametrizes the graph part of the metrized complexes as [AN20] shows and also our assertion follows.…”

Section: Here Mmentioning

confidence: 56%

“…If we try to search analogue of the limit log minimal compactification for the moduli of hyperbolic curves M g , moduli of hyperbolic curves of genera g > 1, one would naturally replace the set of all toroidal compactifications above by the single Deligne-Mumford compactification M g ⊂ M g DM since it is smooth lc model at staky level. Therefore, we do not obtain a similar compactification in the same manner but still there is another compactification which is analogous to some extent (compare with above Theorem 3.6): Amini-Nicolussi [AN20] recently constructed a compactification M g ⊂ M hyb g on whose boundary they parametrize metric complex ([AmB15]) with the ordered partition of the edge sets which they call "layor"s. • the Morgan-Shalen-Boucksom-Jonsson compactification…”

Section: Here Mmentioning

confidence: 99%

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Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures

Odaka¹

2020

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YUJI ODAKA A. For any degenerating Calabi-Yau family, we introduce new limit space which we call galaxy, whose dense subspace is the disjoint union of countably infinite open Calabi-Yau varieties, parametrized by the rational points of the Kontsevich-Soibelman's essential skeleton, while dominated by the Huber adification over the Puiseux series field.Other topics include: projective limits of toroidal compactifications ( §3), locally modelled on limit toric varieties ( §2.4), the way to attach tropicalized family to given Calabi-Yau family ( §4), which are weakly related to each other.Here, B denotes the essential skeleton (as 1.2 again) and B(Q) means its rational points with respect to the Q-affine structure, which does not depend on i by Theorem 2.1 (iii) and, where NKLT stands for the non-klt closed loci.Contents of §3 (Limit toroidal compactifications). In §3 over k = C in turn, we discuss on the projective limit of toroidal compactifications of [AMRT] and its analogue for more general (moduli) varieties. It sounds somewhat independent from §2 but here we observe an analogous phenomenon of the above theorem 1.2, especially the natural continuous maps to their tropical versions.More precisely, what we do there is as follows. Fix a locally Hermitian symmetric space M of non-compact type, i.e., of the ubiquitous form M = Γ\G/K where G is a real valued points of a simple algebraic group over Q, K its maximal compact subgroup with one dimensional center, Γ an arithmetic discrete subgroup of G. Some renowned examples are the moduli space of g-dimensional principally polarized abelian varieties or that of primitively polarized K3 surfaces with possibly ADE singularities (of fixed genera).Recall that for certain combinatorial data i.e., admissible collection of fans Σ = {Σ(F)}, there is an associate toroidal compactification M tor,Σ of M constructed in[AMRT], and its complex analytification M tor,Σ,an . Now, we consider and introduce the projective limit of all of its toroidal compactifications as a locally ringed space and call the limit toroidal compactification: M tor,∞ an := lim ← − Σ

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Section: Here Mmentioning

confidence: 56%

Section: Here Mmentioning

confidence: 99%

Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures

Odaka¹

2020

Preprint

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“…Thus, an analog of Theorem C would be false in the case of Bergman measures2. In fact, to extend the Bergman measures continuously, Amini and Nicolussi construct a large hybrid space which keeps track of the relative orders of the logarithmic rates of approach to each node on a stable curve [AN20].…”

Section: Corollary B the Measures τmentioning

confidence: 99%

“…Tsuji, and Berndtsson-Păun studied the semipositivity of the curvature current of the Narasimhan-Simha metric in families [Tsu07] [BP08]. The asymptotics of the Bergman measure in degenerating families is studied in [HJ96], [Don15], [dJ19], [Shi20] [AN20]. We are interested in computing the asymptotics of the Narasimhan-Simha measure in degenerating families.…”

Section: Introductionmentioning

confidence: 99%

Convergence of Narasimhan-Simha measures on degenerating families of Riemann surfaces

Shivaprasad¹

2020

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Given a compact Riemann surface Y and a positive integer m, Narasimhan and Simha defined a measure on Y associated to the m-th tensor power of the canonical line bundle. We study the limit of this measure on holomorphic families of Riemann surfaces with semistable reduction. The convergence takes place on a hybrid space whose central fiber is the associated metrized curve complex in the sense of Amini and Baker. We also study the limit of the measure induced by the Hermitian pairing defined by Narasimhan-Simha measure. For m = 1, both these measures coincide with the Bergman measure on Y . We also extend the definition of the Narasimhan-Simha measure to the singular curves on the boundary of Mg in such a way that these measures form a continuous family of measures on the universal curve over Mg.

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“…However, in our previous work [AN20] it became clear that the Deligne-Mumford compactification is not the right compactification to study the variations of the Arakelov-Bergman measures µ Ar . Since the Arakelov Green function of a Riemann surface is defined relative to the measure, essentially the same problem arises in the study of the Green functions.…”

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confidence: 99%

Moduli of hybrid curves II: Tropical and hybrid Laplacians

Amini¹,

Nicolussi²

2022

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The present paper is a sequel to our work on hybrid geometry of curves and their moduli spaces. We introduce a notion of hybrid Laplacian, formulate a hybrid Poisson equation, and give a mathematical meaning to the convergence both of the Laplace operator and the solutions to the Poisson equation on Riemann surfaces. As the main theorem of this paper, we then obtain a layered description of the asymptotics of Arakelov Green functions on Riemann surfaces close to the boundary of their moduli spaces. This is done in terms of a suitable notion of hybrid Green functions.As a byproduct of our approach, we obtain other results of independent interest. In particular, we introduce higher rank canonical compactifications of fans and polyhedral spaces and use them to define the moduli space of higher rank tropical curves. Moreover, we develop the first steps of a function theory in higher rank non-Archimedean, hybrid, and tame analysis. Furthermore, we establish the convergence of the Laplace operator on metric graphs toward the tropical Laplace operator on limit tropical curves in the corresponding moduli spaces, leading to new perspectives in operator theory on metric graphs.Our result on the Arakelov Green function is inspired by the works of several authors, in particular those of Faltings, de Jong, Wentworth and Wolpert, and solves a long-standing open problem arising from the Arakelov geometry of Riemann surfaces. The hybrid layered behavior close to the boundary of moduli spaces is expected to be a broad phenomenon and will be explored in our forthcoming work. ContentsPart III: Hybrid curves 13. Hybrid log maps I: towards the hybrid boundary 14. Hybrid log maps II: from Riemann surfaces to hybrid curves

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Moduli of hybrid curves and variations of canonical measures

Cited by 5 publications

References 33 publications

Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures

Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures

Convergence of Narasimhan-Simha measures on degenerating families of Riemann surfaces

Moduli of hybrid curves II: Tropical and hybrid Laplacians

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