The Moduli Space of Twisted Canonical Divisors

Farkas, Gavril; Pandharipande, Rahul

doi:10.1017/s1474748016000128

Cited by 74 publications

(144 citation statements)

References 29 publications

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“…Moduli space of twisted canonical divisors. In [15], Farkas and Pandharipande proposed another compactification of the strata. Let g, n, m be non-integers such that 2g − 2 + n + m > 0.…”

Section: Applications and Related Workmentioning

confidence: 99%

Cohomology classes of strata of differentials

Sauvaget

2019

Geom. Topol.

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We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute the Poincaré-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them.In a second part of the paper we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relations in these Picard groups.

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“…Moduli space of twisted canonical divisors. In [15], Farkas and Pandharipande proposed another compactification of the strata. Let g, n, m be non-integers such that 2g − 2 + n + m > 0.…”

Section: Applications and Related Workmentioning

confidence: 99%

Cohomology classes of strata of differentials

Sauvaget

2019

Geom. Topol.

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“…We then developed the current approach to the problem using our solution of the jump problem. The current paper, and our proof, are completely independent of the concurrent and independent progress on the compactifications of strata of differentials with prescribed zeroes, in [Gen18,Che17,FP18,BCGGM18].…”

Section: Introductionmentioning

confidence: 86%

Real-normalized differentials: limits on stable curves

Grushevsky¹,

Krichever²,

Norton³

2019

Russ. Math. Surv.

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We study the behavior of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration. We describe all possible limits of RN differentials on any stable curve. In particular we prove that the residues at the nodes are solutions of a suitable Kirchhoff problem on the dual graph of the curve. We further show that the limits of zeroes of RN differentials are the divisor of zeroes of a twisted differential an explicitly constructed collection of RN differentials on the irreducible components of the stable curve, with higher order poles at some nodes.Our main tool is a new method for constructing differentials (in this paper, RN differentials, but the method is more general) on smooth Riemann surfaces, in a plumbing neighborhood of a given stable curve. To accomplish this, we think of a smooth Riemann surface as the complement of a neighborhood of the nodes in a stable curve, with boundary circles identified pairwise. Constructing a differential on a smooth surface with prescribed singularities is then reduces to a construction of a suitably normalized holomorphic differential with prescribed "jumps" (mismatches along the identified circles). We solve this additive analog of the multiplicative Riemann-Hilbert problem in a new way, by using iteratively the Cauchy integration kernels on the irreducible components of the stable curve, instead of using the Cauchy kernel on the plumbed smooth surface. As the stable curve is fixed, this provides explicit estimates for the differential constructed, and allows a precise degeneration analysis.

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“…Uniqueness of the r-spin CohFT in genus 0 follows easily from the initial conditions (4) and the axioms of a CohFT with unit. The genus 0 sector of the CohFT W r defines a quantum product 14 • on V r . The resulting algebra (V r , •, 1), even after extension to C, is not semisimple.…”

Section: And 4 Markingsmentioning

confidence: 99%

Cohomological Field Theory Calculations

Pandharipande

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Cohomological field theories (CohFTs) were defined in the mid 1990s by Kontsevich and Manin to capture the formal properties of the virtual fundamental class in Gromov-Witten theory. A beautiful classification result for semisimple CohFTs (via the action of the Givental group) was proven by Teleman in 2012. The Givental-Teleman classification can be used to explicitly calculate the full CohFT in many interesting cases not approachable by earlier methods.My goal here is to present an introduction to these ideas together with a survey of the calculations of the CohFTs obtained from • Witten's classes on the moduli spaces of r-spin curves, • Chern characters of the Verlinde bundles on the moduli of curves, • Gromov-Witten classes of Hilbert schemes of points of C 2 . The subject is full of basic open questions.The boundary 1 of the Deligne-Mumford compactification is the closed locus parameterizing curves with a least one node, ∂M g,n = M g,n \ M g,n .By identifying the last two markings of a single (n + 2)-pointed curve of genus g − 1, we obtain a morphismSimilarly, by identifying the last markings of separate pointed curves, we obtainwhere n = n 1 + n 2 and g = g 1 + g 2 . The images of both q and r lie in the boundary ∂M g,n ⊂ M g,n .The cohomology and Chow groups of the moduli space of curves are H * (M g,n , Q) and A * (M g,n , Q) .While there has been considerable progress in recent years, many basic questions about the cohomology and algebraic cycle theory remain open. 2 Gromov-Witten classesLet X be a nonsingular projective variety over C, and let

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The Moduli Space of Twisted Canonical Divisors

Cited by 74 publications

References 29 publications

Cohomology classes of strata of differentials

Cohomology classes of strata of differentials

Real-normalized differentials: limits on stable curves

Cohomological Field Theory Calculations

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