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An arithmetic Hilbert–Samuel theorem for pointed stable curves

Gerard Freixas i Montplet (2012)

Journal of the European Mathematical Society

Let ( 𝒪 , , F ) be an arithmetic ring of Krull dimension at most 1 , S = Spec ( 𝒪 ) and ( 𝒳 S ; σ 1 , ... , σ n ) a pointed stable curve. Write 𝒰 = 𝒳 j σ j ( S ) . For every integer k > 0 , the invertible sheaf ω 𝒳 / S k + 1 ( k σ 1 + ... + k σ n ) inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface 𝒰 . In this article we define a Quillen type metric · Q on the determinant line λ k + 1 = λ ω 𝒳 / S k + 1 ( k ...

Chen–Ruan Cohomology of 1 , n and ¯ 1 , n

Nicola Pagani (2013)

Annales de l’institut Fourier

In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable n -pointed curves of genus 1 . In the first part of the paper we study and describe stack theoretically the twisted sectors of 1 , n and ¯ 1 , n . In the second part, we study the orbifold intersection theory of ¯ 1 , n . We suggest a definition for an orbifold tautological ring in genus 1 , which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.

Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations

Dan Abramovich, Steffen Marcus, Jonathan Wise (2014)

Annales de l’institut Fourier

We consider four approaches to relative Gromov–Witten theory and Gromov–Witten theory of degenerations: J. Li’s original approach, B. Kim’s logarithmic expansions, Abramovich–Fantechi’s orbifold expansions, and a logarithmic theory without expansions due to Gross–Siebert and Abramovich–Chen. We exhibit morphisms relating these moduli spaces and prove that their virtual fundamental classes are compatible by pushforward through these morphisms. This implies that the Gromov–Witten invariants associated...

de Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities

Matthew Satriano (2012)

Annales de l’institut Fourier

We prove that the Hodge-de Rham spectral sequence for smooth proper tame Artin stacks in characteristic p (as defined by Abramovich, Olsson, and Vistoli) which lift mod p 2 degenerates. We push the result to the coarse spaces of such stacks, thereby obtaining a degeneracy result for schemes which are étale locally the quotient of a smooth scheme by a finite linearly reductive group scheme.

Moduli of unipotent representations I: foundational topics

Ishai Dan-Cohen (2012)

Annales de l’institut Fourier

With this work and its sequel, Moduli of unipotent representations II, we initiate a study of the finite dimensional algebraic representations of a unipotent group over a field of characteristic zero from the modular point of view. Let G be such a group. The stack n ( G ) of all representations of dimension n is badly behaved. In this first installment, we introduce a nondegeneracy condition which cuts out a substack n nd ( G ) which is better behaved, and, in particular, admits a coarse algebraic space, which...

On the motives of moduli of chains and Higgs bundles

Oscar García-Prada, Jochen Heinloth, Alexander Schmitt (2014)

Journal of the European Mathematical Society

We take another approach to Hitchin’s strategy of computing the cohomology of moduli spaces of Higgs bundles by localization with respect to the circle action. Our computation is done in the dimensional completion of the Grothendieck ring of varieties and starts by describing the classes of moduli stacks of chains rather than their coarse moduli spaces. As an application we show that the n -torsion of the Jacobian acts trivially on the middle dimensional cohomology of the moduli space of twisted...

Quantum Singularity Theory for A ( r - 1 ) and r -Spin Theory

Huijun Fan, Tyler Jarvis, Yongbin Ruan (2011)

Annales de l’institut Fourier

We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the r -spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity W of type A our construction of the stack of W -curves is canonically isomorphic to the stack of r -spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an r -spin virtual class. Therefore, the Faber-Shadrin-Zvonkine proof of the...

Representation Theorem for Stacks

Grzegorz Bancerek (2011)

Formalized Mathematics

In the paper the concept of stacks is formalized. As the main result the Theorem of Representation for Stacks is given. Formalization is done according to [13].

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