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1. Projectivity of the moduli of curves

   https://doi.org/10.1017/9781009051897.003  

   Cheng, Raymond; Lian, Carl; Murayama, Takumi (October 2022, Cambridge University Press)

   Belmans, Pieter; Ho, Wei; de_Jong, Aise Johan (Ed.)

   In this expository paper, we show that the Deligne–Mumford moduli space of stable curves is projective over Spec(Z). The proof we present is due to Kollár. Ampleness of a line bundle is deduced from the nefness of a related vector bundle via the ampleness lemma, a classifying map construction. The main positivity result concerns the pushforward of relative dualizing sheaves on families of stable curves over a smooth projective curve. 

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2. Stability in the homology of Deligne–Mumford compactifications

   https://doi.org/10.1112/S0010437X21007582  

   Tosteson, Philip (December 2021, Compositio Mathematica)

   Abstract Using the theory of $${\mathbf {FS}} {^\mathrm {op}}$$ modules, we study the asymptotic behavior of the homology of $${\overline {\mathcal {M}}_{g,n}}$$ , the Deligne–Mumford compactification of the moduli space of curves, for $$n\gg 0$$ . An $${\mathbf {FS}} {^\mathrm {op}}$$ module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via copies that glue on marked projective lines, we give the homology of $${\overline {\mathcal {M}}_{g,n}}$$ the structure of an $${\mathbf {FS}} {^\mathrm {op}}$$ module and bound its degree of generation. As a consequence, we prove that the generating function $$\sum _{n} \dim (H_i({\overline {\mathcal {M}}_{g,n}})) t^n$$ is rational, and its denominator has roots in the set $$\{1, 1/2, \ldots, 1/p(g,i)\},$$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$ . We also obtain restrictions on the decomposition of the homology of $${\overline {\mathcal {M}}_{g,n}}$$ into irreducible $$\mathbf {S}_n$$ representations. 

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3. Holomorphic Floer theory and Donaldson-Thomas invariants

   Bousseau, Pierrick (April 2024, Proceedings of symposia in pure mathematics)

   We present several expected properties of the holomorphic Floer theory of a holomorphic symplectic manifold. In particular, we propose a conjecture relating holomorphic Floer theory of Hitchin integrable systems and Donaldson-Thomas invariants of non-compact Calabi-Yau 3-folds. More generally, we conjecture that the BPS spectrum of a 4-dimensional N = 2 quantum field theory can be recovered from the holomorphic Floer theory of the corresponding Seiberg-Witten integrable system. 

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4. Title: String-Math 2022 Article: Holomorphic Floer theory and Donaldson-Thomas invariants

   Bousseau, Pierrick (April 2024, AMS)

   Donag, Ron; Langer, Adrian; Sułkowski, Piotr; Wendland, Katrin (Ed.)

   We present several expected properties of the holomorphic Floer theory of a holomorphic symplectic manifold. In particular, we propose a conjecture relating holomorphic Floer theory of Hitchin integrable systems and Donaldson-Thomas invariants of non-compact Calabi-Yau 3-folds. More generally, we conjecture that the BPS spectrum of a 4-dimensional N = 2 quantum field theory can be recovered from the holomorphic Floer theory of the corresponding Seiberg-Witten integrable system. 

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5. Moduli Spaces of Generalized Hyperpolygons

   https://doi.org/10.1093/qmath/haaa036  

   Rayan, Steven; Schaposnik, Laura P (December 2020, The Quarterly Journal of Mathematics)

   null (Ed.)

   Abstract We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genus-g Riemann surface (generally with positive codimension). We show that, under certain assumptions on flag types, the space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system of Gelfand–Tsetlin type, inherited from the reduction of partial flag varieties. In the case where all flags are complete, we present the Hamiltonians explicitly. We also remark upon the discretization of the Hitchin equations given by hyperpolygons, the construction of triple branes (in the sense of Kapustin–Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents). 

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