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1. THE INTEGRAL COHOMOLOGY OF THE HILBERT SCHEME OF TWO POINTS

   https://doi.org/10.1017/fms.2016.5  

   TOTARO, BURT (January 2016, Forum of Mathematics, Sigma)

   null (Ed.)

   The Hilbert scheme $$X^{[a]}$$ of points on a complex manifold $$X$$ is a compactification of the configuration space of $$a$$ -element subsets of $$X$$ . The integral cohomology of $$X^{[a]}$$ is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of $$X^{[2]}$$ for any complex manifold $$X$$ , and the integral cohomology of $$X^{[2]}$$ when $$X$$ has torsion-free cohomology. 

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2. OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

   https://doi.org/10.1017/fms.2018.16  

   ASOK, A.; FASEL, J.; HOPKINS, M. (March 2019, Forum of mathematics)

   Suppose is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension , it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general. 

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3. The essential dimension of congruence covers

   https://doi.org/10.1112/S0010437X21007594  

   Farb, Benson; Kisin, Mark; Wolfson, Jesse (November 2021, Compositio Mathematica)

   Abstract Consider the algebraic function $$\Phi _{g,n}$$ that assigns to a general $$g$$ -dimensional abelian variety an $$n$$ -torsion point. A question first posed by Klein asks: What is the minimal $$d$$ such that, after a rational change of variables, the function $$\Phi _{g,n}$$ can be written as an algebraic function of $$d$$ variables? Using techniques from the deformation theory of $$p$$ -divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $$p$$ -dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $$p$$ -dimension of congruence covers of the moduli space of genus $$g$$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $$M$$ is proper . As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety. 

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4. On the v -Picard Group of Stein Spaces

   https://doi.org/10.1093/imrn/rnae185  

   Ertl, Veronika; Gilles, Sally; Nizioł, Wiesława (September 2024, International Mathematics Research Notices)

   Abstract We study the image of the Hodge–Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious $$\mathbf{Z}_{p}$$-module that maps, via the Bloch–Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit disks of dimension strictly greater than $$1$$. 

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5. Modular functions and resolvent problems: With an appendix by Nate Harman

   https://doi.org/10.1007/s00208-022-02395-8  

   Farb, Benson; Kisin, Mark; Wolfson, Jesse (January 2022, Mathematische Annalen)

   The link between modular functions and algebraic functions was a driving force behind the 19th century study of both. Examples include the solutions by Hermite and Klein of the quintic via elliptic modular functions and the general sextic via level 2 hyperelliptic functions. This paper aims to apply modern arithmetic techniques to the circle of “resolvent problems” formulated and pursued by Klein, Hilbert and others. As one example, we prove that the essential dimension at 𝑝=2 for the symmetric groups 𝑆𝑛 is equal to the essential dimension at 2 of certain 𝑆𝑛-coverings defined using moduli spaces of principally polarized abelian varieties. Our proofs use the deformation theory of abelian varieties in characteristic p, specifically Serre-Tate theory, as well as a family of remarkable mod 2 symplectic 𝑆𝑛-representations constructed by Jordan. As shown in an appendix by Nate Harman, the properties we need for such representations exist only in the 𝑝=2 case. In the second half of this paper we introduce the notion of ℰ-versality as a kind of generalization of Kummer theory, and we prove that many congruence covers are ℰ-versal. We use these ℰ-versality result to deduce the equivalence of Hilbert’s 13th Problem (and related conjectures) with problems about congruence covers. 

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