Markus Szymik
The notion of a K3 spectrum is introduced in analogy with that of an elliptic spectrum and it is shown that there are "enough" K3 spectra in the sense that for all K3 surfaces X in a suitable moduli stack of K3 surfaces there is a K3 spectrum whose underlying ring is isomorphic to the local ring of the moduli stack in X with respect to the étale topology, and similarly for the ring of formal functions on the formal deformation space.
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We define a filtration by open substacks on the non-connective spectral moduli stack of formal oriented groups, which simultaneously encodes and relates the chromatic filtration of spectra and the height stratification of the classical moduli stack of formal groups. Using this open filtration, we express various classical constructions in chromatic homotopy theory, such as chromatic localization, the monochromatic layer, and $K(n)$-localization, in terms of restriction and completion of sheaves in non-connective spectral algebraic geometry.
The central aim of this monograph is to provide decomposition results for quasi-coherent sheaves on the moduli stack of one-dimensional formal groups. These results will be based on the geometry of the stack itself, particularly the height filtration and an analysis of the formal neighborhoods of the geometric points. The main theorems are algebraic chromatic convergence results and fracture square decompositions. There is a major technical hurdle in this story, as the moduli stack of formal groups does not have the finitness properties required of an algebraic stack as usually defined. This i…
Cohomology of the Moduli Stacks of Real Vector Bundles on Type I Real Algebraic Curves
研究第一类实代数曲线上固定秩和度的实向量丛模空间，并用示性类确定其模2上同调代数。
We study the singular homology (with field coefficients) of the moduli stack of stable n-pointed complex curves of genus g (the Deligne-Mumford compactification). Each of its irreducible boundary components determines via the Pontrjagin-Thom construction a map to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This implies the existence of many new torsion classes in the homology of the moduli stack.
Rational Homotopy and Hodge Theory of Moduli Stacks of principal $G$-bundles
本文利用拓扑栈的同伦理论，确定了特殊类型复射影簇上主$G$-丛模栈的有理上同调与Hodge-Tate结构。
Using the machinery of etale homotopy theory a' la Artin-Mazur we determine the etale homotopy types of moduli stacks over $\bar{\Q}$ parametrizing families of algebraic curves of genus g greater than 1 endowed with an action of a finite group G of automorphisms, which comes with a fixed embedding in the mapping class group, such that in the associated complex analytic situation the action of G is precisely the differentiable action induced by this specified embedding of G in the mapping class group.
We exhibit a relationship between motivic homotopy theory and spectral algebraic geometry, based on the motivic $τ$-deformation picture of Gheorghe, Isaksen, Wang, Xu. More precisely, we identify cellular motivic spectra over $\mathbf C$ with ind-coherent sheaves (in a slightly non-standard sense) on a certain spectral stack $τ_{\ge 0}(\mathcal M_\mathrm{FG}^\mathrm{or})$. The latter is the connective cover of the non-connective spectral stack $\mathcal M_\mathrm{FG}^\mathrm{or}$, the moduli stack of oriented formal groups, which we have introduced previously and studied in connection with chr…
In this paper, we study the so-called Getzler-Kapranov complexes and their relation to the cohomology of moduli stacks of curves.
We introduce and study the non-connective spectral stack $\mathcal M_\mathrm{FG}^\mathrm{or}$, the moduli stack of oriented formal groups. We realize some results of chromatic homotopy theory in terms of the geometry of this stack. For instance, we show that its descent spectral sequence recovers the Adams-Novikov spectral sequence. For two $\mathbb E_\infty$-forms of periodic complex bordism $\mathrm{MP}$, the Thom spectrum and Snaith construction model, we describe the universal property of the cover $\mathrm{Spec}(\mathrm{MP})\to\mathcal M_\mathrm{FG}^\mathrm{or}$. We show that Quillen's ce…
We study the inertia stack of [M_{0,n}/S_n], the quotient stack of the moduli space of smooth genus 0 curves with n marked points via the action of the symmetric group S_n. Then we see how from this analysis we can obtain a description of the inertia stack of H_g, the moduli stack of hyperelliptic curves of genus g. From this, we can compute additively the Chen-Ruan (or orbifold) cohomology of H_g.
We develop homotopical algebraic geometry (see math.AG/0207028) in the special context where the base symmetric monoidal model category is the category S of spectra, i.e. what might be called, after Waldhausen, ``brave new algebraic geometry''. We discuss various model topologies on the model category of commutative algebras in S, the associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry), and finally show how to define global moduli spaces of associative ring spectra structures and a moduli space related to topo…
Moduli stack of stable curves from a stratified homotopy viewpoint
本文加强了Charney和Lee的结果，证明了稳定曲线范畴能捕获模栈的分层同伦类型，并通过外行进等价对可构造层进行分类。
Let $\mathscr{V}\mathrm{ect}_n$ be the moduli stack of vector bundles of rank $n$ on schemes. We prove that, if $E$ is a Zariski sheaf of ring spectra which is equipped with finite quasi-smooth transfers and satisfies the projective bundle formula, then $E^*(\mathscr{V}\mathrm{ect}_{n,S})$ is freely generated by Chern classes $c_1,\dotsc,c_n$ over $E^*(S)$ for any scheme $S$. Examples include all multiplicative localizing invariants.
We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate diagram category form affine stacks in the sense of Toen-Vezzosi's homotopical algebraic geometry. This includes simplicial moduli spaces of algebraic structures over a given object (for instance a cochain complex). When these algebraic structures are parametrised by properads, the tangent complexes give the known cohomology theory for such structures and there i…
Exotic periodic phenomena in the cohomology of the moduli stack of $1$-dimensional formal group laws
本文研究一维形式群律模栈上同调中的新周期结构，区别于$v_n$周期性，通过$\mathbb{C}$-动机Adams谱序列在动机稳定同伦中揭示类似现象。
On the differential $K$-theory of moduli stacks
本文计算了主 $G$-丛带联络模空间的连通微分 $K$-理论和微分上同调，结果以不变多项式和表示环表述，并利用空间与谱预层的同伦理论证明。
In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the "underlying" diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left adjoint: the functor sending any stack to its diffeological coarse moduli space. As an application, we restrict our attention to differentiable stacks and examine the geometry behind the coarse moduli space construction in terms of Lie groupoids and their principal bundles. With…
The higher gauge field in 11-dimensional supergravity -- the C-field -- is constrained by quantum effects to be a cocycle in some twisted version of differential cohomology. We argue that it should indeed be a cocycle in a certain twisted nonabelian differential cohomology. We give a simple and natural characterization of the full smooth moduli 3-stack of configurations of the C-field, the gravitational field/background, and the (auxiliary) E8-field. We show that the truncation of this moduli 3-stack to a bare 1-groupoid of field configurations reproduces the differential integral Wu structure…
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