Recent zbMATH articles in MSC 14Fhttps://zbmath.org/atom/cc/14F2022-10-04T19:40:27.024758ZWerkzeugBook review of: B. Poonen, Rational points on varieties; S. D. Cutkosky, Introduction to algebraic geometryhttps://zbmath.org/1492.000272022-10-04T19:40:27.024758Z"Oort, Frans"https://zbmath.org/authors/?q=ai:oort.frans|oort.frans-jReview of [Zbl 1387.14004; Zbl 1396.14001].Derived functor cohomology groups with Yoneda producthttps://zbmath.org/1492.130202022-10-04T19:40:27.024758Z"Husain, Hafiz Syed"https://zbmath.org/authors/?q=ai:husain.hafiz-syed"Sultana, Mariam"https://zbmath.org/authors/?q=ai:sultana.mariamSummary: This work presents an exposition of both the internal structure of derived category of an abelian category \(D^*(\mathcal{A})\) and its contribution in solving problems, particularly in algebraic geometry. Calculation of some morphisms will be presented between objects in \(D^*(\mathcal{A})\) as elements in appropriate cohomology groups along with their compositions with the help of Yoneda construction under the assumption that the homological dimension of \(D^*(\mathcal{A})\) is greater than or equal to 2. These computational settings will then be considered under sheaf cohomological context with a particular case from projective geometry.The derived deformation theory of a pointhttps://zbmath.org/1492.140052022-10-04T19:40:27.024758Z"Booth, Matt"https://zbmath.org/authors/?q=ai:booth.matthewThis article provides a comprehensive and detailed study of the derived deformation theory of modules over a noncommutative algebra. The classical (not derived), commutative and characteristic zero counterpart to this story is very well-known and admits a nice description in terms of local algebras, which serve as prorepresenting objects. For example, studying deformations of some \(k\)-scheme over a one-dimensional base, one finds the ring of formal power series \(k[[t]]\), which can be seen either as a pro-object in commutative Artin algebras, or as a commutative complete local \(k\)-algebra.
In the setting of this paper, that is, modules over a noncommutative algebra, the situation is more complicated, since the category of pro-objects in connective Artin dgas does not admit this embedding as local algebras, forcing one to work with pro-Artin dgas. After an exposition of the model category theory and Koszul duality (Sections 3 and 4) in that setting, the author proceeds to presenting the pro-representing object in general, given by the `continuous nonunital Koszul dual' (Prop. 5.4.7).
Section 6 is devoted to the problem of `deforming the point' mentioned by the title: describing the deformations of a one-dimensional module. It is shown that the usual Koszul dual (or more precisely, its pro-version) is a representing object for the problem of framed deformations, that is, of deforming the pair made up of the object and an isomorphism to \(k\). Using this result, the author recovers and extends some results of E. Segal, and gives a deformation-theoretic explanation of a certain quotient construction (Theorem 6.3.9).
As mentioned before, in this derived noncommutative world, it becomes necessary to work with pro-objects, so Section 7 presents some technical results on the deformation theory in that setting. Finally, Section 8 deals with multi-pointed deformations, that is, tuples of deformations with some compatibility data, giving deformations of an object together with multiple augmentations. This is motivated by the appearance of such objects in the study of CY 3-folds; and the author presents a succinct, but rather complete, summary of what would change in the previous sections of the paper for these multi-pointed analogues.
Reading this paper, I particularly enjoyed two parts, which I believe are of interest for the reader searching for precise summaries of results: in Section 4, when discussing Koszul duality, the author is very clear about which conditions must hold for certain quasi-isomorphisms involving the Koszul double dual to exist, see remarks 4.1.5--4.1.15. I think this will be of good use to whoever needs to use these constructions with algebras that fail the traditional finiteness conditions. Also, Section 5.1 presents a convenient summary of the main results involving the Deligne groupoid and the Maurer-Cartan simplicial set, which otherwise are spread around the literature spanning many years. The interested reader can find precise references for all these results in that section of this paper.
Reviewer: Alex Takeda (Bures-sur-Yvette)Prime thick subcategories on elliptic curveshttps://zbmath.org/1492.140272022-10-04T19:40:27.024758Z"Hirano, Yuki"https://zbmath.org/authors/?q=ai:hirano.yuki"Ouchi, Genki"https://zbmath.org/authors/?q=ai:ouchi.genkiSummary: We classify all prime thick subcategories in the derived categories of coherent sheaves on elliptic curves, and determine the Serre invariant locus of Matsui spectra of derived categories of coherent sheaves on any smooth projective curves.On the derived category of a weighted projective threefoldhttps://zbmath.org/1492.140282022-10-04T19:40:27.024758Z"Kawamata, Yujiro"https://zbmath.org/authors/?q=ai:kawamata.yujiroSummary: We calculate a semi-orthogonal decomposition of the bounded derived category of coherent sheaves on \(\mathbb{P}(1,1,1,3)\) using a tilting bundle.The integral Hodge conjecture for two-dimensional Calabi-Yau categorieshttps://zbmath.org/1492.140292022-10-04T19:40:27.024758Z"Perry, Alexander"https://zbmath.org/authors/?q=ai:perry.alexander-rThis paper formulates a version of the integral Hodge conjecture for categories and its variational cousin, and proves the conjectures for CY2 (two-dimensional Calabi-Yau) categories which are deformations of the derived category of a \(K3\) or abelian surface in a suitable sense. See Theorems 1.1, Corollary 1.2, and Theorem 1.3 for the precise definition and statements).
The main notion introduced in this paper is the Mukai structure \(\widetilde{\operatorname{H}}(\mathcal{C},\mathbb{Z})\) for any CY2 category \(\mathcal{C}\), which is the categorical analogue of the integral Hodge structure of a variety. It is defined as the weight 2 Tate twist of Blanc's topological K-theory \(\operatorname{K}_0^{\text{top}}(\mathcal{C})\) introduced in [\textit{A. Blanc}, Compos. Math. 152, No. 3, 489--555 (2016; Zbl 1343.14003)]. In the case \(\mathcal{C}=\operatorname{D}_{\text{perf}}(X)\) for a K3 or abelian surface \(X\), it recovers the usual Mukai Hodge structure of \(X\). See \S5.2 and \S6 for the detail.
The main argument is to study the variational property of the topological K-theory, using a relative topological K-theory \(\operatorname{K}_0^{\text{top}}(\mathcal{C}/S)\) for a CY2 category over a \(\mathbb{C}\)-scheme \(S\) from [\textit{T. Moulinos}, Adv. Math. 351, 761--803 (2019; Zbl 1460.14006)]. Using the moduli spaces of objects in categories, the paper proves that the class of simple universal gluable object of \(\mathcal{C}_0\) stays algebraic at every other \(s \in S(\mathbb{C})\). See Theorem 1.3 for the detail.
There are several applications of the argument to the structure of intermediate Jacobians, one of which is a criterion in terms of derived categories for when they split as a sum of Jacobians of curves. See Theorem 1.6 for the detail.
Reviewer: Shintaro Yanagida (Nagoya)The universal de Rham / Spencer double complex on a supermanifoldhttps://zbmath.org/1492.140302022-10-04T19:40:27.024758Z"Cacciatori, Sergio Luigi"https://zbmath.org/authors/?q=ai:cacciatori.sergio-luigi"Noja, Simone"https://zbmath.org/authors/?q=ai:noja.simone"Re, Riccardo"https://zbmath.org/authors/?q=ai:re.riccardoLet \((M,\mathcal{O}_M)\) be a (real or complex) supermanifold. Its de Rham complex \(\textit{DR}(M)\) is usually not bounded from above, as the \(k\)-fold product of ``even'' 1-forms is no longer zero when \(k\) is bigger than the dimension. Associated to the supermanifold \(M\) is also a ``complex of integral forms'': \(\mathcal{H}\textit{om}_{\mathcal{O}_{M}}(\textit{DR}(M),\mathcal{B}\textit{er}(M))\) (\(\mathcal{B}\textit{er}(M)\) being the Berezinian). This complex is not bounded from below.
In the article under review, the authors introduce a double complex, which they call the ``universal de Rham/Spencer double complex''. The \(E_{1}\) pages of two associated spectral sequences of this double complex are \(\textit{DR}(M)\) and the complex of integral forms, respectively, both of which degenerate at \(E_2\).
In particular, both the de Rham complex and the complex of integral forms are quasi-isomorphic to the constant sheaf (actually, the proof of the above mentioned result relies on these Poincaré lemmas). For de Rham complex, the Poincaré lemma is easy in the super context; for the complex of integral forms, the proof of its Poincaré lemma does not seem to be documented elsewhere, so the authors provide a proof of it.
The authors also point out that if the reduced manifold of \(M\) is a compact Kähler manifold, the Hodge-to-de Rham spectral sequence may not be \(E_1\)-degenerate. A counterexample is given: the underlying manifold is an elliptic curve \(E\), and super structure is split: \(\mathcal{O}_M = \mathcal{O}_E \oplus \mathcal{O}_{E}\).
Reviewer: Dingxin Zhang (Beijing)Intermediate extensions and crystalline distribution algebrashttps://zbmath.org/1492.140312022-10-04T19:40:27.024758Z"Huyghe, Christine"https://zbmath.org/authors/?q=ai:huyghe.christine"Schmidt, Tobias"https://zbmath.org/authors/?q=ai:schmidt.tobiasIn a series of works, the authors introduced and studied the crystalline distribution algebra \(D^{\dagger}(G)\) for a connected split reductive group \(G\) over a complete discrete valuation ring of mixed characteristic \(\mathfrak{o}\). Moreover, they defined a functor from the category of certain admissible representation of the \(p\)-adic group \(G(K)\) to the category of coherent modules over \(D^{\dagger}(G)\), where \(K=\mathfrak{o}[\frac{1}{p}]\).
In this article, the authors use the intermediate extension and an arithmetic Beilinson-Bernstein localization functor to classify irreducible modules over \(D^{\dagger}(G)\) in terms of overconvergent \(F\)-isocrystals on locally closed subschemes in the (formal) flag variety of \(G\). Moreover, they studies two examples in the \(SL_2\)-case.
Reviewer: Daxin Xu (Beijing)Deformations of overconvergent isocrystals on the projective linehttps://zbmath.org/1492.140322022-10-04T19:40:27.024758Z"Agrawal, Shishir"https://zbmath.org/authors/?q=ai:agrawal.shishirA local system \(L\) on a nonempty Zariski open subset \(U\) of \(\mathbb{P}^{1,\mathrm{an}}_{\mathbb{C}}\) that is determined by its local monodromy is called a physically rigid local system. \textit{N. M. Katz} [Rigid local systems. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0864.14013)] proved that if \(L\) is irreducible, then being physically rigid is equivalent to the vanishing of the intersection (= parabolic) cohomology \(H^1_p(\mathrm{End}(L)):=H^{1}(\mathbb{P}^{1,\mathrm{an}}_{\mathbb{C}}, j_{\ast}\mathrm{End}(L))\), where \(j\colon U \to \mathbb{P}^{1}_{\mathbb{C}}\) is the open immersion. In a precise sense, the intersection cohomology space can be regarded as the tangent space to \(L\) in a suitable moduli stack.
The analogue of a local system in rigid cohomology is an overconvergent isocrystal. It is also reasonable to regard the restriction of the isocrystal to the Robba rings of the punctures as a replacement of data of local monodromy. Thus one can define the notion of a physically rigid isocrystal. For an absolutely irreducible overconvergent F-isocrystal, \textit{R. Crew} [Doc. Math. 22, 287--296 (2017; Zbl 1391.14038)] showed that ``vanishing parabolic cohomology'' \(\Longrightarrow\) ``physically rigid'', i.e., the analogue of Katz's theorem holds.
In the article under review, the author studies the local deformation theory of isocrystals. The author proves that the rigid cohomology groups \(H^1(\mathrm{End}(L))\), \(H^1_c(\mathrm{End}(L))\), \(H^1_{p}(\mathrm{End}(L))\) are Zariski tangent spaces of some functors of artin rings, and these deformation functors have hulls if, for example, \(\mathrm{End}(L)\) admits a Frobenius structure. If moreover \(L\) is absolutely irreducible, some smoothness and pro-representability results about these functors are proved.
Reviewer: Dingxin Zhang (Beijing)Zeroth \(\mathbb{A}^1\)-homology of smooth proper varietieshttps://zbmath.org/1492.140332022-10-04T19:40:27.024758Z"Koizumi, Junnosuke"https://zbmath.org/authors/?q=ai:koizumi.junnosukeSummary: We give an explicit formula for the zeroth \(\mathbb{A}^1\)-homology sheaf of a smooth proper variety. We also provide a simple proof of a theorem of Kahn-Sujatha which describes hom sets in the birational localization of the category of smooth varieties.Tensor structures in the theory of modulus presheaves with transfershttps://zbmath.org/1492.140342022-10-04T19:40:27.024758Z"Rülling, Kay"https://zbmath.org/authors/?q=ai:rulling.kay"Sugiyama, Rin"https://zbmath.org/authors/?q=ai:sugiyama.rin"Yamazaki, Takao"https://zbmath.org/authors/?q=ai:yamazaki.takaoConsider the diagram
\[
\begin{tikzcd} \textbf{Cor} \arrow[r, equal] \arrow[d, "\mathbb{Z}_{tr}"] & \textbf{Cor} \arrow[d, "\mathbb{Z}_{tr}"] & \textbf{MCor} \arrow[l, "\omega"] \arrow[d, "\mathbb{Z}_{tr}"] \\
\textbf{PST} \arrow[d, shift left= 1ex, "h_0^{\mathbb{A}^1}"] \arrow[r, equal] & \textbf{PST} \arrow[r, shift left= 1ex, "\omega^*"] \arrow[d, shift left= 1ex, "\rho"] & \textbf{MPST} \arrow[l, shift left= 1ex, "\omega_!"] \arrow[d, shift left= 1ex, "h_0^{\overline{\boxempty}}"] \\
\textbf{HI} \arrow[u, hookrightarrow, shift left= 1ex, "\iota^{\mathbb{A}^1}"] \arrow[r, hookrightarrow, shift left= 1ex] & \textbf{RSC} \arrow[u, hookrightarrow, shift left= 1ex, "\iota^{\natural}"] \arrow[l, shift left= 1ex, "h_0^{rec}"] \arrow[r, shift left= 1ex, "\omega^{\textbf{CI}}"] & \textbf{CI} \arrow[u, hookrightarrow, shift left= 1ex, "\iota^{\overline{\boxempty}}"] \arrow[l, shift left= 1ex, "\omega_{\textbf{CI}}=\omega_!"] \end{tikzcd}
\]
where:
\begin{itemize}
\item \(\textbf{Cor}\) is Voedosky's category of finite correspondences whose objects are smooth varieties over a perfect base field \(k\),
\item \(\textbf{PST}\) the category of presheaves over \(\textbf{Cor}\),
\item \(\textbf{HI}\) is the full subcategory of \(\textbf{PST}\) consisting of \(\mathbb{A}^1\)-invariant presheaves,
\item \(\textbf{MCor}\) is the category whose objects are proper modulus pairs \(M=(\overline{M}, M^{\infty})\),
\item \(\textbf{MPST}\) the category of presheaves over \(\textbf{MCor}\),
\item \(\textbf{CI}\) the full subcategory of \(\textbf{MPST}\) consisting of \(\overline{\boxempty}\)-invariant presheaves (with \(\overline{\boxempty} = ( \mathbb{P}^1, (\infty))\)),
\item \(\textbf{RSC}\) is the full subcategory of \(\textbf{PST}\) consisting of presheaves having SC-reciprocity,
\item and where the functors are the obvious ones.
\end{itemize}
The column on the left consists of well-known categories defined by \textit{V. Voevodsky} [Ann. Math. Stud. 143, 87--137 (2000; Zbl 1019.14010)].
The other categories were developed in the work of Bruno Kahn, Hiroyasu Miyazaki, Shuji Saito, and Takao Yamazaki, see e.g. [\textit{B. Kahn} et al., Compos. Math. 152, No. 9, 1851--1898 (2016; Zbl 1419.19001)]. These categories contain many interesting non-homotopy invariant presheaves with transfers (such as \(\mathbb{G}_a\) or \(\Omega^i_{/\mathbb{Z}}\)).
The present paper focuses on generalizing the tensor product of \(\mathbb{A}^1\)-invariant presheaves to reciprocity sheaves via the theory of modulus presheaves with transfers.
One of the main result is that, for any \(F_i \in \textbf{HI}\) and any regular local \(k\)-algebra \(R\), we have an isomorphism
\[
\omega_! h_0^{\overline{\boxempty}} ( \omega^{\textbf{CI}}F_1 \otimes^{\textbf{MPST}} \dots \otimes^{\textbf{MPST}} \omega^{\textbf{CI}}F_n ) (R) \simeq (F_1 \otimes^{\textbf{HI}} \dots \otimes^{\textbf{HI}} F_n) (R).
\]
In particular, this result says that the product
\[
\textbf{RSC}^{\times n} \to \textbf{RSC}, (F_1,\dots, F_n) \mapsto \omega_! h_0^{\overline{\boxempty}}(\omega^{\textbf{CI}}F_1 \otimes \dots \otimes \omega^{\textbf{CI}}F_n)
\]
is an extension to \(\textbf{RSC}\) of the tensor structure on \(\textbf{HI}\). \par An important ingredient in the proofs is Saito's result that Nisnevich sheafification preserves \(\textbf{RSC}\), which generalizes the analogue statement for \(\textbf{HI}\) by Voevodsky, see [\textit{S. Saito}, Adv. Math. 366, Article ID 107067, 70 p. (2020; Zbl 1437.19001)].
Another key point is the injectivity property of reciprocity sheaves proved in [\textit{B. Kahn} et al., Compos. Math. 152, No. 9, 1851--1898 (2016; Zbl 1419.19001)] which implies that a morphism between reciprocity sheaves is an isomorphism if it is so on all function fields.
As an application of these methods, the authors obtain that certain higher Chow groups of zero-cycles with modulus condition defined via either the sum or the ssup convention do agree.
Reviewer: Niels Feld (Toulouse)Topology of subvarieties of complex semi-abelian varietieshttps://zbmath.org/1492.140352022-10-04T19:40:27.024758Z"Liu, Yongqiang"https://zbmath.org/authors/?q=ai:liu.yongqiang"Maxim, Laurentiu"https://zbmath.org/authors/?q=ai:maxim.laurentiu-g"Wang, Botong"https://zbmath.org/authors/?q=ai:wang.botongA semi-abelian variety \(G\) is an algebraic group given as an extension \(1\to T\to G\to A\to 1\) with \(T\) a complex affine torus and \(A\) an abelian variety.
The authors study properties of closed submanifolds \(X\) of semi-abelian varieties via non-proper Morse Theory as developed by Palais-Smale [\textit{R. S. Palais} and \textit{S. Smale}, Bull. Am. Math. Soc. 70, 165--172 (1964; Zbl 0119.09201)]. This provides proofs of topological properties of \(X\), with a more classical topological emphasis than in existing literature. For example, in [\textit{J. Franecki} and \textit{M. Kapranov}, Duke Math. J. 104, No. 1, 171--180 (2000; Zbl 1021.14016)] some overlapping results are obtained by using perverse sheaves.
One of the main theorems asserts that for a smooth closed connected \(n\)-dimensional subvariety \(X\) of a semi-abelian variety \(G\) and a generic homomorphism \(\xi:\pi_1(X)\to \mathbb{Z}\) the corresponding abelian cover \(X^\xi\) has finite homotopy type with possible infinitely many \(n\)-cells attached. From this, the authors obtain that:
\begin{itemize}
\item[1.] the topological Euler characteristic is signed: \((-1)^n\chi(X)\geq 0 \),
\item[2.] the generic vanishing holds, this is, for all but finitely many \(s\in \mathbb{C}^*\) composing \(\xi\) with \(1\mapsto s\), the corresponding rank-one \(\mathbb{C}\)-local system \(L_s^\xi\) satisfies \(H_i(X,L_s^\xi)=0\) for all \(i\not =n\),
\item[3.] the integral Alexander modules \(H_i(X^\xi,\mathbb{Z})\) are finitely generated abelian groups for \(i\not=n\).
\end{itemize}
Their methods also allow them to generalize June Hu's extension of Varchenko conjecture [\textit{J. Huh}, Compos. Math. 149, No. 8, 1245--1266 (2013; Zbl 1282.14007)] to the case of \(X\) as above, a closed smooth subvariety of a semi-abelian variety \(G\).
Reviewer: Rodolfo Aguilar Aguilar (Sofia)Brauer-Manin obstructions on degree 2 \(K3\) surfaceshttps://zbmath.org/1492.140372022-10-04T19:40:27.024758Z"Corn, Patrick"https://zbmath.org/authors/?q=ai:corn.patrick"Nakahara, Masahiro"https://zbmath.org/authors/?q=ai:nakahara.masahiroSummary: We analyze the Brauer-Manin obstruction to rational points on the \(K3\) surfaces over \(\mathbb{Q}\) given by double covers of \(\mathbb{P}^{2}\) ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer-Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by \textit{E. Ieronymou} and \textit{A. N. Skorobogatov} [Adv. Math. 270, 181--205 (2015; Zbl 1388.14071)].On reduction of moduli schemes of abelian varieties with definite quaternion multiplicationshttps://zbmath.org/1492.140422022-10-04T19:40:27.024758Z"Yu, Chia-Fu"https://zbmath.org/authors/?q=ai:yu.chia-fuLet \(p>2\) be a prime number. In this paper the author studies the special fibre at \(p\) of certain moduli spaces of abelian varieties with additional structures (namely of PEL type D), particularly in the case when \(p\) ramifies in the defining datum. In view of this, part of the paper is dedicated to the classification of the isogeny classes of \(p\)-divisible groups over an algebraically closed field of characteristic \(p\), endowed with a local version of the additional structures. Explicit conclusions about the moduli spaces under consideration are reached in the case of minimal dimension.
To be precise, let \(F\) be a totally real field with ring of integers \(\mathcal{O}_F\), let \(B\) be a totally definite quaternion algebra over \(F\) with canonical involution \(*\), and let \(\mathcal{O}_B\subseteq B\) be a \(*\)-stable \(\mathcal{O}_F\)-order which is maximal at \(p\). For any integer \(m\ge1\), let then \(\mathcal{M}\) be the coarse moduli scheme over \(\mathbb{Z}_{(p)}\) classifying \(2[F\colon\mathbb{Q}_p]m\)-dimensional polarised abelian schemes, endowed with a faithful \(\mathcal{O}_B\)-action which is compatible with the polarisation and which exchanges \(*\) with the Rosati involution. The \(\mathbb{C}\)-valued points of this space form a disjoint union of locally symmetric Hermitian spaces, a subset of which corresponds to a usual Shimura variety. The author also considers the subspace \(\mathcal{M}^{(p)}\) of objects with prime-to-\(p\) polarisation, as well as the subspace \(\mathcal{M}_K\), where a ``determinant condition'' is imposed. Let us denote the geometric special fibre of these spaces with the subscript \(\bar{\mathbb{F}}_p\). The main theorem is that if \(m=1\) and \(F=\mathbb{Q}\), then:
\begin{itemize}
\item[1.] if \(p\) is unramified in \(B\), then \(\dim\mathcal{M}_{\bar{\mathbb{F}}_p}=0\);
\item[2.] if \(p\) is ramified in \(B\), then \(\dim\mathcal{M}_{\bar{\mathbb{F}}_p}=1\);
\item[3.] we have \(\dim\mathcal{M}_{\bar{\mathbb{F}}_p}^{(p)}=0\);
\item[4.] if \(p\) is ramified in \(B\), then \(\dim\mathcal{M}_{K,\bar{\mathbb{F}}_p}=1\).
\end{itemize}
In addition, thanks to an analysis of the corresponding Rapoport-Zink local model, part \((3)\) can be upgraded to the result, for general \(F\), that \(\mathcal{M}^{(p)}\) is flat over \(\mathbb{Z}_{(p)}\) with every connected component projective of relative dimension \(0\). However, a consideration on the dimensions shows that \(\mathcal{M}\) and \(\mathcal{M}_K\) are not flat if \(p\) is ramified in \(B\). The author also observes that in the ramified case the determinant condition is not automatic, i.e.\ \(\mathcal{M}_K\subsetneq\mathcal{M}\) (unlike in the Hilbert-Siegel setting).
The main external inputs to the proof of these results are, on the one hand, the construction of families of objects over \(\mathbb{P}^1\) following \textit{L. Moret-Bailly} [Astérisque 86, 109--124 (1981; Zbl 0515.14006)]. On the other hand, a finite map to the moduli space of polarised abelian varieties allows to reduce some questions to the work of \textit{P. Norman} and \textit{F. Oort} [Ann. Math. (2) 112, 413--439 (1980; Zbl 0483.14010)]. As for the classification of isogeny classes of \(p\)-divisible groups, which is carried out for general \(m\ge1\), Dieudonné theory allows the author to perform this through elementary algebraic methods. A criterion for the nonemptiness of the \(\mu\)-ordinary locus is obtained on the way (cf.\ Corollary 7.5).
Reviewer: Andrea Marrama (Paris)Fundamental group schemes of \(n\)-fold symmetric product of a smooth projective curvehttps://zbmath.org/1492.140772022-10-04T19:40:27.024758Z"Paul, Arjun"https://zbmath.org/authors/?q=ai:paul.arjun"Sebastian, Ronnie"https://zbmath.org/authors/?q=ai:sebastian.ronnieLet \(X\) be a complete connected reduced scheme defined over an algebraically closed field \(k\). Fix \(x\in X(k)\). Recall that the \(S\)-fundamental group scheme of \((X, x)\) is the Tannaka dual to the neutral Tannakian category of numerically flat vector bundles over \(X\), where the fibre functor is given by \(E\mapsto E(x)\) [\textit{A. Langer}, Ann. Inst. Fourier 61, No. 5, 2077--2119 (2011; Zbl 1247.14019)].
Let \(C\) be a smooth projective curve defined over an algebraically closed field \(k\) of positive characteristic. Let \(S^n(C)\) be the \(n\)-fold symmetric product of \(C\). The main result of the paper under review states that the natural morphism \(\tilde{\psi^S_*}\colon \pi^S(C, x)_{\mathrm{ab}}\rightarrow \pi^S(S^n(C), nx)\) is an isomorphism of affine \(k\)-group schemes. To prove this, they first show that the morphism \(\tilde{\psi^S_*}\) is faithfully flat, using the equivalent characterization in terms of representation categories of respective affine group schemes. To show that \(\tilde{\psi^S_*}\) is closed immersion, they use the Albanese morphism \(\mathrm{alb}_X \colon X \rightarrow \mathrm{Alb}(X)\). As a consequence of the main result, they obtain the similar result for the Nori's fundamental group scheme \(\pi^N\) and the etale fundament group \(\pi^{\mathrm{et}}\).
Reviewer: Sanjay Amrutiya (Gandhinagar)