Perspectives on the Construction and Compactification of Moduli Spaces
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Models of Shimura Varieties in Mixed Characteristics by Ben Moonen
Models of Shimura varieties in mixed characteristics by Ben Moonen Contents Introduction .............................. 1 1 Shimuravarieties ........................... 5 2 Canonical models of Shimura varieties. 16 3 Integralcanonicalmodels . 27 4 Deformation theory of p-divisible groups with Tate classes . 44 5 Vasiu’s strategy for proving the existence of integral canonical models 52 6 Characterizing subvarieties of Hodge type; conjectures of Coleman andOort................................ 67 References ............................... 78 Introduction At the 1996 Durham symposium, a series of four lectures was given on Shimura varieties in mixed characteristics. The main goal of these lectures was to discuss some recent developments, and to familiarize the audience with some of the techniques involved. The present notes were written with the same goal in mind. It should be mentioned right away that we intend to discuss only a small number of topics. The bulk of the paper is devoted to models of Shimura varieties over discrete valuation rings of mixed characteristics. Part of the discussion only deals with primes of residue characteristic p such that the group G in question is unramified at p, so that good reduction is expected. Even at such primes, however, many technical problems present themselves, to begin with the “right” definitions. There is a rather large class of Shimura data—those called of pre-abelian type—for which the corresponding Shimura variety can be related, if maybe somewhat indirectly, to a moduli space of abelian varieties. At present, this seems the only available tool for constructing “good” integral models. Thus, 1 if we restrict our attention to Shimura varieties of pre-abelian type, the con- struction of integral canonical models (defined in 3) divides itself into two § parts: Formal aspects. -
The Minimal Model Program for the Hilbert Scheme of Points on P2 and Bridgeland Stability
THE MINIMAL MODEL PROGRAM FOR THE HILBERT SCHEME OF POINTS ON P2 AND BRIDGELAND STABILITY DANIELE ARCARA, AARON BERTRAM, IZZET COSKUN, AND JACK HUIZENGA 2[n] 2 Abstract. In this paper, we study the birational geometry of the Hilbert scheme P of n-points on P . We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n ≤ 9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions. Contents 1. Introduction 2 2. Preliminaries on the Hilbert scheme of points 4 2[n] 3. Effective divisors on P 5 2[n] 4. The stable base locus decomposition of the effective cone of P 8 5. Preliminaries on Bridgeland stability conditions 15 6. Potential walls 18 7. The quiver region 23 8. Moduli of stable objects are GIT quotients 26 9. Walls for the Hilbert scheme 27 10. Explicit Examples 30 2[2] 10.1. The walls for P 30 2[3] 10.2. The walls for P 31 2[4] 10.3. The walls for P 32 2[5] 10.4. The walls for P 33 2[6] 10.5. The walls for P 34 2[7] 10.6. -
Siegel Modular Varieties and Borel-Serre Compactification of Modular Curves
Siegel modular varieties and Borel-Serre compactification of modular curves Michele Fornea December 11, 2015 1 Scholze’s Theorem on Torsion classes To motivate the study of Siegel modular varieties and Borel-Serre compactifica- tions let me recall Scholze’s theorem on torsion classes. i We start from an Hecke eigenclass h 2 H (Xg=Γ; Fp) where Xg is the locally symmetric domain for GLg, and we want to attach to it a continuous semisimple Galois representation ρh : GQ ! GLg(Fp) such that the characteristic polyno- mials of the Frobenius classes at unramified primes are determined by the Hecke eigenvalues of h. Problem: Xg=Γ is not algebraic in general, in fact it could be a real manifold of odd dimension; while the most powerful way we know to construct Galois representations is by considering étale cohomology of algebraic varieties. Idea(Clozel): Find the cohomology of Xg=Γ in the cohomology of the boundary BS of the Borel-Serre compactification Ag of Siegel modular varieties. 1.0.1 Properties of the Borel-Serre compactification BS • Ag is a real manifold with corners. BS • The inclusion Ag ,! Ag is an homotopy equivalence (same cohomology). BS • The boundary of Ag is parametrized by parabolic subgroups of Sp2g and consists of torus bundles over arithmetic domains for the Levi subgroups of each parabolic. BS It follows from the excision exact sequence for the couple (Ag ;@) that i one can associate to the Hecke eigenclass h 2 H (Xg=Γ; Fp) another eigenclass 0 i h 2 H (Ag; Fp) whose eigenvalues are precisely related to those of h. -
The Lci Locus of the Hilbert Scheme of Points & the Cotangent Complex
The lci locus of the Hilbert scheme of points & the cotangent complex Peter J. Haine September 21, 2019 Abstract Weintroduce the basic algebraic geometry background necessary to understand the main results of the work of Elmanto, Hoyois, Khan, Sosnilo, and Yakerson on motivic infinite loop spaces [5]. After recalling some facts about the Hilbert func- tor of points, we introduce local complete intersection (lci) and syntomic morphisms. We then give an overview of the cotangent complex and discuss the relationship be- tween lci morphisms and the cotangent complex (in particular, Avramov’s charac- terization of lci morphisms). We then introduce the lci locus of the Hilbert scheme of points and the Hilbert scheme of framed points, and prove that the lci locus of the Hilbert scheme of points is formally smooth. Contents 1 Hilbert schemes 2 2 Local complete intersections via equations 3 Relative global complete intersections ...................... 3 Local complete intersections ........................... 4 3 Background on the cotangent complex 6 Motivation .................................... 6 Properties of the cotangent complex ....................... 7 4 Local complete intersections and the cotangent complex 9 Avramov’s characterization of local complete intersections ........... 11 5 The lci locus of the Hilbert scheme of points 13 6 The Hilbert scheme of framed points 14 1 1 Hilbert schemes In this section we recall the Hilbert functor of points from last time as well as state the basic representability properties of the Hilbert functor of points. 1.1 Recollection ([STK, Tag 02K9]). A morphism of schemes 푓∶ 푌 → 푋 is finite lo- cally free if 푓 is affine and 푓⋆풪푌 is a finite locally free 풪푋-module. -
Abelian Varieties
Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties . -
Arxiv:1507.05922V1
Torsion in the Coherent Cohomology of Shimura Varieties and Galois Representations A dissertation presented by George Andrew Boxer to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts arXiv:1507.05922v1 [math.NT] 21 Jul 2015 April 2015 c 2015 – George A. Boxer All rights reserved. Dissertation Advisor: Richard Taylor George Andrew Boxer Torsion in the Coherent Cohomology of Shimura Varieties and Galois Representations Abstract We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties. The congruences are produced using some “generalized Hasse invariants” adapted to the Ekedahl-Oort stratification of the special fiber. iii Contents 1 Introduction 1 1.1 Motivation: Weight 1 Modular Forms mod p .................. 1 1.2 Constructing Congruences Using Hasse Invariants . ...... 3 1.3 GeometricSiegelModularForms ........................ 5 1.4 The Ekedahl-Oort Stratification and Generalized Hasse Invariants...... 9 1.5 HasseInvariantsattheBoundary . .. 16 1.6 ConstructingCongruences . 17 1.7 OverviewandAdvicefortheReader . 22 1.8 NotationandConventions ............................ 22 2 Good Reduction PEL Modular Varieties 25 2.1 PELDatum.................................... 25 2.2 PELModularVarieties.............................. 31 2.3 AutomorphicVectorBundles. 37 2.4 HeckeAction ................................... 38 2.5 MorphismtoSiegelSpace ............................ 39 3 Arithmetic Compactifications of PEL Modular Varieties 42 3.1 ToroidalBoundaryCharts . .. .. 43 3.2 Arithmetic Compactifications . 51 3.2.1 ToroidalCompactifications. 51 iv 3.2.2 Minimal Compactifications . 52 3.3 Extensions of Automorphic Vector Bundles . ... 54 3.3.1 Canonical and Subcanonical Extensions . 54 3.3.2 HigherDirectImages........................... 57 3.3.3 Pushforwards to the Minimal Compactification . -
Further Pathologies in Algebraic Geometry Author(S): David Mumford Source: American Journal of Mathematics , Oct., 1962, Vol
Further Pathologies in Algebraic Geometry Author(s): David Mumford Source: American Journal of Mathematics , Oct., 1962, Vol. 84, No. 4 (Oct., 1962), pp. 642-648 Published by: The Johns Hopkins University Press Stable URL: http://www.jstor.com/stable/2372870 JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics This content downloaded from 132.174.252.179 on Sat, 01 Aug 2020 11:36:26 UTC All use subject to https://about.jstor.org/terms FURTHER PATHOLOGIES IN ALGEBRAIC GEOMETRY.*1 By DAVID MUMFORD. The following note is not strictly a continuation of our previous note [1]. However, we wish to present two more examples of algebro-geometric phe- nomnena which seem to us rather startling. The first relates to characteristic ]) behaviour, and the second relates to the hypothesis of the completeness of the characteristic linear system of a maximal algebraic family. We will use the same notations as in [1]. The first example is an illustration of a general principle that might be said to be indicated by many of the pathologies of characteristic p: A non-singular characteristic p variety is analogous to a general non- K:ahler complex manifold; in particular, a projective embedding of such a variety is not as "strong" as a Kiihler metric on a complex manifold; and the Hodge-Lefschetz-Dolbeault theorems on sheaf cohomology break down in every possible way. -
Algebraic Surfaces with Minimal Betti Numbers
P. I. C. M. – 2018 Rio de Janeiro, Vol. 2 (717–736) ALGEBRAIC SURFACES WITH MINIMAL BETTI NUMBERS JH K (금종해) Abstract These are algebraic surfaces with the Betti numbers of the complex projective plane, and are called Q-homology projective planes. Fake projective planes and the complex projective plane are smooth examples. We describe recent progress in the study of such surfaces, singular ones and fake projective planes. We also discuss open questions. 1 Q-homology Projective Planes and Montgomery-Yang problem A normal projective surface with the Betti numbers of the complex projective plane CP 2 is called a rational homology projective plane or a Q-homology CP 2. When a normal projective surface S has only rational singularities, S is a Q-homology CP 2 if its second Betti number b2(S) = 1. This can be seen easily by considering the Albanese fibration on a resolution of S. It is known that a Q-homology CP 2 with quotient singularities (and no worse singu- larities) has at most 5 singular points (cf. Hwang and Keum [2011b, Corollary 3.4]). The Q-homology projective planes with 5 quotient singularities were classified in Hwang and Keum [ibid.]. In this section we summarize progress on the Algebraic Montgomery-Yang problem, which was formulated by J. Kollár. Conjecture 1.1 (Algebraic Montgomery–Yang Problem Kollár [2008]). Let S be a Q- homology projective plane with quotient singularities. Assume that S 0 := S Sing(S) is n simply connected. Then S has at most 3 singular points. This is the algebraic version of Montgomery–Yang Problem Fintushel and Stern [1987], which was originated from pseudofree circle group actions on higher dimensional sphere. -
The Kuga-Satake Construction: a Modular Interpretation
The Kuga-Satake Construction: A Modular Interpretation Alice Pozzi A Thesis for The Department of Mathematics and Statistics Presented in Partial Fulfillment of the Requirements for the Degree of Master of Science (Mathematics) at Concordia University Montreal, Quebec, Canada August 2013 c Alice Pozzi, 2013 CONCORDIA UNIVERSITY School of Graduate Studies This is to certify that the thesis prepared By: Alice Pozzi Entitled: The Kuga-Satake Construction: A Modular Interpretation and submitted in partial fulfillment of the requirements for the degree of Master of Science (Mathematics) complies with the regulations of the University and meets the accepted standards with respect to originality and quality. Signed by the final examining committee: Examiner Dr. C. Cummins Examiner Dr. C. David Thesis Supervisor Dr. A. Iovita Approved by Chair of Department or Graduate Program Director Dean of Faculty Date Abstract The Kuga-Satake Construction: A Modular Interpretation Alice Pozzi Given a polarized complex K3 surface, one can attach to it a complex abelian variety, called Kuga-Satake variety. The Kuga-Satake variety is determined by the singular cohomology of the K3 surface; on the other hand, this singular cohomology can be recovered by means of the weight 1 Hodge structure associated to the Kuga-Satake variety. Despite the transcendental origin of this construction, Kuga-Satake varieties have interesting arithmetic properties. Kuga-Satake varieties of K3 surfaces defined over number fields descend to finite extension of the field of definition. This property suggests that the Kuga-Satake construction can be interpreted as a map between moduli spaces. More precisely, one can define a morphism, called Kuga-Satake map, between the moduli space of K3 surfaces and the moduli space of abelian varieties with polarization and level structure. -
Arxiv:2109.01112V1 [Math.AG]
A NOTE ON MURPHY’S LAW FOR HILBERT SCHEMES FABIAN REEDE Abstract. Let S be a smooth projective surface with pg = q = 0. We prove that there is a polynomial p(t) such that the Hilbert scheme Hilbp(t)(S[n]) of the Hilbert scheme S[n] of length n subschemes contains a smooth connected component isomorphic to S. Introduction Hilbert schemes are ubiquitous in modern algebraic geometry. But even in good situations these schemes can exhibit strange behavior. This became clear with Mumford’s famous example, which shows that there is an irreducible component of the Hilbert scheme of smooth irreducible curves in P3 of degree 14 and genus 24 that is generically non-reduced, see [14]. More exactly Mumford constructs a 56-dimensional irreducible family of such curves, such that the tangent space at each point of this component has dimension 57 and proves that this family is not contained in any other irreducible family of dimension > 56. This and other examples led Harris and Morrison to formulate their (informal) “Murphy’s law for Hilbert schemes”, see [7, §1.D]: There is no geometric possibility so horrible that it cannot be found generically on some component of some Hilbert scheme. For a rigorous study of Murphy’s law in algebraic geometry see [16]. In this note we prove that one can also find plenty of Hilbert schemes which contain at least one smooth connected component. Thus following [10, 31.5] the statement of Murphy’s law for Hilbert schemes should possibly be: Anything that can happen will happen, good and bad. -
Report No. 26/2008
Mathematisches Forschungsinstitut Oberwolfach Report No. 26/2008 Classical Algebraic Geometry Organised by David Eisenbud, Berkeley Joe Harris, Harvard Frank-Olaf Schreyer, Saarbr¨ucken Ravi Vakil, Stanford June 8th – June 14th, 2008 Abstract. Algebraic geometry studies properties of specific algebraic vari- eties, on the one hand, and moduli spaces of all varieties of fixed topological type on the other hand. Of special importance is the moduli space of curves, whose properties are subject of ongoing research. The rationality versus general type question of these spaces is of classical and also very modern interest with recent progress presented in the conference. Certain different birational models of the moduli space of curves have an interpretation as moduli spaces of singular curves. The moduli spaces in a more general set- ting are algebraic stacks. In the conference we learned about a surprisingly simple characterization under which circumstances a stack can be regarded as a scheme. For specific varieties a wide range of questions was addressed, such as normal generation and regularity of ideal sheaves, generalized inequalities of Castelnuovo-de Franchis type, tropical mirror symmetry constructions for Calabi-Yau manifolds, Riemann-Roch theorems for Gromov-Witten theory in the virtual setting, cone of effective cycles and the Hodge conjecture, Frobe- nius splitting, ampleness criteria on holomorphic symplectic manifolds, and more. Mathematics Subject Classification (2000): 14xx. Introduction by the Organisers The Workshop on Classical Algebraic Geometry, organized by David Eisenbud (Berkeley), Joe Harris (Harvard), Frank-Olaf Schreyer (Saarbr¨ucken) and Ravi Vakil (Stanford), was held June 8th to June 14th. It was attended by about 45 participants from USA, Canada, Japan, Norway, Sweden, UK, Italy, France and Germany, among of them a large number of strong young mathematicians. -
THE HILBERT SCHEME of INFINITE AFFINE SPACE and ALGEBRAIC K-THEORY 3 Cohomology Theories and the Corresponding Transfers Is Given in [?, §1.1]
THE HILBERT SCHEME OF INFINITE AFFINE SPACE AND ALGEBRAIC K-THEORY MARC HOYOIS, JOACHIM JELISIEJEW, DENIS NARDIN, BURT TOTARO, AND MARIA YAKERSON ∞ 1 Abstract. We study the Hilbert scheme Hilbd(A ) from an A -homotopical viewpoint and obtain appli- ∞ 1 cations to algebraic K-theory. We show that the Hilbert scheme Hilbd(A ) is A -equivalent to the Grass- ∞ 1 n mannian of (d − 1)-planes in A . We then describe the A -homotopy type of Hilbd(A ) in a range, for n n large compared to d. For example, we compute the integral cohomology of Hilbd(A )(C) in a range. We also deduce that the forgetful map FFlat → Vect from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an A1-equivalence after group completion. This implies that the moduli stack FFlat, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum kgl representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the kgl-homology of smooth proper schemes over a perfect field. Contents 1. Introduction 1 1 2. Thestacks FFlatd and Vectd−1 are A -equivalent 4 3. A1-equivalence between the group completions of the stacks FFlat and Vect 6 4. Consequences for the Hilbert scheme of A∞ 9 5. Theeffective motivic K-theory spectrum 14 6. Comparison with algebraic cobordism 17 7. The Hilbert scheme of finite lci schemes of degree 3 18 8. Stability theorems for the Hilbert scheme 20 References 25 1.