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Arxiv:2008.05229V2 [Math.AG]
A TORELLI THEOREM FOR MODULI SPACES OF PARABOLIC VECTOR BUNDLES OVER AN ELLIPTIC CURVE THIAGO FASSARELLA AND LUANA JUSTO Abstract. Let C be an elliptic curve, w ∈ C, and let S ⊂ C be a finite subset of cardinality at least 3. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle OC (w) over (C,S) which are semistable with respect to a weight vector 1 1 2 ,..., 2 . 1. Introduction Let C be a smooth complex curve of genus g ≥ 2 and fix w ∈ C. Let M be the corresponding moduli space of semistable rank two vector bundles having OC (w) as determinant line bundle. A classical Torelli type theorem of D. Mumford and P. Newstead [MN68] says that the isomorphism class of M determines the isomorphism class of C. This result has been extended first in [Tyu70, NR75, KP95] to higher rank and later to the parabolic context, which we now describe. We now assume g ≥ 0. Let S ⊂ C be a finite subset of cardinality n ≥ 1. Let Ma be the moduli space of rank two parabolic vector bundles on (C,S) with fixed determinant line bundle OC (w), and which are µa-semistable, see Section 2. The subscript a refers to a particular choice of a weight vector a = (a1, . , an) of real numbers, 0 ≤ ai ≤ 1, which gives the slope-stability condition. The moduli 1 1 space associated to the central weight aF = 2 ,..., 2 is particularly interesting, for instance when g = 0 and n ≥ 5 it is a Fano variety that is smooth if n is odd and has isolated singularities if n is even, see [Muk05, Cas15, AM16, AFKM19]. -
A Torelli Theorem for Moduli Spaces of Principal Bundles Over a Curve Tome 62, No 1 (2012), P
R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Indranil BISWAS & Norbert HOFFMANN A Torelli theorem for moduli spaces of principal bundles over a curve Tome 62, no 1 (2012), p. 87-106. <http://aif.cedram.org/item?id=AIF_2012__62_1_87_0> © Association des Annales de l’institut Fourier, 2012, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 62, 1 (2012) 87-106 A TORELLI THEOREM FOR MODULI SPACES OF PRINCIPAL BUNDLES OVER A CURVE by Indranil BISWAS & Norbert HOFFMANN (*) 0 Abstract. — Let X and X be compact Riemann surfaces of genus > 3, and 0 d let G and G be nonabelian reductive complex groups. If one component MG(X) of the coarse moduli space for semistable principal G–bundles over X is isomorphic d0 0 0 to another component MG0 (X ), then X is isomorphic to X . Résumé. — Soient X et X0 des surfaces de Riemann compactes de genre au moins 3, et G et G0 des groupes complexes réductifs non abéliens. -
Weighted Fano Varieties and Infinitesimal Torelli Problem 3
WEIGHTED FANO VARIETIES AND INFINITESIMAL TORELLI PROBLEM ENRICO FATIGHENTI, LUCA RIZZI AND FRANCESCO ZUCCONI Abstract. We solve the infinitesimal Torelli problem for 3-dimensional quasi-smooth Q-Fano hypersurfaces with at worst terminal singularities. We also find infinite chains of double coverings of increasing dimension which alternatively distribute themselves in examples and counterexamples for the infinitesimal Torelli claim and which share the analogue, and in some cases the same, Hodge-diagram properties as the length 3 Gushel- Mukai chain of prime smooth Fanos of coindex 3 and degree 10. 1. Introduction Minimal model program suggests us to formulate inside the category of varieties with terminal singularities many questions which were initially asked for smooth varieties. The construction of the period map, and the related Torelli type problems are definitely among these problems. 1.1. Infinitesimal Torelli. A Q-Fano variety is a projective variety X such that X has at worst Q-factorial terminal singularities, −KX is ample and Pic(X) has rank 1; cf. [CPR00]. In this paper we give a full answer to the infinitesimal Torelli problem in the case of quasi- smooth Q-Fano hypersurfaces of dimension 3 with terminal singularities and with Picard number 1. In subsections 2.1 of this paper the reader can find a basic dictionary and up to date references needed to understand the statement of the infinitesimal Torelli problem and where to find the meaning of infinitesimal variation of Hodge structures (IVHS; to short). Here we stress only that the method à la Griffiths based on the extension of the Macaulay’s theorem in algebra to the weighted case, cf. -
Algebraic Curves and Surfaces
Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces . Let me know when you find errors or typos. I'm sure there are plenty. 1 Curves on a surface 1 1.1 Topological invariants . 1 1.2 Holomorphic invariants . 2 1.3 Divisors . 3 1.4 Algebraic intersection theory . 4 1.5 Arithmetic genus . 6 1.6 Riemann{Roch formula . 7 1.7 Hodge index theorem . 7 1.8 Ample and nef divisors . 8 1.9 Ample cone and its closure . 11 1.10 Closure of the ample cone . 13 1.11 Div and Num as functors . 15 2 Birational geometry 17 2.1 Blowing up and down . 17 2.2 Numerical invariants of X~ ...................................... 18 2.3 Embedded resolutions for curves on a surface . 19 2.4 Minimal models of surfaces . 23 2.5 More general contractions . 24 2.6 Rational singularities . 26 2.7 Fundamental cycles . 28 2.8 Surface singularities . 31 2.9 Gorenstein condition for normal surface singularities . 33 3 Examples of surfaces 36 3.1 Rational ruled surfaces . 36 3.2 More general ruled surfaces . 39 3.3 Numerical invariants . 41 3.4 The invariant e(V ).......................................... 42 3.5 Ample and nef cones . 44 3.6 del Pezzo surfaces . 44 3.7 Lines on a cubic and del Pezzos . 47 3.8 Characterization of del Pezzo surfaces . 50 3.9 K3 surfaces . 51 3.10 Period map . 54 a 3.11 Elliptic surfaces . -
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 . -
The Derived Torelli Theorem
The derived Torelli theorem Pieter Belmans Abstract These are the notes for my lecture for the 2012–2013 course Surfaces K3 et théorie des déformations at Université Paris-Sud 11, and provide some back- ground to the actual lecture. The goal is to discuss Fourier-Mukai transforms, and apply the obtained techniques to K3 surfaces. The main result that will be discussed is a version of the global Torelli theorem in the context of derived categories, the so called derived Torelli theorem, which is theorem 40. The main reference for these notes (and everything else that is related to Fourier-Mukai transforms) is the excellent book [Huy06]. It is followed closely, to add a bit of originality some of the calculations left as an exercise are done explicitly. Contents 1 Derived categories in algebraic geometry2 1.1 Derived categories...............................2 1.2 Total derived functors.............................3 b 1.3 Relationship between D (X ) and the geometry of X ...........5 2 Fourier-Mukai transforms6 2.1 Definition.....................................6 2.2 Some examples.................................8 2.3 Properties of Fourier-Mukai transforms...................9 2.4 Bondal-Orlov theorem............................. 10 3 The derived Torelli theorem 11 3.1 Preliminaries: the K-theoretic Fourier-Mukai transform......... 11 3.2 Preliminaries: the cohomological Fourier-Mukai transform....... 12 3.3 Preliminaries: the Hodge structure..................... 14 3.4 Derived Torelli.................................. 15 3.5 Auto-equivalences of K3 surfaces...................... 15 1 1 Derived categories in algebraic geometry The subject of these notes is the notion of derived equivalence (see definition1), applied to the case of K3 surfaces (see §3). -
Moduli of K3 Surfaces and Irreducible Symplectic Manifolds
Moduli of K3 Surfaces and Irreducible Symplectic Manifolds V. Gritsenko, K. Hulek and G.K. Sankaran Abstract The name “K3 surfaces” was coined by A. Weil in 1957 when he formu- lated a research programme for these surfaces and their moduli. Since then, irreducible holomorphic symplectic manifolds have been intro- duced as a higher dimensional analogue of K3 surfaces. In this paper we present a review of this theory starting from the definition of K3 surfaces and going as far as the global Torelli theorem for irreducible holomorphic symplectic manifolds as recently proved by M. Verbitsky. For many years the last open question of Weil’s programme was that of the geometric type of the moduli spaces of polarised K3 surfaces. We explain how this problem has been solved. Our method uses algebraic geometry, modular forms and Borcherds automorphic products. We collect and discuss the relevant facts from the theory of modular forms with respect to the orthogonal group O(2,n). We also give a detailed description of quasi pull-back of automorphic Borcherds products. This part contains previously unpublished results. Contents 1 Introduction 2 2 Periods of K3 surfaces and the Torelli theorem 4 3 Irreducible symplectic manifolds 15 4 Projective models 24 arXiv:1012.4155v2 [math.AG] 26 May 2011 5 Compactifications 26 6 Modular forms and Kodaira dimension 39 7 Orthogonal groups and reflections 46 8 The quasi pull-back of modular forms 51 9 Arithmetic of root systems 59 1 1 Introduction The history of K3 surfaces goes back to classical algebraic geometry. In modern complex algebraic geometry a K3 surface is a compact complex surface S whose canonical bundle is trivial, i.e. -
HEEL and TOE ONLINE the Official
HEEL AND TOE ONLINE The official organ of the Victorian Race Walking Club 2015/2016 Number 36 7 June 2016 VRWC Preferred Supplier of Shoes, clothes and sporting accessories. Address: RUNNERS WORLD, 598 High Street, East Kew, Victoria (Melways 45 G4) Telephone: 03 9817 3503 Hours: Monday to Friday: 9:30am to 5:30pm Saturday: 9:00am to 3:00pm Website: http://www.runnersworld.com.au Facebook: http://www.facebook.com/pages/Runners-World/235649459888840 WALKER OF THE WEEK My Walkers of the Week this time around go to Rhydian Cowley, Tanya Holliday and Regan Lamble who have all just been added to our 2016 Austrlian Olympic Team. The big announcement came on Friday with Rhydian being added to the men's 20km team, along with Jared Tallent who will now do the double, and Tanya and Regan being added to the women's 20km team. The full Australian walks team now reads as follows 20km Men Dane Bird-Smith (QLD), Jared Tallent (SA), Rhydian Cowley (VIC) 20km Women Rachel Tallent (VIC), Regan Lamble (VIC), Tanya Holliday (SA) 50km Jared Tallent (SA), Chris Erickson (VIC), Brendon Reading (ACT) Dane, Rhydian, Rachel, Tanya and Brendon will make their Olympic debuts, Regan will be going to her second Olympics and Chris and Jared will front for their third Olympics. But today it's timely to zoom in on Rhydian, Tanya and Regan. • 24 year old Regan was outstanding on Olympic debut four years ago, placing 16 th in London with a huge PB of 1:30:03. But her world then fell apart with the ridiculous decision to terminate the AIS scholarship program. -
Algebraic Boundaries of Hilbert's SOS Cones
COMPOSITIO MATHEMATICA Algebraic boundaries of Hilbert's SOS cones Grigoriy Blekherman, Jonathan Hauenstein, John Christian Ottem, Kristian Ranestad and Bernd Sturmfels Compositio Math. 148 (2012), 1717{1735. doi:10.1112/S0010437X12000437 FOUNDATION COMPOSITIO MATHEMATICA Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.93, on 29 Sep 2021 at 03:34:16, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X12000437 Compositio Math. 148 (2012) 1717{1735 doi:10.1112/S0010437X12000437 Algebraic boundaries of Hilbert's SOS cones Grigoriy Blekherman, Jonathan Hauenstein, John Christian Ottem, Kristian Ranestad and Bernd Sturmfels Abstract We study the geometry underlying the difference between non-negative polynomials and sums of squares (SOS). The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether{Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized, respectively, by the Severi variety of plane rational sextics and by the variety of quartic symmetroids. 1. Introduction A fundamental object in convex algebraic geometry is the cone Σn;2d of homogeneous polynomials of degree 2d in R[x1; : : : ; xn] that are sums of squares (SOS). Hilbert [Hil88] showed that the cones Σ3;6 and Σ4;4 are strictly contained in the corresponding cones P3;6 and P4;4 of non-negative polynomials. -
The Moduli of Curves of Genus Six and K3 Surfaces
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 3, March 2011, Pages 1445–1462 S 0002-9947(2010)05126-8 Article electronically published on October 25, 2010 THE MODULI OF CURVES OF GENUS SIX AND K3 SURFACES MICHELA ARTEBANI AND SHIGEYUKI KONDO¯ Abstract. We prove that the coarse moduli space of curves of genus six is birational to an arithmetic quotient of a bounded symmetric domain of type IV by giving a period map to the moduli space of some lattice-polarized K3 surfaces. Introduction This paper gives a birational period map between the coarse moduli space M6 of curves of genus six and a bounded symmetric domain of type IV, parametrizing some lattice-polarized K3 surfaces. A similar correspondence, between a moduli space of curves and a period space of K3 surfaces, was given by the second author for curves of genus three and genus four in [Ko2] and [Ko3]. Some of the results in this paper were announced in [Ko3]. Realizing a moduli space as an arithmetic quotient of a bounded symmetric domain is interesting for several reasons. The first reason is that this gives a connection with the theory of automorphic forms. Recently Borcherds [Bor] gave a systematic method to construct automor- phic forms on a bounded symmetric domain of type IV. Allcock and Freitag [AF] and the second author [Ko4], [Ko5] used his theory to obtain explicit embeddings in a projective space for the moduli spaces of cubic surfaces, Enriques surfaces and curves of genus three, respectively. Secondly, an arithmetic quotient of a bounded symmetric domain has several compactifications, that is, Satake-Baily-Borel’s, Mumford’s toroidal and Looijenga’s compactifications [Lo1], [Lo2]. -
On a Proof of Torelli's Theorem
Contents 0.1 Assumed Knowledge . 5 0.2 Outline . 5 Chapter 1 Sheaves and sheaf cohomology 7 1.1 Sheaves . 7 1.2 Sheaves associated with functions . 12 1.3 Sheaf cohomology . 14 1.4 Vanishing theorems . 22 n n 1.5 Cohomology of C and P ........................... 24 Chapter 2 Riemann surfaces 27 2.1 Properties of Riemann surfaces . 28 2.2 Examples of Riemann surfaces . 29 2.3 Cohomology of Riemann surfaces . 31 2.4 The Riemann-Hurwitz formula . 32 2.5 Hyperellipticity . 33 Chapter 3 The classical theorems of Abel and Jacobi 35 3.1 Divisors . 35 3.2 The Abel-Jacobi map and the Jacobian variety . 38 3.3 Line bundles . 41 3.4 Pic(S)...................................... 45 Chapter 4 Linear systems and the Riemann-Roch theorem 51 4.1 The Riemann-Roch theorem . 57 4.2 Application and Examples . 59 Chapter 5 Complex tori 65 5.1 Cohomology of complex tori . 66 5.2 Line bundles on complex tori . 67 1 5.3 Theta functions . 72 Chapter 6 The Jacobian Variety 77 6.1 Motivation: Abelian integrals . 77 6.2 Properties of the Jacobian variety . 79 6.3 Riemann’s theorem . 83 Chapter 7 The Torelli theorem 91 7.1 Proof of the Torelli theorem . 91 Chapter 8 Concluding remarks 98 8.1 The Schottky problem . 98 Chapter 9 Background material 100 9.1 Group cohomology . 100 9.2 Major theorems . 101 2 Acknowledgements I would like to extend my gratitude to all those who helped in the production of this thesis, and to those who supported me throughout this year. -
2019 World Championships Statistics
2019 World Championships Statistics - Women’s 20Km Walk by K Ken Nakamura The records to look for in Doha: 1) No nation ever swept medals; can CHN do it? 2) Can Morejon win first medal for ECU at W20kmWalk? Summary: All time Performance List at the World Championships Performance Performer Time Name Nat Pos Venue Year 1 1 1:25:41 Olimpiada Ivanova RUS 1 Helsinki 2005 2 2 1:26:18 Yang Jiayu CHN 1 London 2017 3 3 1:26:19 Mari a Guadalupe Gonzalez MEX 2 London 2017 4 4 1:26:36 Antonella Palmisano ITA 3 London 2017 5 5 1:26:52 Yelena Nikolayeva RUS 1 Paris 2003 6 6 1:26:59 Erica De Sena BRA 4 London 2017 7 7 1:27:05 Rita Turova BLR 2 Helsinki 2005 8 8 1:27:08 Elena L ashmanova RUS 1 Moskva 2013 9 9 1:27:11 Anisya Kirdyapkina RUS 2 Moskva 20 13 10 10 1:27:34 Gillian O’Sullivan IRL 2 Paris 2003 Margin of Victory Difference Winning time Name Nat Venue Year Max 1 min 24 second s 1:25:41 Olimpiada Ivanova RUS Helsinki 2005 1 min 1 second 1:27:48 Olimpiada Ivanova RUS Edmonton 2001 Min 0 second 1:27:45 Liu Hong CHN Beijing 2015 1 second 1:26:18 Yang Jiayu CHN London 2017 2 seconds 1:30:50 Liu Hongyu CHN Sevilla 1999 3 seconds 1:27:08 Elena Lashmanova RUS Moskva 2013 Best Marks for Places in the World Championships Pos Time Name Nat Venue Year 1 1:25:41 Olimpiada Ivanova RUS Helsinki 2005 2 1:26:19 Maria Guadalupe Gonzalez MEX London 2017 1:27:05 Rita Turova BLR Helsinki 2005 3 1:26:36 Antonella Palmisano ITA London 2017 1:28:10 Valentina Tsybulskaya BLR Paris 2003 Liu Hong CHN Moskva 2013 4 1:26:59 Erica De Sena BRA London 2017 1:28:32 Sun Huanhuan