GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY. Sampei Usui

Size: px
Start display at page:

Download "GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY. Sampei Usui"

Transcription

1 GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY Sampei Usui Abstract. This article is a geometric application of polarized logarithmic Hodge theory of Kazuya Kato and Sampei Usui. We prove generic Torelli theorem for the well-known quintic-mirror family in two ways by using different logarithmic points at the boundary of the fine moduli of polarized logarithmic Hodge structures. 1. Review of quntic-mirror family We review the construction of the mirror family of the pencil joining quintic hypersurface of Fermat type and the simplex consisting of the coordinate hyperplanes in a 4-dimensional complex projective space after [M1]. Let ψ P 1, and let (1) Q ψ = { 5 j=1 x5 j 5ψ 5 j=1 x j = 0} P 4. Let µ 5 = {α C α 5 = 1}. The singular members of the pencil (1) are listed as follows: (2) Over each point ψ µ 5 C P 1, Q ψ contains 5 3 = 125 ordinary double points (α 1,..., α 5 ) (µ 5 ) 5 /µ 5 P 4 with ψα 1 α 5 = 1. (3) Over P 1, Q = j {x j = 0}. This is a 4-dimensional simplex. Let G = {α (µ 5 ) 5 α 1 α 5 = 1}/µ 5. By multiplication, this becomes a group which acts on P 4 coordinate-wise. This action of G preserves Q ψ. Taking the quotient Q ψ /G, the following canonical singularities appear: (4) For each pair of distinct indices j, k, a compound du Val singularity ca 4 appears as the quotient of the curve Q ψ {x j = x k = 0} \ ( m j,k {x m = 0}). (5) For each triple of distinct indices j, k, l, the point which is the quotient of the five points Q ψ {x j = x k = x l = 0} belongs to the closure of three curves in (4). Key words and phrases. quintic-mirror family, logarithmic Hodge theory, moduli, Torelli theorem Mathematics Subject Classification. Primary 14C30; Secondary 14D07, 32G20. Partly supported by the Grant-in-Aid for Scientific Research (B) No , Japan Society for the Promotion of Science 1 Typeset by AMS-TEX

2 Moreover, we see that holomorphic 3-forms on Q ψ are G-invariant for every ψ C by adjunction formula. For ψ C, it is known that there is a simultaneous minimal desingularization W ψ of these quotient singularities, and that holomorphic 3-forms extend to nowhere vanishing forms on W ψ. We thus have a pencil (W ψ ) ψ P 1 whose singular fibers are listed as follows: (6) Over each point ψ µ 5 C P 1, W ψ has one ordinary double point. (7) Over ψ = P 1, W ψ is a normal crossing divisor in the total space, whose components are all rational. The other members W ψ are smooth with Hodge numbers h p,q = 1 for p + q = 3. By the action of α µ 5, (x 1,..., x 5 ) (α 1 x 1, x 2,..., x 5 ), we have W αψ W ψ. Let λ = ψ 5, and let (W λ ) λ ((W ψ ) ψ )/µ 5 (λ-plan) (ψ-plan)/µ 5. This pencil (W λ ) λ P 1 is the mirror of the original pencil (1). (For more details of the above construction, see e.g. [M1].) 2. Review of fine moduli space Γ\D Ξ We review some facts in [KU] that are necessary in the present article. Let w = 3, and h p,q = 1 (p + q = 3, p, q 0). Let H 0 = 4 j=1 Ze j, and e 3, e 1 0 = e 4, e 2 0 = 1. Let D be the corresponding classifying space of polarized Hodge structures, and Ď the compact dual. Let S = (square free positive integers), and let m S. Define N α, N β, N m End(H 0,, 0 ) as follows. N α (e 3 ) = e 1, N α (e j ) = 0 (j 3); N β (e 4 ) = e 3, N β (e 3 ) = e 1, N β (e 1 ) = e 2, N β (e 2 ) = 0; N m (e 1 ) = e 3, N m (e 4 ) = me 2, N m (e 2 ) = N m (e 3 ) = 0. Then the respective Hodge diamonds are 0 : N α : (3, 0) (2, 1) (1, 2) (0, 3) (2, 2) (3, 0) (0, 3) (1, 1) 2

3 N β : N m : (3, 3) (2, 2) (1, 1) (0, 0) (3, 1) (1, 3) (2, 0) (0, 2) Let σ α = R 0 N α, σ β = R 0 N β, and σ m = R 0 N m (m S). Proposition ([KU, 12.3]). Let Ξ = (rational nilpotent cones in g R of rank 1). Then Ξ = {Ad(g)σ σ = {0}, σ α, σ β, σ m (m S), g G Q }. This is a complete fan, i.e., if there exists Z Ď such that (σ, Z) is a nilpotent orbit, then σ Ξ. Here a pair (σ, Z) is a nilpotent orbit if Z = exp(cn)f, NF p F p 1 ( p) and exp(iyn)f D (y 0) hold for R 0 N = σ and F Z. In [KU], the fine moduli space Γ\D Ξ of polarized logarithmic Hodge structures is constructed. We briefly explain this according to the present case. Let Γ be a neat subgroup of G Z := Aut(H 0,, 0 ) of finite index. As a set, D Ξ = {(σ, Z) : nilpotent orbit σ Ξ, Z Ď}. Let σ Ξ, and let Γ(σ) = Γ exp(σ). If σ = {0}, then D {({0}, F ) F D} D Ξ. If σ {0}, then Γ(σ) N. Let γ be its generator and let N = log γ. Define E σ = and the map { } (q, F ) C Ď exp((2πi) 1 log(q)n)f D if q 0, and, exp(cn)f is a σ-nilpotent orbit if q = 0 { exp((2πi) 1 (1) E σ Γ(σ) gp log(q)n)f mod Γ(σ) gp if q 0, \D σ, (q, F ) (σ, exp(cn)f ) mod Γ(σ) gp if q = 0. Here Γ(σ) gp is the subgroup of Γ generated by Γ(σ). C Ď is obviously an analytic manifold. We endow this with the logarithmic structure M associated to the divisor {0} Ď. The strong topology of E σ in C Ď is defined as follows; a subset U of E σ 3

4 is open if, for any analytic space Y and any analytic morphsm f : Y C Ď such that f(y ) E σ, f 1 (U) is open in Y. By (1), the quotient topology, the sheaf O of local rings over C, and the logarithmic structure M are introduced on Γ(σ) gp \D σ. Introduce the corresponding structures O, M on Γ\D Ξ so that Γ(σ) gp \D σ Γ\D Ξ (σ Ξ) are local isomorphisms and form an open covering. Then, the resulting Γ\D Ξ is a logarithmic manifold, which is nearly a logarithmic analytic space but has slits at the boundaries. In fact, in the present case, we have in [KU, 12.3] that dim C D = 4, but dim C Γ(σ α ) gp \(D σα D) = 2, dim C Γ(σ β ) gp \(D σβ D) = 1 and dim C Γ(σ m ) gp \(D σm D) = Period map Let be a unit disc, = {0}, and h, z q = e 2πiz, the universal covering. Let ϕ : γ \D be a period map, where γ is the monodromy group generated by a unipotent element γ, and ϕ : h D a lifting. Let N = log γ. The map exp( zn) ϕ(z) from h to D drops down to the map ψ : D. Then the nilpotent orbit theorem of Schmid asserts that there exists the limit ψ(0), denoted by F, and that ϕ(z) and exp(zn)f are closing as Im z. Let σ = R 0 N. In our space γ \D σ we have, moreover, exp(zn)f ((σ, exp(σ C )F ) mod γ ) as Im z. Hence ϕ(q) ((σ, exp(σ C )F ) mod γ ) as q 0 in γ \D σ. (For details, see [KU].) Fix a point b P 1 {0, 1, } on the λ-plane, identify H 3 (W b, Z) = H 0, and let (1) Γ = Image(π 1 (P 1 {0, 1, }) G Z ). This Γ is not neat. In fact, the local monodromy around 0 is of order 5. Let (2) P 1 {0, 1, } Γ\D be the period map. Since K Wλ is trivial, the differential of (2) is injective everywhere. Endow P 1 with the logarithmic structure associated to the divisor {1, }. Then, by 1, 2 and [KU], (2) extends to a morphism (3) ϕ : P 1 Γ\D Ξ, 0 (point) mod Γ Γ\D, 1 (Ad(g)(σ α )-nilpotent orbit for some g G Q ) mod Γ, (Ad(g)(σ β )-nilpotent orbit for some g G Q ) mod Γ. of logarithmic ringed spaces (cf. [KU, (i)]). The image of the extended period map ϕ is an analytic curve, which is not affected by the slits of the space Γ\D Ξ. Let X = Γ\D Ξ. Let P 1 = 1, P = P 1, and let Q 1 = ϕ(p 1 ), Q = ϕ(p ) X. Then, by the observation of local monodromy and holomorphic 3-form basing on the descriptions in 1 and 2, we have (4) ϕ 1 (Q λ ) = {P λ } for λ = 1,. 4

5 4. Generic Torelli theorem We use the notation in the previous sections. Theorem. The period map ϕ in 3 (2) is the normalization of analytic spaces over its image. Proof. We use the fs logarithmic point P 1 and its image Q 1 at the boundaries ( 3). Since ϕ 1 (Q 1 ) = {P 1 } ( 3 (4)), it is enough to show that the local ramification index at Q 1 is one, i.e., Claim. (M X /O X ) Q 1 (M P 1/O P 1 ) P1 is surjective. Let N := N α be the nilpotent endomorphism introduced in 2. Let q be a local coordinate on a neighborhood U of P 1 = 1 in P 1, and let z = (2πi) 1 log q be a branch over U {P 1 }. Then exp( zn)e 3 = e 3 ze 1 is single-valued. Let ω( q) be a local frame of the locally free O P 1-module F 3. Write ω( q) = 4 j=1 a j( q)e j, and define t = a 1 ( q)/a 3 ( q). Then t = e 3, ω( q) 0 e 1, ω( q) 0 = exp( zn)e 3, ω( q) 0 + z e 1, ω( q) 0 e 1, ω( q) 0 = z + (single-valued holomorphic function in q). Let q = e 2πit. Then q = u q for some u O P 1,P 1. Let V be a neighborhood of Q 1 in X = Γ\D Ξ. We have a composite morphism of fs logarithmic local ringed spaces U V C, q q = e 2πi( a 1/a 3 ) (= u q). Hence the composite morphism (M P 1/O P 1 ) P1 (M X /O X ) Q 1 (M C /O C ) 0 of reduced logarithmic structures is an isomorphism. The claim follows. In fact, that is an isomorphism since the rank of (M X /O X ) Q 1 is one in the present case. 5. The second proof, and logarithmic generic Torelli theorem In this section, we give another proof of the generic Torelli theorem in 4 by using the fs logarithmic points P and Q at the boundaries. Since ϕ 1 (Q ) = {P } ( 3 (4)), it is enough to show the following: Claim. (M X /O X ) Q (M P 1/O P 1 ) P is surjective. Proof. Let N be the logarithm of the local monodromy at λ =. Let β 1, β 2, α 1, α 2 be the integral symplectic basis of H 0 given in [M1, Appendix C], and let g 0 = α 2, g 1 = 2α 1 + β 1, g 2 = β 2, g 3 = α 1 + β 1. Then g 3, g 2, g 1, g 0 is another integral symplectic basis such that g 0, g 1 is a good integral basis of Image(N 2 ) in the sense of [M1, 2, Appendix C]. Using the result there, we have (N(g 3 ), N(g 2 ), N(g 1 ), N(g 0 )) = (g 3, g 2, g 1, g 0 ). 5 9/ /2 25/

6 Let q be a local coordinate on a neighborhood U of P = in P 1, and let z = (2πi) 1 log q be a branch over U {P }. Then exp( zn)g 1 = g 1 zg 0 is single-valued. Let ω( q) be a local frame of the locally free O P 1-module F 3. Write ω( q) = 3 j=0 b j( q)g j, and define t = b 3 ( q)/b 2 ( q) and q = e 2πit. Then, as in 4, we see q = u q for some u O P 1,P, and the claim follows. Combining the results in 4 and 5, we have the following refinement. Theorem. The period map ϕ in 3 (3) is the normalization of fs logarithmic analytic spaces over its image. Here the normalization of the image of ϕ is endowed with the pull-back logarithmic structure. Note. Canonical coordinates in [M1], [M2], like q in the proofs of Claims in 4 and 5, can be understood more naturally in the context of PLH. Problem 1. Eliminate the possibility that the image ϕ(p 1 ) would have singularities in Theorems in 4 and 5. Problem 2. Describe the global monodromy group Γ explicitly. Generators of Γ are computed ([COGP], [M1, Appendix C]), and Γ is known to be Zariski dense in G Z = Sp(4, Z) ([COGP], [D, 13]). Acknowledgement. The author is grateful to Kazuya Kato and Chikara Nakayama. The author is also grateful to the referee for the useful comments. References [COGP] P. Candelas, C. de la Ossa, P. S. Green, and L. Parks, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Phys. Lett. B 258 (1991), [D] P. Deligne, Local behavior of Hodge structures at infinity, AMS/IP Studies in Advanced Mathematics 1 (1997), [KU] K. Kato and S. Usui, Classifying spaces of degenerating polarized Hodge structures, Ann. Math. Studies, Princeton Univ. Press, Princeton, 2009, pp [M1] D. Morrison, Mirror symmetry and rational curves on quintic threefolds: A guide for mathematicians, J. of AMS 6-1 (1993), [M2], Compactifications of moduli spaces inspired by mirror symmetry, Astérisque 218 (1993), [S] B. Szendröi, Calabi-Yau threefolds with a curve of singularities and counterexamples to the Torelli problem; ibid II, Internat. J. Math. 11; Math. Proc. Cambridge Philos. Soc. 129 (2000), ; [V] C. Voisin, A generic Torelli theorem for the quintic threefold, New Trends in Algebraic Geometry, London Math. Soc., Lecture Note Ser. 264 (1999), Sampei USUI Graduate School of Science Osaka University Toyonaka, Osaka, , Japan 6

1 Moduli spaces of polarized Hodge structures.

1 Moduli spaces of polarized Hodge structures. 1 Moduli spaces of polarized Hodge structures. First of all, we briefly summarize the classical theory of the moduli spaces of polarized Hodge structures. 1.1 The moduli space M h = Γ\D h. Let n be an

More information

Monodromy of the Dwork family, following Shepherd-Barron X n+1. P 1 λ. ζ i = 1}/ (µ n+1 ) H.

Monodromy of the Dwork family, following Shepherd-Barron X n+1. P 1 λ. ζ i = 1}/ (µ n+1 ) H. Monodromy of the Dwork family, following Shepherd-Barron 1. The Dwork family. Consider the equation (f λ ) f λ (X 0, X 1,..., X n ) = λ(x n+1 0 + + X n+1 n ) (n + 1)X 0... X n = 0, where λ is a free parameter.

More information

Mirror Symmetry: Introduction to the B Model

Mirror Symmetry: Introduction to the B Model Mirror Symmetry: Introduction to the B Model Kyler Siegel February 23, 2014 1 Introduction Recall that mirror symmetry predicts the existence of pairs X, ˇX of Calabi-Yau manifolds whose Hodge diamonds

More information

SUBJECT I. TOROIDAL PARTIAL COMPACTIFICATIONS AND MODULI OF POLARIZED LOGARITHMIC HODGE STRUCTURES

SUBJECT I. TOROIDAL PARTIAL COMPACTIFICATIONS AND MODULI OF POLARIZED LOGARITHMIC HODGE STRUCTURES Introduction This book is the full detailed version of the paper [KU1]. In [G1], Griffiths defined and studied the classifying space D of polarized Hodge structures of fixed weight w and fixed Hodge numbers

More information

arxiv: v3 [math.ag] 15 Oct 2010

arxiv: v3 [math.ag] 15 Oct 2010 ON THE BOUNDARY OF THE MODULI SPACES OF LOG HODGE STRUCTURES: TRIVIALITY OF THE TORSOR TATSUKI HAYAMA arxiv:0806.4965v3 [math.ag] 15 Oct 2010 Abstract. In this paper we will study the moduli spaces of

More information

THE QUANTUM CONNECTION

THE QUANTUM CONNECTION THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,

More information

EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION

EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION JAN CHRISTIAN ROHDE Introduction By string theoretical considerations one is interested in Calabi-Yau manifolds since Calabi-Yau 3-manifolds

More information

Periods and generating functions in Algebraic Geometry

Periods and generating functions in Algebraic Geometry Periods and generating functions in Algebraic Geometry Hossein Movasati IMPA, Instituto de Matemática Pura e Aplicada, Brazil www.impa.br/ hossein/ Abstract In 1991 Candelas-de la Ossa-Green-Parkes predicted

More information

The geometry of Landau-Ginzburg models

The geometry of Landau-Ginzburg models Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror

More information

Math 797W Homework 4

Math 797W Homework 4 Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

More information

Introduction To K3 Surfaces (Part 2)

Introduction To K3 Surfaces (Part 2) Introduction To K3 Surfaces (Part 2) James Smith Calf 26th May 2005 Abstract In this second introductory talk, we shall take a look at moduli spaces for certain families of K3 surfaces. We introduce the

More information

CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents

CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS Contents 1. The Lefschetz hyperplane theorem 1 2. The Hodge decomposition 4 3. Hodge numbers in smooth families 6 4. Birationally

More information

Topics in Geometry: Mirror Symmetry

Topics in Geometry: Mirror Symmetry MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:

More information

ALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3

ALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3 ALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3 IZZET COSKUN AND ERIC RIEDL Abstract. We prove that a curve of degree dk on a very general surface of degree d 5 in P 3 has geometric

More information

THE EXTENDED LOCUS OF HODGE CLASSES

THE EXTENDED LOCUS OF HODGE CLASSES THE EXTENDED LOCUS OF HODGE CLASSES CHRISTIAN SCHNELL Abstract. We introduce an extended locus of Hodge classes that also takes into account integral classes that become Hodge classes in the limit. More

More information

Kähler manifolds and variations of Hodge structures

Kähler manifolds and variations of Hodge structures Kähler manifolds and variations of Hodge structures October 21, 2013 1 Some amazing facts about Kähler manifolds The best source for this is Claire Voisin s wonderful book Hodge Theory and Complex Algebraic

More information

ON THE MODULI B-DIVISORS OF LC-TRIVIAL FIBRATIONS

ON THE MODULI B-DIVISORS OF LC-TRIVIAL FIBRATIONS ON THE MODULI B-DIVISORS OF LC-TRIVIAL FIBRATIONS OSAMU FUJINO AND YOSHINORI GONGYO Abstract. Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial

More information

SMOOTH FINITE ABELIAN UNIFORMIZATIONS OF PROJECTIVE SPACES AND CALABI-YAU ORBIFOLDS

SMOOTH FINITE ABELIAN UNIFORMIZATIONS OF PROJECTIVE SPACES AND CALABI-YAU ORBIFOLDS SMOOTH FINITE ABELIAN UNIFORMIZATIONS OF PROJECTIVE SPACES AND CALABI-YAU ORBIFOLDS A. MUHAMMED ULUDAĞ Dedicated to Mehmet Çiftçi Abstract. We give a classification of smooth complex manifolds with a finite

More information

Part II. Riemann Surfaces. Year

Part II. Riemann Surfaces. Year Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 96 Paper 2, Section II 23F State the uniformisation theorem. List without proof the Riemann surfaces which are uniformised

More information

HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

More information

Arithmetic Mirror Symmetry

Arithmetic Mirror Symmetry Arithmetic Mirror Symmetry Daqing Wan April 15, 2005 Institute of Mathematics, Chinese Academy of Sciences, Beijing, P.R. China Department of Mathematics, University of California, Irvine, CA 92697-3875

More information

arxiv:alg-geom/ v1 29 Jul 1993

arxiv:alg-geom/ v1 29 Jul 1993 Hyperkähler embeddings and holomorphic symplectic geometry. Mikhail Verbitsky, verbit@math.harvard.edu arxiv:alg-geom/9307009v1 29 Jul 1993 0. ntroduction. n this paper we are studying complex analytic

More information

Special cubic fourfolds

Special cubic fourfolds Special cubic fourfolds 1 Hodge diamonds Let X be a cubic fourfold, h H 2 (X, Z) be the (Poincaré dual to the) hyperplane class. We have h 4 = deg(x) = 3. By the Lefschetz hyperplane theorem, one knows

More information

On log flat descent. Luc Illusie, Chikara Nakayama, and Takeshi Tsuji

On log flat descent. Luc Illusie, Chikara Nakayama, and Takeshi Tsuji On log flat descent Luc Illusie, Chikara Nakayama, and Takeshi Tsuji Abstract We prove the log flat descent of log étaleness, log smoothness, and log flatness for log schemes. Contents 1. Review of log

More information

Generalized Tian-Todorov theorems

Generalized Tian-Todorov theorems Generalized Tian-Todorov theorems M.Kontsevich 1 The classical Tian-Todorov theorem Recall the classical Tian-Todorov theorem (see [4],[5]) about the smoothness of the moduli spaces of Calabi-Yau manifolds:

More information

where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism

where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism 8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the

More information

MODULI OF ALGEBRAIC SL 3 -VECTOR BUNDLES OVER ADJOINT REPRESENTATION

MODULI OF ALGEBRAIC SL 3 -VECTOR BUNDLES OVER ADJOINT REPRESENTATION Masuda, K. Osaka J. Math. 38 (200), 50 506 MODULI OF ALGEBRAIC SL 3 -VECTOR BUNDLES OVER ADJOINT REPRESENTATION KAYO MASUDA (Received June 2, 999). Introduction and result Let be a reductive complex algebraic

More information

ON EMBEDDABLE 1-CONVEX SPACES

ON EMBEDDABLE 1-CONVEX SPACES Vâjâitu, V. Osaka J. Math. 38 (2001), 287 294 ON EMBEDDABLE 1-CONVEX SPACES VIOREL VÂJÂITU (Received May 31, 1999) 1. Introduction Throughout this paper all complex spaces are assumed to be reduced and

More information

Chern numbers and Hilbert Modular Varieties

Chern numbers and Hilbert Modular Varieties Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point

More information

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ

More information

Hodge Theory of Maps

Hodge Theory of Maps Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar

More information

Logarithmic geometry and rational curves

Logarithmic geometry and rational curves Logarithmic geometry and rational curves Summer School 2015 of the IRTG Moduli and Automorphic Forms Siena, Italy Dan Abramovich Brown University August 24-28, 2015 Abramovich (Brown) Logarithmic geometry

More information

HODGE THEORY AND QUANTUM COHOMOLOGY. Gregory J Pearlstein

HODGE THEORY AND QUANTUM COHOMOLOGY. Gregory J Pearlstein HODGE THEORY AND QUANTUM COHOMOLOGY Gregory J Pearlstein 1 Introduction: Axiomatically, a Hodge structure of weight k consists of a finite dimensional R-vector space V R equipped with a decreasing filtration,

More information

CANONICAL BUNDLE FORMULA AND VANISHING THEOREM

CANONICAL BUNDLE FORMULA AND VANISHING THEOREM CANONICAL BUNDLE FORMULA AND VANISHING THEOREM OSAMU FUJINO Abstract. In this paper, we treat two different topics. We give sample computations of our canonical bundle formula. They help us understand

More information

MOTIVES OF SOME ACYCLIC VARIETIES

MOTIVES OF SOME ACYCLIC VARIETIES Homology, Homotopy and Applications, vol.??(?),??, pp.1 6 Introduction MOTIVES OF SOME ACYCLIC VARIETIES ARAVIND ASOK (communicated by Charles Weibel) Abstract We prove that the Voevodsky motive with Z-coefficients

More information

Period Domains. Carlson. June 24, 2010

Period Domains. Carlson. June 24, 2010 Period Domains Carlson June 4, 00 Carlson - Period Domains Period domains are parameter spaces for marked Hodge structures. We call Γ\D the period space, which is a parameter space of isomorphism classes

More information

3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).

3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X). 3. Lecture 3 3.1. Freely generate qfh-sheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)

More information

THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3

THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3 THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3 KAZUNORI NAKAMOTO AND TAKESHI TORII Abstract. There exist 26 equivalence classes of k-subalgebras of M 3 (k) for any algebraically closed field

More information

Topics in Geometry: Mirror Symmetry

Topics in Geometry: Mirror Symmetry MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:

More information

Overview of classical mirror symmetry

Overview of classical mirror symmetry Overview of classical mirror symmetry David Cox (notes by Paul Hacking) 9/8/09 () Physics (2) Quintic 3-fold (3) Math String theory is a N = 2 superconformal field theory (SCFT) which models elementary

More information

SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS

SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e in a general complete intersection of multidegree

More information

Useful theorems in complex geometry

Useful theorems in complex geometry Useful theorems in complex geometry Diego Matessi April 30, 2003 Abstract This is a list of main theorems in complex geometry that I will use throughout the course on Calabi-Yau manifolds and Mirror Symmetry.

More information

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES N. MOHAN KUMAR, A. P. RAO, AND G. V. RAVINDRA Abstract. We prove that any rank two arithmetically Cohen- Macaulay vector bundle on a general hypersurface

More information

TitleOn manifolds with trivial logarithm. Citation Osaka Journal of Mathematics. 41(2)

TitleOn manifolds with trivial logarithm. Citation Osaka Journal of Mathematics. 41(2) TitleOn manifolds with trivial logarithm Author(s) Winkelmann, Jorg Citation Osaka Journal of Mathematics. 41(2) Issue 2004-06 Date Text Version publisher URL http://hdl.handle.net/11094/7844 DOI Rights

More information

Kähler Potential of Moduli Space. of Calabi-Yau d-fold embedded in CP d+1

Kähler Potential of Moduli Space. of Calabi-Yau d-fold embedded in CP d+1 March 2000 hep-th/000364 Kähler Potential of Moduli Space arxiv:hep-th/000364v 20 Mar 2000 of Calabi-Yau d-fold embedded in CP d+ Katsuyuki Sugiyama Department of Fundamental Sciences Faculty of Integrated

More information

Takao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,...

Takao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,... J. Korean Math. Soc. 40 (2003), No. 4, pp. 667 680 HOMOGENEOUS POLYNOMIAL HYPERSURFACE ISOLATED SINGULARITIES Takao Akahori Abstract. The mirror conjecture means originally the deep relation between complex

More information

On exceptional completions of symmetric varieties

On exceptional completions of symmetric varieties Journal of Lie Theory Volume 16 (2006) 39 46 c 2006 Heldermann Verlag On exceptional completions of symmetric varieties Rocco Chirivì and Andrea Maffei Communicated by E. B. Vinberg Abstract. Let G be

More information

Complex Algebraic Geometry: Smooth Curves Aaron Bertram, First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool

Complex Algebraic Geometry: Smooth Curves Aaron Bertram, First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool Complex Algebraic Geometry: Smooth Curves Aaron Bertram, 2010 12. First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool for classifying smooth projective curves, i.e. giving

More information

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9

More information

arxiv: v3 [math.ag] 26 Jun 2017

arxiv: v3 [math.ag] 26 Jun 2017 CALABI-YAU MANIFOLDS REALIZING SYMPLECTICALLY RIGID MONODROMY TUPLES arxiv:50.07500v [math.ag] 6 Jun 07 CHARLES F. DORAN, ANDREAS MALMENDIER Abstract. We define an iterative construction that produces

More information

COMPLEX ALGEBRAIC SURFACES CLASS 9

COMPLEX ALGEBRAIC SURFACES CLASS 9 COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution

More information

RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES RIMS A Note on an Anabelian Open Basis for a Smooth Variety. Yuichiro HOSHI.

RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES RIMS A Note on an Anabelian Open Basis for a Smooth Variety. Yuichiro HOSHI. RIMS-1898 A Note on an Anabelian Open Basis for a Smooth Variety By Yuichiro HOSHI January 2019 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan A Note on an Anabelian Open Basis

More information

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections The generalized Hodge and Bloch conjectures are equivalent for general complete intersections Claire Voisin CNRS, Institut de mathématiques de Jussieu 0 Introduction Recall first that a weight k Hodge

More information

RUSSELL S HYPERSURFACE FROM A GEOMETRIC POINT OF VIEW

RUSSELL S HYPERSURFACE FROM A GEOMETRIC POINT OF VIEW Hedén, I. Osaka J. Math. 53 (2016), 637 644 RUSSELL S HYPERSURFACE FROM A GEOMETRIC POINT OF VIEW ISAC HEDÉN (Received November 4, 2014, revised May 11, 2015) Abstract The famous Russell hypersurface is

More information

Delzant s Garden. A one-hour tour to symplectic toric geometry

Delzant s Garden. A one-hour tour to symplectic toric geometry Delzant s Garden A one-hour tour to symplectic toric geometry Tour Guide: Zuoqin Wang Travel Plan: The earth America MIT Main building Math. dept. The moon Toric world Symplectic toric Delzant s theorem

More information

A Note on Dormant Opers of Rank p 1 in Characteristic p

A Note on Dormant Opers of Rank p 1 in Characteristic p A Note on Dormant Opers of Rank p 1 in Characteristic p Yuichiro Hoshi May 2017 Abstract. In the present paper, we prove that the set of equivalence classes of dormant opers of rank p 1 over a projective

More information

SINGULARITIES OF LOW DEGREE COMPLETE INTERSECTIONS

SINGULARITIES OF LOW DEGREE COMPLETE INTERSECTIONS METHODS AND APPLICATIONS OF ANALYSIS. c 2017 International Press Vol. 24, No. 1, pp. 099 104, March 2017 007 SINGULARITIES OF LOW DEGREE COMPLETE INTERSECTIONS SÁNDOR J. KOVÁCS Dedicated to Henry Laufer

More information

a double cover branched along the smooth quadratic line complex

a double cover branched along the smooth quadratic line complex QUADRATIC LINE COMPLEXES OLIVIER DEBARRE Abstract. In this talk, a quadratic line complex is the intersection, in its Plücker embedding, of the Grassmannian of lines in an 4-dimensional projective space

More information

K3 Surfaces and Lattice Theory

K3 Surfaces and Lattice Theory K3 Surfaces and Lattice Theory Ichiro Shimada Hiroshima University 2014 Aug Singapore 1 / 26 Example Consider two surfaces S + and S in C 3 defined by w 2 (G(x, y) ± 5 H(x, y)) = 1, where G(x, y) := 9

More information

The Canonical Sheaf. Stefano Filipazzi. September 14, 2015

The Canonical Sheaf. Stefano Filipazzi. September 14, 2015 The Canonical Sheaf Stefano Filipazzi September 14, 015 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will go over

More information

On the GIT moduli of semistable pairs consisting of a plane cubic curve and a line

On the GIT moduli of semistable pairs consisting of a plane cubic curve and a line On the GIT moduli of semistable pairs consisting of a plane cubic curve and a line Masamichi Kuroda at Tambara Institute of Mathematical Sciences The University of Tokyo August 26, 2015 Masamichi Kuroda

More information

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion

More information

VERY STABLE BUNDLES AND PROPERNESS OF THE HITCHIN MAP

VERY STABLE BUNDLES AND PROPERNESS OF THE HITCHIN MAP VERY STABLE BUNDLES AND PROPERNESS OF THE HITCHIN MAP CHRISTIAN PAULY AND ANA PEÓN-NIETO Abstract. Let X be a smooth complex projective curve of genus g 2 and let K be its canonical bundle. In this note

More information

On the Virtual Fundamental Class

On the Virtual Fundamental Class On the Virtual Fundamental Class Kai Behrend The University of British Columbia Seoul, August 14, 2014 http://www.math.ubc.ca/~behrend/talks/seoul14.pdf Overview Donaldson-Thomas theory: counting invariants

More information

Mirror symmetry. Mark Gross. July 24, University of Cambridge

Mirror symmetry. Mark Gross. July 24, University of Cambridge University of Cambridge July 24, 2015 : A very brief and biased history. A search for examples of compact Calabi-Yau three-folds by Candelas, Lynker and Schimmrigk (1990) as crepant resolutions of hypersurfaces

More information

SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS

SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e on a general complete intersection of multidegree

More information

MODULI SPACES OF CURVES

MODULI SPACES OF CURVES MODULI SPACES OF CURVES SCOTT NOLLET Abstract. My goal is to introduce vocabulary and present examples that will help graduate students to better follow lectures at TAGS 2018. Assuming some background

More information

RESEARCH STATEMENT FEI XU

RESEARCH STATEMENT FEI XU RESEARCH STATEMENT FEI XU. Introduction Recall that a variety is said to be rational if it is birational to P n. For many years, mathematicians have worked on the rationality of smooth complete intersections,

More information

A NEW FAMILY OF SYMPLECTIC FOURFOLDS

A NEW FAMILY OF SYMPLECTIC FOURFOLDS A NEW FAMILY OF SYMPLECTIC FOURFOLDS OLIVIER DEBARRE This is joint work with Claire Voisin. 1. Irreducible symplectic varieties It follows from work of Beauville and Bogomolov that any smooth complex compact

More information

LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS

LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS JON WOLFSON Abstract. We show that there is an integral homology class in a Kähler-Einstein surface that can be represented by a lagrangian twosphere

More information

An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations

An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations Mircea Mustaţă University of Michigan Mainz July 9, 2018 Mircea Mustaţă () An overview of D-modules Mainz July 9, 2018 1 The

More information

Porteous s Formula for Maps between Coherent Sheaves

Porteous s Formula for Maps between Coherent Sheaves Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for

More information

SETS OF DEGREES OF MAPS BETWEEN SU(2)-BUNDLES OVER THE 5-SPHERE

SETS OF DEGREES OF MAPS BETWEEN SU(2)-BUNDLES OVER THE 5-SPHERE SETS OF DEGREES OF MAPS BETWEEN SU(2)-BUNDLES OVER THE 5-SPHERE JEAN-FRANÇOIS LAFONT AND CHRISTOFOROS NEOFYTIDIS ABSTRACT. We compute the sets of degrees of maps between SU(2)-bundles over S 5, i.e. between

More information

Periods, Galois theory and particle physics

Periods, Galois theory and particle physics Periods, Galois theory and particle physics Francis Brown All Souls College, Oxford Gergen Lectures, 21st-24th March 2016 1 / 29 Reminders We are interested in periods I = γ ω where ω is a regular algebraic

More information

arxiv: v2 [math.ag] 29 Aug 2009

arxiv: v2 [math.ag] 29 Aug 2009 LOGARITHMIC GEOMETRY, MINIMAL FREE RESOLUTIONS AND TORIC ALGEBRAIC STACKS arxiv:0707.2568v2 [math.ag] 29 Aug 2009 ISAMU IWANARI Abstract. In this paper we will introduce a certain type of morphisms of

More information

PRIME EXCEPTIONAL DIVISORS ON HOLOMORPHIC SYMPLECTIC VARIETIES AND MONODROMY-REFLECTIONS

PRIME EXCEPTIONAL DIVISORS ON HOLOMORPHIC SYMPLECTIC VARIETIES AND MONODROMY-REFLECTIONS PRIME EXCEPTIONAL DIVISORS ON HOLOMORPHIC SYMPLECTIC VARIETIES AND MONODROMY-REFLECTIONS EYAL MARKMAN Abstract. Let X be a projective irreducible holomorphic symplectic variety. The second integral cohomology

More information

ALGORITHMS FOR ALGEBRAIC CURVES

ALGORITHMS FOR ALGEBRAIC CURVES ALGORITHMS FOR ALGEBRAIC CURVES SUMMARY OF LECTURE 7 I consider the problem of computing in Pic 0 (X) where X is a curve (absolutely integral, projective, smooth) over a field K. Typically K is a finite

More information

IN POSITIVE CHARACTERISTICS: 3. Modular varieties with Hecke symmetries. 7. Foliation and a conjecture of Oort

IN POSITIVE CHARACTERISTICS: 3. Modular varieties with Hecke symmetries. 7. Foliation and a conjecture of Oort FINE STRUCTURES OF MODULI SPACES IN POSITIVE CHARACTERISTICS: HECKE SYMMETRIES AND OORT FOLIATION 1. Elliptic curves and their moduli 2. Moduli of abelian varieties 3. Modular varieties with Hecke symmetries

More information

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be

More information

BEZOUT S THEOREM CHRISTIAN KLEVDAL

BEZOUT S THEOREM CHRISTIAN KLEVDAL BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this

More information

Counting problems in Number Theory and Physics

Counting problems in Number Theory and Physics Counting problems in Number Theory and Physics Hossein Movasati IMPA, Instituto de Matemática Pura e Aplicada, Brazil www.impa.br/ hossein/ Encontro conjunto CBPF-IMPA, 2011 A documentary on string theory

More information

COURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA

COURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties

More information

FLABBY STRICT DEFORMATION QUANTIZATIONS AND K-GROUPS

FLABBY STRICT DEFORMATION QUANTIZATIONS AND K-GROUPS FLABBY STRICT DEFORMATION QUANTIZATIONS AND K-GROUPS HANFENG LI Abstract. We construct examples of flabby strict deformation quantizations not preserving K-groups. This answers a question of Rieffel negatively.

More information

Toric Varieties and the Secondary Fan

Toric Varieties and the Secondary Fan Toric Varieties and the Secondary Fan Emily Clader Fall 2011 1 Motivation The Batyrev mirror symmetry construction for Calabi-Yau hypersurfaces goes roughly as follows: Start with an n-dimensional reflexive

More information

Introduction Curves Surfaces Curves on surfaces. Curves and surfaces. Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway

Introduction Curves Surfaces Curves on surfaces. Curves and surfaces. Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway Curves and surfaces Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway What is algebraic geometry? IMA, April 13, 2007 Outline Introduction Curves Surfaces Curves on surfaces

More information

Lecture 1. Toric Varieties: Basics

Lecture 1. Toric Varieties: Basics Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture

More information

Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves. Yuichiro Hoshi

Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves. Yuichiro Hoshi Hokkaido Mathematical Journal ol. 45 (2016) p. 271 291 Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves uichiro Hoshi (Received February 28, 2014; Revised June 12, 2014) Abstract.

More information

Vanishing theorems for toric polyhedra

Vanishing theorems for toric polyhedra RIMS Kôkyûroku Bessatsu 4x (200x), 000 000 Vanishing theorems for toric polyhedra By Osamu Fujino Abstract A toric polyhedron is a reduced closed subscheme of a toric variety that are partial unions of

More information

ASIAN J. MATH. cfl 2002 International Press Vol. 6, No. 1, pp , March THE WEAK LEFSCHETZ PRINCIPLE IS FALSE FOR AMPLE CONES Λ BRENDAN

ASIAN J. MATH. cfl 2002 International Press Vol. 6, No. 1, pp , March THE WEAK LEFSCHETZ PRINCIPLE IS FALSE FOR AMPLE CONES Λ BRENDAN ASIAN J. MATH. cfl 2002 International Press Vol. 6, No. 1, pp. 095 100, March 2002 005 THE WEAK LEFSCHETZ PRINCIPLE IS FALSE FOR AMPLE CONES Λ BRENDAN HASSETT y, HUI-WEN LIN z, AND CHIN-LUNG WANG x 1.

More information

Hodge Structures. October 8, A few examples of symmetric spaces

Hodge Structures. October 8, A few examples of symmetric spaces Hodge Structures October 8, 2013 1 A few examples of symmetric spaces The upper half-plane H is the quotient of SL 2 (R) by its maximal compact subgroup SO(2). More generally, Siegel upper-half space H

More information

A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY. We also have an isomorphism of holomorphic vector bundles

A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY. We also have an isomorphism of holomorphic vector bundles A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY LIVIU I. NICOLAESCU ABSTRACT. These are notes for a talk at a topology seminar at ND.. GENERAL FACTS In the sequel, for simplicity we denote the complex

More information

arxiv: v1 [math.ag] 13 Mar 2019

arxiv: v1 [math.ag] 13 Mar 2019 THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show

More information

Quotients of E n by a n+1 and Calabi-Yau manifolds

Quotients of E n by a n+1 and Calabi-Yau manifolds Quotients of E n by a n+1 and Calabi-Yau manifolds Kapil Paranjape and Dinakar Ramakrishnan Abstract. We give a simple construction, for n 2, of an n- dimensional Calabi-Yau variety of Kummer type by studying

More information

FAKE PROJECTIVE SPACES AND FAKE TORI

FAKE PROJECTIVE SPACES AND FAKE TORI FAKE PROJECTIVE SPACES AND FAKE TORI OLIVIER DEBARRE Abstract. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to P n is isomorphic to P n.

More information

Crash Course on Toric Geometry

Crash Course on Toric Geometry Crash Course on Toric Geometry Emily Clader RTG Workshop on Mirror Symmetry February 2012 The Kähler cone If X Σ is a simplicial projective toric variety, then A n 1 (X Σ ) R = H 2 (X Σ ; R), so H 1,1

More information

Logarithmic geometry and moduli

Logarithmic geometry and moduli Logarithmic geometry and moduli Lectures at the Sophus Lie Center Dan Abramovich Brown University June 16-17, 2014 Abramovich (Brown) Logarithmic geometry and moduli June 16-17, 2014 1 / 1 Heros: Olsson

More information

Nodal symplectic spheres in CP 2 with positive self intersection

Nodal symplectic spheres in CP 2 with positive self intersection Nodal symplectic spheres in CP 2 with positive self intersection Jean-François BARRAUD barraud@picard.ups-tlse.fr Abstract 1 : Let ω be the canonical Kähler structure on CP 2 We prove that any ω-symplectic

More information

Stable bundles on CP 3 and special holonomies

Stable bundles on CP 3 and special holonomies Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26, 2010 1 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M

More information

arxiv: v1 [math.ag] 30 Apr 2018

arxiv: v1 [math.ag] 30 Apr 2018 QUASI-LOG CANONICAL PAIRS ARE DU BOIS OSAMU FUJINO AND HAIDONG LIU arxiv:1804.11138v1 [math.ag] 30 Apr 2018 Abstract. We prove that every quasi-log canonical pair has only Du Bois singularities. Note that

More information

Curve counting and generating functions

Curve counting and generating functions Curve counting and generating functions Ragni Piene Università di Roma Tor Vergata February 26, 2010 Count partitions Let n be an integer. How many ways can you write n as a sum of two positive integers?

More information