Eduard Looijenga

This is a survey paper that also contains some new results. It will appear in the proceedings of the AMS summer research institute on Algebraic Geometry at Santa Cruz. This is a survey paper that also contains some new results. It will appear in the proceedings of the AMS summer research institute on Algebraic Geometry at Santa Cruz.

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The period map for cubic fourfolds

2009-01-30

Eduard Looijenga

The period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20. We determine the image of this period map (thus confirming a … The period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20. We determine the image of this period map (thus confirming a conjecture of Hassett) and give at the same time a new proof of the theorem of Voisin that asserts that this period map is an open embedding. An algebraic version of our main result is an identification of the algebra of SL (6,ℂ)-invariant polynomials on the representation space Sym 3(ℂ6)* with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20. We also describe the stratification of the moduli space of semistable cubic fourfolds in terms of a Vinberg-Dynkin diagram.

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$L^2$ -cohomology of locally symmetric varieties

1988-01-01

Eduard Looijenga

Smooth deligne-mumford compactifications by means of prym level structures

1994-01-01

Eduard Looijenga

We show that the Deligne Mumford compacti cation of the moduli space of smooth complex curves of genus g admits a smooth Galois covering whose general point classi es curves … We show that the Deligne Mumford compacti cation of the moduli space of smooth complex curves of genus g admits a smooth Galois covering whose general point classi es curves with a level structure on their universal Prym cover We also correct Brylinski s proof that the Teichm uller group rami es universally with respect to the Deligne Mumford compacti cation

Compactifications defined by arrangements, II: Locally symmetric varieties of type IV

2003-09-15

Eduard Looijenga

We define a new class of completions of locally symmetric varieties of type IV which interpolates between the Baily-Borel compactification and Mumford's toric compactifications. An arithmetic arrangement in a locally … We define a new class of completions of locally symmetric varieties of type IV which interpolates between the Baily-Borel compactification and Mumford's toric compactifications. An arithmetic arrangement in a locally symmetric variety of type IV determines such a completion canonically. This completion admits a natural contraction that leaves the complement of the arrangement untouched. The resulting completion of the arrangement complement is very much like a Baily-Borel compactification: it is the Proj of an algebra of meromorphic automorphic forms. When that complement has a moduli-space interpretation, then what we get is often a compactification obtained by means of geometric invariant theory. We illustrate this with several examples: moduli spaces of polarized K3 and Enriques surfaces and the semiuniversal deformation of a triangle singularity. We also discuss the question of when a type IV arrangement is definable by an automorphic form.

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On the semi-universal deformation of a simple-elliptic hypersurface singularity

1977-01-01

Eduard Looijenga

Geometric Structures on the Complement of a Projective Arrangement

2005-06-01

Wim Couwenberg , Gert Heckman , Eduard Looijenga

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A note on polynomial isolated singularities

1971-01-01

Eduard Looijenga

Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map

1994-01-01

Eduard Looijenga

This replacement corrects statement and proof of the main result. Also, a section on the universal Abel-Jacobi map has been added. This replacement corrects statement and proof of the main result. Also, a section on the universal Abel-Jacobi map has been added.

The smoothing components of a triangle singularity. II

1984-10-01

Eduard Looijenga

Cellular Decompositions of Compactified Moduli Spaces of Pointed Curves

1995-01-01

Eduard Looijenga

To a closed connected oriented surface S of genus g and a nonempty finite subset P of S is associated a simplicial complex (the arc complex) that plays a basic … To a closed connected oriented surface S of genus g and a nonempty finite subset P of S is associated a simplicial complex (the arc complex) that plays a basic role in understanding the mapping class group of the pair (S, P). It is known that this arc complex contains in a natural way the product of the Teichmüller space of (S, P) with an open simplex. In this paper we give an interpretation for the whole arc complex and prove that it is a quotient of a Deligne-Mumford extension of this Teichmüller space and a closed simplex. We also describe a modification of the arc complex in the spirit of Deligne-Mumford.

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On the semi-universal deformation of a simple-elliptic hypersurface singularity Part II: the discriminant

1978-01-01

Eduard Looijenga

Compactifications defined by arrangements, I: The ball quotient case

2003-05-15

Eduard Looijenga

We define a natural compactification of an arrangement complement in a ball quotient. We show that when this complement has a moduli space interpretation, then this compactification is often one … We define a natural compactification of an arrangement complement in a ball quotient. We show that when this complement has a moduli space interpretation, then this compactification is often one that appears naturally by means of geometric invariant theory. We illustrate this with the moduli spaces of smooth quartic curves and rational elliptic surfaces.

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Uniformization by Lauricella Functions — An Overview of the Theory of Deligne-Mostow

2007-06-27

Eduard Looijenga

This is a survey of the Deligne-Mostow theory of Lauricella functions, or what almost amounts to the same, of the period map for cyclic coverings of the Riemann sphere. This is a survey of the Deligne-Mostow theory of Lauricella functions, or what almost amounts to the same, of the period map for cyclic coverings of the Riemann sphere.

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The period map for cubic threefolds

2007-07-01

Eduard Looijenga , Rogier Swierstra

Allcock, Carlson and Toledo defined a period map for cubic threefolds which takes values in a ball quotient of dimension 10. A theorem of Voisin implies that this is an … Allcock, Carlson and Toledo defined a period map for cubic threefolds which takes values in a ball quotient of dimension 10. A theorem of Voisin implies that this is an open embedding. We determine its image and show that on the algebraic level this amounts to identification of the algebra of with an explicitly described algebra of meromorphic automorphic forms on the complex 10-ball.

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Cohomology of ℳ₃ and ℳ¹₃

1993-01-01

Eduard Looijenga

Moduli of Curves and Abelian Varieties

1999-01-01

Carel Faber , Eduard Looijenga

Riemann-Roch and smoothings of singularities

1986-01-01

Eduard Looijenga

Milnor number and Tjurina number of complete intersections

1985-03-01

Eduard Looijenga , J. H. M. Steenbrink

Artin groups and the fundamental groups of some moduli spaces

2007-10-28

Eduard Looijenga

We define for every affine Coxeter graph, a certain factor group of the associated Artin group and prove that some of these groups appear as orbifold fundamental groups of moduli … We define for every affine Coxeter graph, a certain factor group of the associated Artin group and prove that some of these groups appear as orbifold fundamental groups of moduli spaces. Examples are the moduli space of nonsingular cubic algebraic surfaces and the universal nonhyperelliptic smooth genus three curve. We use this to obtain a simple presentation of the mapping class group of a compact genus three topological surface with connected boundary. This leads to a modification of Wajnryb's presentation of the mapping class groups in the higher genus case that can be understood in algebro-geometric terms.

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The Moduli Space of Rational Elliptic Surfaces

2019-01-26

Gert Heckman , Eduard Looijenga

We show that the moduli space of rational elliptic surfaces admitting a section is locally a complex hyperbolic variety of dimension 8. We compare …  We show that the moduli space of rational elliptic surfaces admitting a section is locally a complex hyperbolic variety of dimension 8. We compare its Satake-Baily-Borel compactification with a compactification obtained by means of geometric invariant theory, considered by Miranda.

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Intersection theory on deligne-mumford compactifications (after Witten and Kontsevich)

1993-01-01

Eduard Looijenga

Physicists have developed two approaches to quantum gravity in dimension two One involves an a priori ill de ned integral over all conformal structures on a surface which after a … Physicists have developed two approaches to quantum gravity in dimension two One involves an a priori ill de ned integral over all conformal structures on a surface which after a suitable renormalization procedure produces a well de ned integral over moduli spaces of curves In another they consider a weighted average over piecewise at metrics on that surface and take a suitable limit of such expressions The belief that these two approaches yield the same answer led Witten to make a number of conjectures about the intersection numbers of certain natural classes that live on the moduli space of stable pointed curves One of these conjectures has been rigourously proved by Kontsevich

The discriminant of a real simple singularity

1977-01-01

Eduard Looijenga

Geometry Symposium Utrecht 1980

1981-01-01

A Oold , Beno Eckmann , Eduard Looijenga +3 more

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A Torelli theorem for Kähler-Einstein K3 surfaces

1981-01-01

Eduard Looijenga

The dimension of smoothing components

1985-03-01

Gert–Martin Greuel , Eduard Looijenga

Intersection theory on Deligne-Mumford compactifications

1993-01-01

Eduard Looijenga

The fine structure of the moduli space of abelian differentials in genus 3

2013-04-02

Eduard Looijenga , Gabriele Mondello

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Discrete automorphism groups of convex cones of finite type

2014-09-10

Eduard Looijenga

We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some … We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on selfdual cones, Weyl groups of certain Kac-Moody algebras and do occur in algebraic geometry as the automorphism groups of projective manifolds acting on their ample cones.

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Classification of Algebraic Varieties

2010-12-15

Carel Faber , Gerard van der Geer , Eduard Looijenga

The classification of projective algebraic surfaces was obtained by G. Castelnuovo and The classification of projective algebraic surfaces was obtained by G. Castelnuovo and

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Unitarity of SL(2)-conformal blocks in genus zero

2009-02-16

Eduard Looijenga

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On period maps that are open embeddings

2008-01-01

Eduard Looijenga , Rogier Swierstra

For certain complex projective manifolds (such as K3 surfaces and their higher dimensional analogues, the complex symplectic projective manifolds) the period map takes values in a locally symmetric variety of … For certain complex projective manifolds (such as K3 surfaces and their higher dimensional analogues, the complex symplectic projective manifolds) the period map takes values in a locally symmetric variety of type IV. It is often an open embedding and in such cases it has been observed that the image is the complement of a locally symmetric divisor. We explain that phenomenon and get our hands on the complementary divisor in terms of geometric data.

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Geometry of the Wiman–Edge pencil, I: algebro-geometric aspects

2018-04-02

Igor Dolgachev , Benson Farb , Eduard Looijenga

In 1981 William L. Edge discovered and studied a pencil $$\mathscr {C}$$ of highly symmetric genus 6 projective curves with remarkable properties. Edge's work was based on an 1895 paper … In 1981 William L. Edge discovered and studied a pencil $$\mathscr {C}$$ of highly symmetric genus 6 projective curves with remarkable properties. Edge's work was based on an 1895 paper of Anders Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel Geometry of the Wiman–Edge pencil, II: hyperbolic, conformal and modular aspects (in preparation), we consider $$\mathscr {C}$$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.

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Cohomology and intersection homology of algebraic varieties

1997-03-04

Eduard Looijenga

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The smoothing components of a triangle singularity. I

1983-01-01

Eduard Looijenga

Affine Artin groups and the fundamental groups of some moduli spaces

1998-01-01

Eduard Looijenga

We define for every affine Coxeter graph a certain factor group of the associated Artin group and prove that some of these groups appear as orbifold fundamental groups of moduli … We define for every affine Coxeter graph a certain factor group of the associated Artin group and prove that some of these groups appear as orbifold fundamental groups of moduli spaces. Examples are the moduli space of nonsingular cubic algebraic surfaces and the universal nonhyperelliptic smooth genus three curve. We use this to obtain a simple presentation of the mapping class group of a compact genus three topological surface with connected boundary. This leads to a modification of Wajnryb's presentation of the mapping class groups in the higher genus case that can be understood in algebro-geometric terms.

Stable cohomology of the mapping class group with symplectic coefficients

1996-01-01

Eduard Looijenga

Mapping Class Groups and Moduli Spaces of Curves

1996-01-01

Richard Hain , Eduard Looijenga

Conformal blocks revisited

2005-01-01

Eduard Looijenga

We give a simple coordinate free description of the WZW connection and derive its main properties. We give a simple coordinate free description of the WZW connection and derive its main properties.

Del Pezzo Moduli via Root Systems

2009-01-01

Elisabetta Colombo , Bert van Geemen , Eduard Looijenga

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A perfect stratification of $${\mathcal M_g}$$ for g ≤ 5

2008-07-11

Claudio Fontanari , Eduard Looijenga

Arithmetic representations of mapping class groups

2025-05-16

Eduard Looijenga

Let $S$ be a closed oriented surface and $G$ a finite group of orientation preserving automorphisms of $S$ whose orbit space has genus at least $2$. There is a natural … Let $S$ be a closed oriented surface and $G$ a finite group of orientation preserving automorphisms of $S$ whose orbit space has genus at least $2$. There is a natural group homomorphism from the $G$-centralizer in $Diff^+(S)$ to the $G$-centralizer in $Sp(H_1(S))$. We give a sufficient condition for its image to be a subgroup of finite index.

On the motivic description of truncated fundamental group rings

2025-05-03

Eduard Looijenga

A topological theorem due to Beilinson (and which appears in a paper by Deligne–Goncharov) states that for a path connected pointed space with a reasonable topology its fundamental group ring … A topological theorem due to Beilinson (and which appears in a paper by Deligne–Goncharov) states that for a path connected pointed space with a reasonable topology its fundamental group ring with field coefficients and truncated by a power of the augmentation ideal, is naturally isomorphic to a relative cohomology group that is functorial in terms of that space. We generalize this to integral coefficients, give a more directly proof and also express the maps that define the Hopf algebra structure on these truncations in these terms.

Hodge decomposition of L_2-cohomology and intersection cohomology of a Shimura variety

2025-01-31

Eduard Looijenga

Classical Hodge theory endows the square integrable cohomology of a Shimura variety X with values in a locally homogeneous polarized variation of Hodge structure E with a natural Hodge decomposition. … Classical Hodge theory endows the square integrable cohomology of a Shimura variety X with values in a locally homogeneous polarized variation of Hodge structure E with a natural Hodge decomposition. The theory of Morihiko Saito does the same for the E-valued intersection cohomology of the Baily-Borel compactification of X. Existing proofs of the Zucker conjecture identify these cohomology groups, but do not claim this for their Hodge decompositions. We show that one of the proofs yields that as well.

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The Nielsen realization problem for K3 surfaces

2024-06-01

Benson Farb , Eduard Looijenga

The smooth (resp. metric and complex) Nielsen Realization Problem for K3 surfaces $M$ asks: when can a finite group $G$ of mapping classes of $M$ be realized by a finite … The smooth (resp. metric and complex) Nielsen Realization Problem for K3 surfaces $M$ asks: when can a finite group $G$ of mapping classes of $M$ be realized by a finite group of diffeomorphisms (resp. isometries of a Ricci-flat metric, or automorphisms of a complex structure)? We solve the metric and complex versions of Nielsen Realization, and we solve the smooth version for involutions. Unlike the case of 2-manifolds, some $G$ are realizable and some are not, and the answer depends on the category of structure preserved. In particular, Dehn twists are not realizable by finite order diffeomorphisms. We introduce a computable invariant $L_G$ that determines in many cases whether $G$ is realizable or not, and apply this invariant to construct an $S_4$ action by isometries of some Ricci-flat metric on $M$ that preserves no complex structure. We also show that the subgroups of $\mathrm{Diff}(M)$ of a given prime order $p$ which fix pointwise some positive-definite 3-plane in $H^2(M; \mathbb{R})$ and preserve some complex structure on $M$ form a single conjugacy class in $\mathrm{Diff}(M)$ (it is known that then $p \in \{2, 3, 5, 7\}$).

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Retraction Note: Curves with prescribed symmetry and associated representations of mapping class groups

2024-04-09

Marco Boggi , Eduard Looijenga

Specifically, the proofs of Lemmas 1.6 and 1.8 are both incomplete. Specifically, the proofs of Lemmas 1.6 and 1.8 are both incomplete.

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The smooth Mordell-Weil group and mapping class groups of elliptic surfaces

2024-03-23

Benson Farb , Eduard Looijenga

This is a paper in smooth $4$-manifold topology, inspired by the Mordell-Weil Theorem in number theory. More precisely, we prove a smooth version of the Mordell-Weil Theorem and apply it … This is a paper in smooth $4$-manifold topology, inspired by the Mordell-Weil Theorem in number theory. More precisely, we prove a smooth version of the Mordell-Weil Theorem and apply it to the `unipotent radical' case of a Thurston-type classification of mapping classes of simply-connected $4$-manifolds $M_d$ that admit the structure of an elliptic complex surface of arithmetic genus $d\geq 1$. Applications include Nielsen realization theorems for $M_d$. By combining this with known results, we obtain the following remarkable consequence: if the singular fibers of such an elliptic fibration are of the simplest (i.e.\ nodal) type, then the fibered structure is unique up topological isotopy. In particular, any diffeomorphism of $M_d,d\geq 3$ is topologically isotopic to a diffeomorphism taking fibers to fibers.

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On the motivic description of truncated fundamental group rings

2024-03-06

Eduard Looijenga

A topological theorem that appears in a paper by Deligne-Goncharov (and which they attribute to Beilinson) states the following. Let $(X,*)$ be a path connected pointed space with a reasonable … A topological theorem that appears in a paper by Deligne-Goncharov (and which they attribute to Beilinson) states the following. Let $(X,*)$ be a path connected pointed space with a reasonable topology and denote by $I$ the augmentation ideal of its fundamental group ring. Then for every field F and positive integer n, the space of F-valued linear forms on $ I/I^{n+1}$ is naturally isomorphic to $H^n(X^n,X(n,*); F)$, where $X(n,*)$ is an explicitly defined subspace of $X^n$. We here construct a simple isomorphism between $I/I^{n+1}$ and $H_n(X^n,X(n,*); \mathbf{Z})$ and express the maps that define the Hopf algebra structure on the $I$-adic completion of the fundamental group ring of $(X,*)$ in these terms.

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A BALL QUOTIENT PARAMETRIZING TRIGONAL GENUS 4 CURVES

2023-09-21

Eduard Looijenga

Abstract We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that … Abstract We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $\frac {1}{2}(3^{10}-1)$ cover of the nine-dimensional Deligne–Mostow ball quotient such that the natural divisors that live on that moduli space become totally geodesic (their normalizations are eight-dimensional ball quotients). This isomorphism differs from the one considered by S. Kondō, and its construction is perhaps more elementary, as it does not involve K3 surfaces and their Torelli theorem: the Deligne–Mostow ball quotient parametrizes certain cyclic covers of degree 6 of a projective line and we show how a level structure on such a cover produces a degree 3 cover of that line with the same discriminant, yielding a genus 4 curve endowed with a $g^1_3$ .

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Monodromy and period map of the Winger pencil

2023-01-09

Eduard Looijenga , Yunpeng Zi

Abstract The sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by … Abstract The sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a sum of six lines and three times a conic). This pencil was first considered in a note by R. M. Winger in 1925 and is nowadays named after him. The second author recently gave this a modern treatment and proved among other things that it contains essentially every smooth genus ten curve with icosahedral symmetry. We here show that the Jacobian of such a curve contains the tensor product of an elliptic curve with a certain integral representation of the icosahedral group. We find that the elliptic curve comes with a distinguished point of order 3, which proves that the monodromy on this part of the homology is the full congruence subgroup and subsequently identify the base of the pencil with the associated modular curve. We also observe that the Winger pencil “accounts” for the deformation of the Jacobian of Bring's curve as a principal abelian fourfold with an action of the icosahedral group.

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Moduli spaces and period mappings of genus one fibered K3 surfaces

2023-01-01

Benson Farb , Eduard Looijenga

In this paper we construct various moduli spaces of K3 surfaces $M$ equipped with a surjective holomorphic map $\pi:M\to\Pb^1$ with generic fiber a complex torus (e.g., an elliptic fibration). Examples … In this paper we construct various moduli spaces of K3 surfaces $M$ equipped with a surjective holomorphic map $\pi:M\to\Pb^1$ with generic fiber a complex torus (e.g., an elliptic fibration). Examples include moduli spaces of such maps with primitive fibers; with reduced, irreducible fibers; equipped with a section; etc. Such spaces are closely related to the moduli space of Ricci-flat metrics on $M$. We construct period mappings relating these moduli spaces to locally symmetric spaces, and use these to compute their (orbifold) fundamental groups. These results lie in contrast to, and exhibit different behavior than, the well-studied case of moduli spaces of polarized K3 surfaces, and are more useful for applications to the mapping class group $\Mod(M)$. Indeed, we apply our results on moduli space to give two applications to the smooth mapping class group of $M$.

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Cubic threefolds moduli and the Monster group

2023-01-01

Eduard Looijenga

Allcock constructed a 13-dimensional complex ball quotient of which he conjectured that it admits a natural covering with covering group isomorphic to the Bimonster. This ball quotient contains the moduli … Allcock constructed a 13-dimensional complex ball quotient of which he conjectured that it admits a natural covering with covering group isomorphic to the Bimonster. This ball quotient contains the moduli space of cubic threefolds as an open dense subset of a 10-dimensional complex subball quotient. We prove that this subball quotient has a neighborhood in the Allcock ball quotient over which the conjectured cover exists.

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A ball quotient parametrizing trigonal genus 4 curves

2022-01-01

Eduard Looijenga

We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it … We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $\frac{1}{2}(3^{10}-1)$ cover of the 9-dimensional Deligne-Mostow ball quotient such that the natural divisors that live on that moduli space become totally geodesic (their normalizations are 8-dimensional ball quotients). This isomorphism differs from the one considered by S. Kond\=o and its construction is perhaps more elementary, as it does not involve K3 surfaces and their Torelli theorem: the Deligne-Mostow ball quotient parametrizes certain cyclic covers of degree 6 of a projective line and we show how a level structure on such a cover produces a degree 3 cover of that line with the same discriminant, yielding a genus 4 curve endowed with a $g^1_3$.

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Arithmeticity of the monodromy of the Wiman–Edge pencil

2021-12-08

Benson Farb , Eduard Looijenga

The Wiman–Edge pencil is the universal family of projective, genus 6, complex-algebraic curves endowed with a faithful action of the icosahedral group. The goal of this paper is to prove … The Wiman–Edge pencil is the universal family of projective, genus 6, complex-algebraic curves endowed with a faithful action of the icosahedral group. The goal of this paper is to prove that its monodromy group is commensurable with a Hilbert modular group; in particular is arithmetic. We then give a modular interpretation of this, as well as a uniformization of its base.

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Conformal blocks and the cohomology of configuration spaces of curves

2021-12-06

Eduard Looijenga

We realize any space of conformal blocks attached to a punctured curve inside the cohomology of a configuration space of that curve and compare the WZW connection with the Gauss-Manin … We realize any space of conformal blocks attached to a punctured curve inside the cohomology of a configuration space of that curve and compare the WZW connection with the Gauss-Manin connection.

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Characteristic forms for holomorphic families of local systems

2021-07-22

Indranil Biswas , Eduard Looijenga

Let $f:\mathcal{X}\to S$ be a proper holomorphic submersion of complex manifolds and $G$ a complex reductive linear algebraic group with Lie algebra $\mathfrak{g}$. Assume also given a holomorphic principal $G$-bundle … Let $f:\mathcal{X}\to S$ be a proper holomorphic submersion of complex manifolds and $G$ a complex reductive linear algebraic group with Lie algebra $\mathfrak{g}$. Assume also given a holomorphic principal $G$-bundle $\mathcal{P}$ over $\mathcal{X}$ which is endowed with a holomorphic connection $\nabla$ relative to $f$ that is flat (this to be thought of as a holomorphic family of compact complex manifolds endowed with a holomorphic principal $G$-bundle with flat connection). We show that a refinement of the Chern-Weil homomorphism yields a graded algebra homomorphism $\mathbb{C}[\mathfrak{g}]^G\to \bigoplus_{n\ge 0} H^0(S,\,\Omega^n_{S,cl}\otimes R^nf_*\mathbb{C})$, where $\Omega^n_{S,cl}$ stands for the sheaf of closed holomorphic $n$-forms on $S$. If the fibers of $f$ are compact Riemann surfaces and we take as our invariant the Killing form, then we recover Goldman's closed holomorphic $2$-form on the base $S$.

Curves with prescribed symmetry and associated representations of mapping class groups

2021-07-21

Marco Boggi , Eduard Looijenga

Abstract Let C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We … Abstract Let C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We prove that if the G -curve C is very general for these properties, then the natural map from the group algebra $${{\mathbb {Q}}}G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> to the algebra of $${{\mathbb {Q}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> -endomorphisms of its Jacobian is an isomorphism. We use this to obtain (topological) properties regarding certain virtual linear representations of a mapping class group. For example, we show that the connected component of the Zariski closure of such a representation often acts $${{\mathbb {Q}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> -irreducibly in a G -isogeny space of $$H^1(C; {{\mathbb {Q}}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>;</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and with image a $${{\mathbb {Q}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> -almost simple group.

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Lifting Chern classes by means of Ekedahl–Oort strata

2021-05-13

Gerard van der Geer , Eduard Looijenga

The moduli space of principally polarized abelian varieties $A_g$ of genus g is defined over the integers and admits a minimal compactification $A_g^*$, also defined over the integers. The Hodge … The moduli space of principally polarized abelian varieties $A_g$ of genus g is defined over the integers and admits a minimal compactification $A_g^*$, also defined over the integers. The Hodge bundle over $A_g$ has its Chern classes in the Chow ring of $A_g$ with rational coefficients. We show that over the prime field $F_p$, these Chern classes naturally lift to $A_g^*$ and do so in the best possible way: despite the highly singular nature of $A_g^*$ they are represented by algebraic cycles on $A_g^*\otimes F_p$ which define elements in its bivariant Chow ring. This is in contrast to the situation in the analytic topology, where these Chern classes have canonical lifts to the complex cohomology of the minimal compactification as Goresky-Pardon classes, which are known to define nontrivial Tate extensions inside the mixed Hodge structure on this cohomology.

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Torelli group action on the configuration space of a surface

2021-04-08

Eduard Looijenga

We show that the Torelli group of a closed surface of genus [Formula: see text] acts nontrivially on the rational cohomology of the space of [Formula: see text]-element subsets of … We show that the Torelli group of a closed surface of genus [Formula: see text] acts nontrivially on the rational cohomology of the space of [Formula: see text]-element subsets of that surface. This implies that for a Riemann surface of genus [Formula: see text], the mixed Hodge structure on the space of its positive, reduced divisors of degree [Formula: see text] does in general not split over [Formula: see text].

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The homotopy type of the Baily–Borel and allied compactifications

2021-01-01

Jiaming Chen , Eduard Looijenga

A number of compactifications familiar in complex-analytic geometry, in particular the Baily-Borel compactification and its toroidal variants, as well as the Deligne-Mumford compactifications, can be covered by open subsets whose … A number of compactifications familiar in complex-analytic geometry, in particular the Baily-Borel compactification and its toroidal variants, as well as the Deligne-Mumford compactifications, can be covered by open subsets whose nonempty intersections are classified by their fundamental groups.We exploit this fact to define a 'stacky homotopy type' for these spaces as the homotopy type of a small category.We thus generalize an old result of Charney-Lee on the Baily-Borel compactification of A g and recover (and rephrase) a more recent one of Ebert-Giansiracusa on the Deligne-Mumford compactifications.We also describe an extension of the period map for Riemann surfaces (going from the Deligne-Mumford compactification to the Baily-Borel compactification of the moduli space of principally polarized varieties) in these terms.

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Monodromy and period map of the Winger Pencil

2021-01-01

Eduard Looijenga , Yunpeng Zi

The sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a … The sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a sum of six lines and a three times a conic). This pencil was first considered in a note by R.~M.~Winger in 1925 and is nowadays named after him. The second author recently gave this a modern treatment and proved among other things that it contains essentially every smooth genus ten curve with icosahedral symmetry. We here show that the Jacobian of such a curve contains the tensor product of an elliptic curve with a certain integral representation of the icosahedral group. We find that the elliptic curve comes with a distinguished point of order $3$, prove that the monodromy on this part of the homology is the full congruence subgroup $\Gamma_1(3)\subset \SL_2(\Zds)$ and subsequently identify the base of the pencil with the associated modular curve. We also observe that the Winger pencil `accounts' for the deformation of the Jacobian of Bring's curve as a principal abelian fourfold with an action of the icosahedral group.

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Conformal blocks and the cohomology of configuration spaces of curves

2021-01-01

Eduard Looijenga

Characteristic forms for holomorphic families of local systems

2021-01-01

Indranil Biswas , Eduard Looijenga

The Nielsen realization problem for K3 surfaces

2021-01-01

Benson Farb , Eduard Looijenga

The smooth (resp. metric and complex) Nielsen Realization Problem for K3 surfaces $M$ asks: when can a finite group $G$ of mapping classes of $M$ be realized by a finite … The smooth (resp. metric and complex) Nielsen Realization Problem for K3 surfaces $M$ asks: when can a finite group $G$ of mapping classes of $M$ be realized by a finite group of diffeomorphisms (resp. isometries of a Ricci-flat metric, or automorphisms of a complex structure)? We solve the metric and complex versions of Nielsen Realization, and we solve the smooth version almost completely for involutions. Unlike the case of $2$-manifolds, some $G$ are realizable and some are not, and the answer depends on the category of structure preserved. In particular, Dehn twists are not realizable by finite order diffeomorphisms. We introduce a computable invariant $L_G$ that determines in many cases whether $G$ is realizable or not, and apply this invariant to construct an $S_4$ action by isometries of some Ricci-flat metric on $M$ that preserves no complex structure. We also show that the subgroups of ${\rm Diff}(M)$ of a given prime order $p$ which fix pointwise some positive-definite $3$-plane in $H_2(M;\mathbb{R})$ and preserve some complex structure on $M$ form a single conjugacy class in ${\rm Diff}(M)$ (it is known that then $p\in \{2,3,5,7\}$).

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Arithmetic representations of mapping class groups

2021-01-01

Eduard Looijenga

Torelli group action on the configuration space of a surface.

2020-08-24

Eduard Looijenga

We show that the Torelli group of a closed surface of genus $\ge 3$ acts nontrivially on the rational cohomology of its space of $3$-element subsets. We show that the Torelli group of a closed surface of genus $\ge 3$ acts nontrivially on the rational cohomology of its space of $3$-element subsets.

Teichmüller spaces and Torelli theorems for hyperkähler manifolds

2020-08-20

Eduard Looijenga

Abstract Kreck and Yang Su recently gave counterexamples to a version of the Torelli theorem for hyperkählerian manifolds as stated by Verbitsky. We extract the correct statement and give a … Abstract Kreck and Yang Su recently gave counterexamples to a version of the Torelli theorem for hyperkählerian manifolds as stated by Verbitsky. We extract the correct statement and give a short proof of it. We also revisit a few of its consequences, some of which are given new (shorter) proofs, and ask some questions.

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Geometry of the Wiman–Edge pencil and the Wiman curve

2020-03-11

Benson Farb , Eduard Looijenga

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Torelli group action on the configuration space of a surface

2020-01-01

Eduard Looijenga

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Geometry of the Wiman-Edge pencil and the Wiman curve

2019-12-27

Benson Farb , Eduard Looijenga

The {\em Wiman-Edge pencil} is the universal family $C_t, t\in\mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The curve $C_0$, discovered … The {\em Wiman-Edge pencil} is the universal family $C_t, t\in\mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The curve $C_0$, discovered by Wiman in 1895 \cite{Wiman} and called the {\em Wiman curve}, is the unique smooth, genus $6$ curve admitting a faithful action of the symmetric group $\Sf_5$. In this paper we give an explicit uniformization of $\mathcal B$ as a non-congruence quotient $\Gamma\backslash \Hf$ of the hyperbolic plane $\Hf$, where $\Gamma<\PSL_2(\Z)$ is a subgroup of index $18$. We also give modular interpretations for various aspects of this uniformization, for example for the degenerations of $C_t$ into $10$ lines (resp. $5$ conics) whose intersection graph is the Petersen graph (resp. $K_5$). In the second half of this paper we give an explicit arithmetic uniformization of the Wiman curve $C_0$ itself as the quotient $\Lambda\backslash \Hf$, where $\Lambda$ is a principal level $5$ subgroup of a certain unit spinor norm group of Mobius transformations. We then prove that $C_0$ is a certain moduli space of Hodge structures, endowing it with the structure of a Shimura curve of indefinite quaternionic type.

Teichm\"uller spaces and Torelli theorems for hyperk\"ahler manifolds

2019-12-05

Eduard Looijenga

Kreck and Yang Su recently gave counterexamples to a version of the Torelli theorem for hyperkahlerian manifolds as stated by Verbitsky. The initial purpose of this document (which was prepared … Kreck and Yang Su recently gave counterexamples to a version of the Torelli theorem for hyperkahlerian manifolds as stated by Verbitsky. The initial purpose of this document (which was prepared for a seminar talk) was to extract the correct statement and to give a short proof of it. We also revisit a few of its consequences, some of which are given new (shorter) proofs.

Arithmeticity of the monodromy of the Wiman-Edge pencil

2019-11-04

Benson Farb , Eduard Looijenga

The {\em Wiman-Edge pencil} is the universal family $\Cs/\mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The goal of this paper … The {\em Wiman-Edge pencil} is the universal family $\Cs/\mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The goal of this paper is to prove that the monodromy of $\Cs/\mathcal B$ is commensurable with a Hilbert modular group; in particular is arithmetic. We then give a modular interpretation of this, as well as a uniformization of $\mathcal B$.

Fermat varieties and the periods of some hypersurfaces

2019-03-07

Eduard Looijenga

The variety of all smooth hypersurfaces of given degree and dimension has the Fermat hypersurface as a natural base point. In order to study …  The variety of all smooth hypersurfaces of given degree and dimension has the Fermat hypersurface as a natural base point. In order to study the period map for such varieties, we first determine the integral polarized Hodge structure of the primitive cohomology of a Fermat hypersurface (as a module over the automorphism group of the hypersurface). We then focus on the degree 3 case and show that the period map for cubic fourfolds as analyzed by R. Laza and the author gives complete information about the period map for cubic hypersurfaces of lower dimension. In particular, we thus recover the results of Allcock–Carlson–Toledo on the cubic surface case.

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The KZ system via polydifferentials

2019-03-07

Eduard Looijenga

We show that the KZ system has a topological interpretation in the sense that it may be understood as a variation of complex mixed …  We show that the KZ system has a topological interpretation in the sense that it may be understood as a variation of complex mixed Hodge structure whose successive pure weight quotients are polarized. This in a sense completes and elucidates work of Schechtman–Varchenko done in the early 1990's. A central ingredient is a new realization of the irreducible highest weight representations of a Lie algebra of Kac–Moody type, namely on an algebra of rational polydifferentials on a countable product of Riemann spheres. We also obtain the kind of properties that in the $\mathfrak{sl} (2)$ case are due to Ramadas and are then known to imply the unitarity of the WZW system in genus zero.

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Moduli spaces and locally symmetric varieties

2019-03-07

Eduard Looijenga

This survey paper is about moduli spaces in algebraic geometry for which a period map gives that space the structure of a (possibly incomplete) …  This survey paper is about moduli spaces in algebraic geometry for which a period map gives that space the structure of a (possibly incomplete) locally symmetric variety and about their natural compactifications. We outline the Baily-Borel compactification for such varieties, and show that it usually differs from the compactifications furnished by the standard techniques in algebraic geometry. It turns out however, that a reconciliation is possible by means of a generalization of the Baily-Borel construction for the class of incomplete locally symmetric varieties that occur here. The emphasis is here on moduli spaces of varieties other than that of polarized abelian varieties.

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The Moduli Space of Rational Elliptic Surfaces

2019-01-26

Gert Heckman , Eduard Looijenga

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Teichmüller spaces and Torelli theorems for hyperkähler manifolds

2019-01-01

Eduard Looijenga

Kreck and Yang Su recently gave counterexamples to a version of the Torelli theorem for hyperk\"ahlerian manifolds as stated by Verbitsky. The initial purpose of this document (which was prepared … Kreck and Yang Su recently gave counterexamples to a version of the Torelli theorem for hyperk\"ahlerian manifolds as stated by Verbitsky. The initial purpose of this document (which was prepared for a seminar talk) was to extract the correct statement and to give a short proof of it. We also revisit a few of its consequences, some of which are given new (shorter) proofs.

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Geometry of the Wiman-Edge pencil and the Wiman curve

2019-01-01

Benson Farb , Eduard Looijenga

The {\em Wiman-Edge pencil} is the universal family $C_t, t\in\mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The curve $C_0$, discovered … The {\em Wiman-Edge pencil} is the universal family $C_t, t\in\mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The curve $C_0$, discovered by Wiman in 1895 \cite{Wiman} and called the {\em Wiman curve}, is the unique smooth, genus $6$ curve admitting a faithful action of the symmetric group $\Sf_5$. In this paper we give an explicit uniformization of $\mathcal B$ as a non-congruence quotient $\Gamma\backslash \Hf$ of the hyperbolic plane $\Hf$, where $\Gamma<\PSL_2(\Z)$ is a subgroup of index $18$. We also give modular interpretations for various aspects of this uniformization, for example for the degenerations of $C_t$ into $10$ lines (resp.\ $5$ conics) whose intersection graph is the Petersen graph (resp.\ $K_5$). In the second half of this paper we give an explicit arithmetic uniformization of the Wiman curve $C_0$ itself as the quotient $\Lambda\backslash \Hf$, where $\Lambda$ is a principal level $5$ subgroup of a certain "unit spinor norm" group of M\"{o}bius transformations. We then prove that $C_0$ is a certain moduli space of Hodge structures, endowing it with the structure of a Shimura curve of indefinite quaternionic type.

Arithmeticity of the monodromy of the Wiman-Edge pencil

2019-01-01

Benson Farb , Eduard Looijenga

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Curves with prescribed symmetry and associated representations of mapping class groups.

2018-11-24

Marco Boggi , Eduard Looijenga

Let C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We prove … Let C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We prove that if the G-curve C is very general for these properties, then the natural map from the group algebra QG to the algebra of Q-endomorphisms of its Jacobian is an isomorphism. We use this to obtain (topological) properties regarding certain virtual linear representations of a mapping class group. For example, we show that the connected component of the Zariski closure of such a representation acts Q-irreducibly in a G-isogeny space of H^1(C; Q)and with image often a Q-almost simple group.

Geometry of the Wiman–Edge pencil, I: algebro-geometric aspects

2018-04-02

Igor Dolgachev , Benson Farb , Eduard Looijenga

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Deforming a Canonical Curve Inside a Quadric

2018-02-02

Marco Boggi , Eduard Looijenga

Journal Article Deforming a Canonical Curve Inside a Quadric Get access Marco Boggi, Marco Boggi Departamento de Matemática, UFMG, Belo Horizonte, Brasil Correspondence to be sent to: email: [email protected] Search … Journal Article Deforming a Canonical Curve Inside a Quadric Get access Marco Boggi, Marco Boggi Departamento de Matemática, UFMG, Belo Horizonte, Brasil Correspondence to be sent to: email: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Eduard Looijenga Eduard Looijenga Yau Mathematical Sciences Center, Tsinghua University Beijing, China, and Mathematisch Instituut, Universiteit Utrecht, Netherlands Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2020, Issue 2, January 2020, Pages 367–377, https://doi.org/10.1093/imrn/rny027 Published: 08 March 2018 Article history Received: 15 March 2017 Revision requested: 30 January 2018 Accepted: 31 January 2018 Published: 08 March 2018

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Curves with prescribed symmetry and associated representations of mapping class groups

2018-01-01

Marco Boggi , Eduard Looijenga

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The stable cohomology of the Satake compactification of 𝒜g

2017-05-19

Jiaming Chen , Eduard Looijenga

Charney and Lee have shown that the rational cohomology of the Satake-Baily-Borel compactification the moduli space of principally polarized abelian varieties of dimension g stabilizes as g grows and they … Charney and Lee have shown that the rational cohomology of the Satake-Baily-Borel compactification the moduli space of principally polarized abelian varieties of dimension g stabilizes as g grows and they computed this stable cohomology as a Hopf algebra. We give a relatively simple algebro-geometric proof of their theorem that also takes into account the mixed Hodge structure that is present here. We find the latter to be impure.

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Goresky–Pardon lifts of Chern classes and associated Tate extensions

2017-05-03

Eduard Looijenga

Let X be an irreducible complex variety, S a stratification of X and F a holomorphic vector bundle on the open statum. We give geometric conditions on S and F … Let X be an irreducible complex variety, S a stratification of X and F a holomorphic vector bundle on the open statum. We give geometric conditions on S and F that produce a natural extension of the k-th Chern class F as a class in the complex cohomology of X of Hodge level at least k. When X is the Baily-Borel compactification of a locally symmetric variety with its stratification by boundary components, and F an automorphic bundle on its interior, then this recovers and refines a theorem of Goresky-Pardon. In passing we define a class of simplicial resolutions of the Baily-Borel compactification that can be used to define its mixed Hodge structure. Finally, we use the Beilinson regulator for the rationals to show that when X is the Satake (=Baily-Borel) compactification of A_g and F the Hodge bundle (with g large compared to k), the Goresky-Pardon k-th Chern class extension has nonzero imaginary part and gives rise to a basic Tate extension. This also shows that the lifts of Chern classes constructed by Goresky and Pardon need not be real and thus answers (negatively) a question asked by these authors.

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Geometry of the Wiman Pencil, I: Algebro-Geometric Aspects

2017-01-01

Igor Dolgachev , Benson Farb , Eduard Looijenga

In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's work was based on an 1895 paper of A. … In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's work was based on an 1895 paper of A. Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel [FL], we consider $\mathcal{C}$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.

Deforming a canonical curve inside a quadric

2017-01-01

Marco Boggi , Eduard Looijenga

Let $C\subset{\mathbb P}^{g-1}$ be a canonically embedded nonsingular nonhyperelliptic curve of genus $g$ and let $X\subset{\mathbb P}^{g-1}$ be a quadric containing $C$. Our main result states among other things that … Let $C\subset{\mathbb P}^{g-1}$ be a canonically embedded nonsingular nonhyperelliptic curve of genus $g$ and let $X\subset{\mathbb P}^{g-1}$ be a quadric containing $C$. Our main result states among other things that the Hilbert scheme of $X$ is at $[C\subset X]$ a local complete intersection of dimension $g^2-1$, and is smooth when $X$ is. It also includes the assertion that the minimal obstruction space for this deformation problem is in fact the full associated $\operatorname{Ext}^1$-group and that in particular the deformations of $C$ in $X$ are obstructed in case $C$ meets the singular locus of $X$. As we will show in a forthcoming paper, this has applications of a topological nature.

Goresky-Pardon extensions of Chern classes and associated Tate extensions

2015-10-14

Eduard Looijenga

The homotopy type of the Baily-Borel and allied compactifications

2015-01-01

Jiaming Chen , Eduard Looijenga

A number of compactifications familiar in complex-analytic geometry, in particular, the Baily-Borel compactification and its toroidal variants, as well as the Deligne-Mumford compactifications, can be covered by open subsets whose … A number of compactifications familiar in complex-analytic geometry, in particular, the Baily-Borel compactification and its toroidal variants, as well as the Deligne-Mumford compactifications, can be covered by open subsets whose nonempty intersections are Eilenberg-MacLane spaces. We exploit this fact to describe the (rational) homotopy type of these spaces and the natural maps between them in terms of the simplicial sets attached to certain categories. We thus generalize an old result of Charney-Lee on the Baily-Borel compactification of A_g and recover (and rephrase) a more recent one of Ebert-Giansiracusa on the Deligne-Mumford compactifications. We also describe an extension of the period map for Riemann surfaces (going from the Deligne-Mumford compactification to the Baily-Borel compactification of the moduli space of principally polarized varieties) in these terms.

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Discrete automorphism groups of convex cones of finite type

2014-09-10

Eduard Looijenga

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Moduli spaces and locally symmetric varieties

2014-04-15

Eduard Looijenga

This is a survey paper on moduli spaces that have a natural structure of a (possibly incomplete) locally symmetric variety. We outline the Baily-Borel compactification for such varieties and compare … This is a survey paper on moduli spaces that have a natural structure of a (possibly incomplete) locally symmetric variety. We outline the Baily-Borel compactification for such varieties and compare it with the compactifications furnished by techniques in algebraic geometry. These differ in general, but we show that a reconciliation is possible by means of a generalization of the Baily-Borel technique for a class of incomplete locally symmetric varieties. The emphasis is here on moduli spaces of varieties other than that of polarized abelian varieties.

Coauthor	Papers Together
Benson Farb	12
Gert Heckman	11
Marco Boggi	6
Wim Couwenberg	5
Rogier Swierstra	4
Jiaming Chen	3
Carel Faber	3
Elisabetta Colombo	2
Richard Hain	2
Dirk Siersma	2
Ezra Getzler	2
Gerard van der Geer	2
Floris Takens	2
Claudio Fontanari	2
Yunpeng Zi	2
Gabriele Mondello	2
Bert van Geemen	2
Igor Dolgachev	2
Indranil Biswas	2
Gert–Martin Greuel	1
Jeremy T. Tyson	1
Alex Boer	1
Shmuel Weinberger	1
A Oold	1
O Siersma	1
Andrew A. du Plessis	1
Valery Alexeev	1
Artur Avila	1
Jang-Mei Wu	1
Otmar Venjakob	1
John Coates	1
Carlos Gustavo Moreira	1
Wilberd van der Kallen	1
Valery A. Lunts	1
太香子深谷	1
C. G. Gibson	1
Robert Kaufman	1
Beno Eckmann	1
Nigel Higson	1
J. H. M. Steenbrink	1
Elham Izadi	1
Erik Guentner	1
Jonathan Wahl	1
R. Sujatha	1
Janós Kollár	1
Klaus Wirthmüller	1
Compact moduli	1
Chris Peters	1
Angela Gibney	1
和也加藤	1

Monodromy of hypergeometric functions and non-lattice integral monodromy

1986-12-01

Pierre Deligne , G. D. Mostow

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Generalized picard lattices arising from half-integral conditions

1986-12-01

G. D. Mostow

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A complex hyperbolic structure for the moduli space of curves of genus three

2000-08-11

Shigeyuki Kondō

Article A complex hyperbolic structure for the moduli space of curves of genus three was published on August 11, 2000 in the journal Journal für die reine und angewandte Mathematik … Article A complex hyperbolic structure for the moduli space of curves of genus three was published on August 11, 2000 in the journal Journal für die reine und angewandte Mathematik (volume 2000, issue 525).

The complex hyperbolic geometry of the moduli space of cubic surfaces

2002-01-01

Daniel Allcock , James A. Carlson , Domingo Toledo

We prove that the moduli space of semistable cubic surfaces over the complex numbers is biholomorphic to the Satake compactification of the quotient of the four-ball by the projective unitary … We prove that the moduli space of semistable cubic surfaces over the complex numbers is biholomorphic to the Satake compactification of the quotient of the four-ball by the projective unitary group of the standard Hermitian form of signature $(4,1)$ with coefficients in the ring of integers of $\mathbb {Q}(\sqrt {-3})$. We also explain the precise relation between the orbifold structures on the moduli space of stable cubic surfaces and on the quotient of the ball.

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Groupes et algèbres de Lie

1971-01-01

Nicolás Bourbaki

Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes … Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algebre de Lie, neuvieme Livre du traite, est consacre aux concepts fondamentaux pour les algebres de Lie. Il comprend les paragraphes: - 1 Definition des algebres de Lie; 2 Algebre enveloppante d une algebre de Lie; 3 Representations; 4 Algebres de Lie nilpotentes; 5 Algebres de Lie resolubles; 6 Algebres de Lie semi-simples; 7 Le theoreme d Ado. Ce volume est une reimpression de l edition de 1971.

Stability of the Homology of the Mapping Class Groups of Orientable Surfaces

1985-03-01

John Harer

The mapping class group of F = Fgs r is F = rgs = wo(A) where A is the topological group of orientation preserving diffeomorphisms of F which are the … The mapping class group of F = Fgs r is F = rgs = wo(A) where A is the topological group of orientation preserving diffeomorphisms of F which are the identity on dF and fix the s punctures. When r = 0 and 2g + s > 3, an important feature of F is that it acts properly discontinuously on the Teichmiller space US; the quotient is the moduli space .#f of isometry classes of complete hyperbolic metrics of finite area on F. Since US is homeomorphic to Euclidean space and F is virtually torsion-free, we have

The moduli space of cubic threefolds

2002-11-18

Daniel Allcock

We describe the moduli space of cubic hypersurfaces in<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C upper P Superscript 4"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:msup><mml:mi>P</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:annotation encoding="application/x-tex">\mathbb {C}P^{4}</mml:annotation></mml:semantics></mml:math></inline-formula>in the sense of geometric invariant theory. That … We describe the moduli space of cubic hypersurfaces in<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C upper P Superscript 4"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:msup><mml:mi>P</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:annotation encoding="application/x-tex">\mathbb {C}P^{4}</mml:annotation></mml:semantics></mml:math></inline-formula>in the sense of geometric invariant theory. That is, we characterize the stable and semistable hypersurfaces in terms of their singularities, and determine the equivalence classes of semistable hypersurfaces under the equivalence relation of their orbit-closures meeting.

The virtual cohomological dimension of the mapping class group of an orientable surface

1986-02-01

John Harer

Compactifications defined by arrangements, I: The ball quotient case

2003-05-15

Eduard Looijenga

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The Leech lattice and complex hyperbolic reflections

2000-05-01

Daniel Allcock

Geometry of Algebraic Curves

1985-01-01

Enrico Arbarello , Maurizio Cornalba , P. A. Griffiths +1 more

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On the Periods of Certain Rational Integrals: I

1969-11-01

Philip A. Griffiths

In this section we want to re-prove the results of ?? 4 and 8 using sheaf cohomology. One reason for doing this is to clarify the discussion in those paragraphs … In this section we want to re-prove the results of ?? 4 and 8 using sheaf cohomology. One reason for doing this is to clarify the discussion in those paragraphs and, in particular, to show how Macaulay's theorem 4.11 is essentially equivalent to a suitable vanishing theorem for sheaf cohomology. We shall also give a proof of the de Rham algebraic theorem used in the proof of Theorem 5.3. However, our principal motivation is to be able to discuss rational integrals in case our hypersurface V c P, has rather simple singularities, and the localization technique of sheaf theory seems to be the best method for doing this (cf. ?? 15, 16 below). Let then V c P, be a non-singular hypersurface. We want to give a sheaf-theoretic version of the Hodge filtration of Hq(V, C) (cf. ? 8). For this we let Qv be the sheaf on V of holomorphic q-forms and & c QV the subsheaf of closed forms. The Poincare' lemma for holomorphic differentials gives an exact sequence

Artin groups and the fundamental groups of some moduli spaces

2007-10-28

Eduard Looijenga

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On the tautological ring of ℳ g

1995-12-01

Eduard Looijenga

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On the semi-universal deformation of a simple-elliptic hypersurface singularity

1977-01-01

Eduard Looijenga

Cohomology of the satake compactification

1983-01-01

Ruth Charney , Ronnie Lee

The Geometry of some special Arithmetic Quotients

1996-01-01

Bruce J. Hunt

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A bound on the dimensions of complete subvarieties of ℳg

1984-06-01

Steven Diaz

The irreducibility of the space of curves of given genus

1969-01-01

Pierre Deligne , D. Mumford

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Coble fourfold, 𝔖6-invariant quartic threefolds, and Wiman–Edge sextics

2020-03-15

Ivan Cheltsov , Alexander Kuznetsov , Constantin Shramov

We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic).We use them to show that all … We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic).We use them to show that all S 6 -invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman-Edge pencil.As an application, we check that S 6 -invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class groups as S 6 -representations.Lemma 2.6.The linear projection P(W 3 ) × P(W 3 ) P 5 with center in the span of the points P 1 , P 2 , P 3 induces an S 3,2 -equivariant commutative diagram Bl P 1 ,P 2 ,P 3 (P(W 3 ) × P(W 3 )) ρ ′

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Intersection homology, hypergeometric functions, and moduli spaces as ball quotients

2003-01-01

Brent Doran

The Chow Ring of the Moduli Space of Curves of Genus 5

1995-01-01

Elham Izadi

Let M g be the moduli spare of smooth curves of genus g over an algebraically closed field (of characteristic differetd, from 2 and 3) and let ${\bar M_g}$ be … Let M g be the moduli spare of smooth curves of genus g over an algebraically closed field (of characteristic differetd, from 2 and 3) and let ${\bar M_g}$ be its compactification by Deligne-Mumford stable curves.

Chow Rings of Moduli Spaces of Curves I: The Chow Ring of $\overbar{\mathcal{M}}_3$

1990-09-01

Carel Faber

Théorie de Hodge, II

1971-12-01

Pierre Deligne

Cellular Decompositions of Compactified Moduli Spaces of Pointed Curves

1995-01-01

Eduard Looijenga

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The moduli of Weierstrass fibrations over ?1

1981-09-01

Rick Miranda

Shapes of polyhedra and triangulations of the sphere

1998-11-30

William P. Thurston

The space of shapes of a polyhedron with given total angles less than 2π at each of its n vertices has a Kähler metric, locally isometric to complex hyperbolic space … The space of shapes of a polyhedron with given total angles less than 2π at each of its n vertices has a Kähler metric, locally isometric to complex hyperbolic space CH n-3 .The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point.The metric completion is a complex hyperbolic cone-manifold.In some interesting special cases, the metric completion is an orbifold.The concrete description of these spaces of shapes gives information about the combinatorial classification of triangulations of the sphere with no more than 6 triangles at a vertex.

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A Complete Moduli Space for K3 Surfaces of Degree 2

1980-11-01

Jayant Shah

A complex ball uniformization of the moduli space of cubic surfaces via periods of<i>K</i>3 surfaces

2005-11-01

Igor Dolgachev , Bert van Geemen , Shigeyuki Kondō

In this paper we show that the moduli space of nodal cubic surfaces is isomorphic to a quotient of a 4-dimensional complex ball by an arithmetic subgroup of the unitary … In this paper we show that the moduli space of nodal cubic surfaces is isomorphic to a quotient of a 4-dimensional complex ball by an arithmetic subgroup of the unitary group. This complex ball uniformization uses the periods of certain K 3 surfaces which are naturally associated to cubic surfaces. A similar uniformization is given for different covers of the moduli space corresponding to geometric markings of the Picard group or a choice of a line on the surface. We also give a detailed description of the boundary components corresponding to singular surfaces.

An abelian quotient of the mapping class groupI g

1980-10-01

Dennis L. Johnson

Compactifications defined by arrangements, II: Locally symmetric varieties of type IV

2003-09-15

Eduard Looijenga

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Arrangements, KZ Systems and Lie Algebra Homology

1999-06-03

Eduard Looijenga

The Knizhnik–Zamolodchikov equations were originally defined in terms of a local system associated to tuples of finite dimensional irreducible representations of SU2, but were soon afterwards generalized to a Kac–Moody … The Knizhnik–Zamolodchikov equations were originally defined in terms of a local system associated to tuples of finite dimensional irreducible representations of SU2, but were soon afterwards generalized to a Kac–Moody setting. The natural question that arises is whether these local systems admit a topological interpretation. A paper by Varchenko–Schechtman [8] comes close to answering this affirmatively and it is this article and related work that we intend to survey here.

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On the invariants of the ternary icosahedral group

1925-12-01

R. M. Winger

Discrete automorphism groups of convex cones of finite type

2014-09-10

Eduard Looijenga

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Character theory of finite groups

1999-07-01

I. M. Isaacs

Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur … Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur index Projective representations Character degrees Character correspondence Linear groups Changing the characteristic Some character tables Bibliographic notes References Index.

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Intersection theory on the moduli space of curves and the matrix airy function

1992-06-01

Maxim Kontsevich

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Mapping class groups and moduli spaces of curves

1997-01-01

Richard Hain , Eduard Looijenga

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Geometric Structures on the Complement of a Projective Arrangement

2005-06-01

Wim Couwenberg , Gert Heckman , Eduard Looijenga

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The Moduli Space of Rational Elliptic Surfaces

2019-01-26

Gert Heckman , Eduard Looijenga

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Chern classes of automorphic vector bundles

2002-03-01

Mark Goresky , William Pardon

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Weierstrass points and moduli of curves

1974-01-01

Enrico Arbarello

On period maps that are open embeddings

2008-01-01

Eduard Looijenga , Rogier Swierstra

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A simple presentation for the mapping class group of an orientable surface

1983-06-01

Bronisław Wajnryb

INTEGRAL SYMMETRIC BILINEAR FORMS AND SOME OF THEIR APPLICATIONS

1980-02-28

Vadim V. Nikulin

We set up the technique of discriminant-forms, which allows us to transfer many results for unimodular symmetric bilinear forms to the nonunimodular case and is convenient in calculations. Further, these … We set up the technique of discriminant-forms, which allows us to transfer many results for unimodular symmetric bilinear forms to the nonunimodular case and is convenient in calculations. Further, these results are applied to Milnor's quadratic forms for singularities of holomorphic functions and also to algebraic geometry over the reals. Bibliography: 57 titles.

Characteristic classes of surface bundles

1987-10-01

Shigeyuki Morita

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Calculating cohomology groups of moduli spaces of curves via algebraic geometry

1998-12-01

Enrico Arbarello , Maurizio Cornalba

The smoothing components of a triangle singularity. II

1984-10-01

Eduard Looijenga

A Conjectural Description of the Tautological Ring of the Moduli Space of Curves

1999-01-01

Carel Faber

We formulate a number of conjectures giving a rather complete description of the tautological ring of M g and we discuss the evidence for these conjectures. We formulate a number of conjectures giving a rather complete description of the tautological ring of M g and we discuss the evidence for these conjectures.

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Théorie de hodge, III

1974-01-01

Pierre Deligne

Arithmetic quotients of the mapping class group

2015-10-01

Fritz Grunewald , Michael Larsen , Alexander Lubotzky +1 more

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