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 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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Following our previous work, we develop an algorithm to compute a presentation of the fundamental group of certain partial compactifications of the complement of a complex arrangement of lines in the projective plane. It applies, in particular, to homology planes arising from arrangements of lines. This permits to determine the infinitude of the fundamental group of homology planes of log-general type that may be the first examples of this type.
We define the wonderful compactification of an arrangement of subvarieties. Given a complex nonsingular algebraic variety $Y$ and certain collection $\mathcal{G}$ of subvarieties of $Y$, the wonderful compactification $Y_\mathcal{G}$ can be constructed by a sequence of blow-ups of $Y$ along the subvarieties of the arrangement. This generalizes the Fulton-MacPherson configuration spaces and the wonderful models given by De Concini and Procesi. We give a condition on the order of blow-ups in the construction of $Y_\mathcal{G}$ such that each blow-up is along a nonsingular center.
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 in…
Let $\mathscr{A}$ be a real projective line arrangement and $M(\mathscr{A})$ its complement in $\mathbb{CP}^2$. We obtain an explicit expression in terms of Randell's generators of the meridians around the exceptional divisors in the blow-up $\bar{X}$ of $\mathbb{CP}^2$ in the singular points of $\mathscr{A}$. We use this to investigate the partial compactifications of $M(\mathscr{A})$ contained in $\bar{X}$ and give a counterexample to a statement suggested by A. Dimca and P. Eyssidieux to the effect that the fundamental group of such an algebraic variety is finite whenever its abelianization…
Consider the moduli space M^0 of arrangements of n hyperplanes in general position in projective (r-1)-space. When r=2 the space has a compactification given by the moduli space of stable curves of genus 0 with n marked points. In higher dimensions, the analogue of the moduli space of stable curves is the moduli space of stable pairs: pairs (S,B) consisting of a variety S (possibly reducible) and a divisor B=B_1+..+B_n, satisfying various additional assumptions. We identify the closure of M^0 in the moduli space of stable pairs as Kapranov's Chow quotient compactification of M^0, and give an e…
The perfect cone compactification is a toroidal compactification which can be defined for locally symmetric varieties. Let $\overline{D_{L}/\widetilde{O}^{+}(L)}^{p}$ be the perfect cone compactification of the quotient of the type IV domain $D_{L}$ associated to an even lattice $L$. In our main theorem we show that the pair ${ (\overline{D_{L}/\widetilde{O}^{+}(L)}^{p}, Δ/2) }$ has klt singularities, where $Δ$ is the closure of the branch divisor of ${ D_{L}/\widetilde{O}^{+}(L) }$. In particular this applies to the perfect cone compactification of the moduli space of $2d$-polarised $K3$ surf…
Compactifications of moduli of del Pezzo surfaces via line arrangement and K-stability
本文利用K-稳定性和线构形研究光滑del Pezzo曲面模空间的紧化，构造了log del Pezzo对的K-模空间，并证明了其在$d=2,3,4$时同构于del Pezzo曲面的K-模空间，而在$d=1$时两者不同。
Let X be a locally symmetric variety. Let EBS(X) and TorE(X) denote its excentric Borel-Serre and excentric toroidal compactifications, resp. We determine their least common modification and use it to prove a conjecture of Goresky and Tai concerning canonical extensions of homogeneous vector bundles. In the process, we see that EBS(X) and TorE(X) are homotopy equivalent.
We give explicit estimates for the volume of the Green-Griffiths jet differentials of any order on a toroidal compactification of a ball quotient. To this end, we first determine the growth of the logarithmic Green-Griffiths jet differentials on these objects, using a natural deformation of the logarithmic jet space of a given order, to a suitable weighted projective bundle. Then, we estimate the growth of the vanishing conditions that a logarithmic jet differential must satisfy over the boundary to be a standard one.
Let X' be the toroidal compactification of the quotient of the complex 2-ball by a torsion free lattice G of SU(2,1). We say that X'is co-abelian if there is an abelian surface, birational to X'. The present work can be viewed as an illustration for the presence of a plenty of non-compact co-abelian torsion free toroidal compactifications. More precisely, it shows that all the admissible values for the volume of a torsion free quotient of the complex 2-ball are attained by co-abelian Picard modular ones over Eisenstein numbers. The article provides three types of infinite series of finite unra…
The MMP singularities of GIT versus Baily-Borel compactifications for the ball quotient case
本文研究球商情形下 GIT 与 Baily-Borel 紧化之间的双有理关系，证明非空排列会使非 log canonical 奇点变为 log canonical 奇点，并用多个模空间例证。
The moduli space of holomorphic maps from Riemann surfaces to the Grassmannian is known to have two kinds of compactifications: Kontsevich's stable map compactification and Marian-Oprea-Pandharipande's stable quotient compactification. Over a non-singular curve, the latter moduli space is Grothendieck's Quot scheme. In this paper, we give the notion of `$ε$-stable quotients' for a positive real number $ε$, and show that stable maps and stable quotients are related by the wall-crossing phenomena. We will also discuss Gromov-Witten type invariants associated to $ε$-stable quotients, and investig…
We study GIT quotients parametrizing n-pointed conics that generalize the GIT quotients $(\mathbb{P}^1)^n//SL2$. Our main result is that $\overline{M}_{0,n}$ admits a morphism to each such GIT quotient, analogous to the well-known result of Kapranov for the simpler $(\mathbb{P}^1)^n$ quotients. Moreover, these morphisms factor through Hassett's moduli spaces of weighted pointed rational curves, where the weight data comes from the GIT linearization data.
Hyperplane Arrangements and Compactifications of Vector Groups
本文研究了超平面配置的 Schubert 簇作为仿射空间等变紧化的充分必要条件，将理论推广至部分紧化及态射，揭示了其与环面簇和多面体扇对应关系的相似性。
We classify compact complex surfaces which contain a Zariski open subset whose universal covering is the cylinder DxC.
We study the number of distinct ways in which a smooth projective surface $X$ can be realized as a smooth toroidal compactification of a ball quotient. It follows from work of Hirzebruch that there are infinitely many distinct ball quotients with birational smooth toroidal compactifications. We take this to its natural extreme by constructing arbitrarily large families of distinct ball quotients with biholomorphic smooth toroidal compactifications.
We introduce and study smooth compactifications of the moduli space of n labeled points with weights in projective space, which have normal crossings boundary and are defined as GIT quotients of the weighted Fulton-MacPherson compactification. We show that the GIT quotient of a wonderful compactification is also a wonderful compactification under certain hypotheses. We also study a weighted version of the configuration spaces parametrizing n points in affine space up to translation and homothety. In dimension one, the above compactifications are isomorphic to Hassett's moduli space of rational…
Some projective wonderful models for the complement of a toric arrangement in a n-dimensional algebraic torus T were constructed in [3]. In this paper we describe their integer cohomology rings by generators and relations.
We study the hyperplane arrangements associated, via the minimal model programme, to symplectic quotient singularities. We show that this hyperplane arrangement equals the arrangement of CM-hyperplanes coming from the representation theory of restricted rational Cherednik algebras. We explain some of the interesting consequences of this identification for the representation theory of restricted rational Cherednik algebras. We also show that the Calogero-Moser space is smooth if and only if the Calogero-Moser families are trivial. We describe the arrangements of CM-hyperplanes associated to sev…
In this paper, we study punctured spheres in two dimensional ball quotient compactifications $(X, D)$. For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded $3$-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness of $K_X + αD$ for $α\in (\frac{1}{4}, 1)$, giving a sharp version of a theorem of the first author with G. Di Cerbo. Finally, we produce the first examples of bielliptic ball quotient compactifications modeled on the Gaussian integers.
Quasilinear tropical compactifications
本文提出拟线性热带紧化这一更广泛的热带紧化类，证明其具有 schön 性质且相交理论完全由热带扇决定，并应用于证明6条直线模空间与带标记三次曲面模空间的拟线性。
For each complex central essential hyperplane arrangement $\mathcal{A}$, let $F_{\mathcal{A}}$ denote its Milnor fiber. We use Tevelev's theory of tropical compactifications to study invariants related to the mixed Hodge structure on the cohomology of $F_{\mathcal{A}}$. We prove that the map taking each arrangement $\mathcal{A}$ to the Hodge-Deligne polynomial of $F_{\mathcal{A}}$ is locally constant on the realization space of any loop-free matroid. When $\mathcal{A}$ consists of distinct hyperplanes, we also give a combinatorial description for the homotopy type of the boundary complex of an…
Let F(X,n):= X^n-Δbe the complementary of the union Δof the diagonals of X^n and let U be a quotient of F(X,n) (possibly trivial) by a subgroup of the symmetric group S_n. We construct compactifications of U in products of Hilbert schemes. Our approach generalizes and unifies classical constructions by Schubert-Semple, Le Barz-Keel, Kleiman and Cheah. An extensive study is done in the case n<4. This includes in particular a complete classification and a description of the quotients by the natural actions.
The minimal model program suggests a compactification of the moduli space of hyperplane arrangements which is a moduli space of stable pairs. Here, a stable pair consists of a scheme X which is a degeneration of projective space and a divisor D=D_1+..+D_n on X which is a limit of hyperplane arrangements. For example, in the 1-dimensional case, the stable pairs are stable curves of genus 0 with n marked points. Kapranov has defined an alternative compactification using his Chow quotient construction, which may be described fairly explicitly. We prove that these two compactifications coincide. W…
An explicit wall crossing for the moduli space of hyperplane arrangements
本文研究超平面排列模空间中从环面紧化到非环面紧化的第一个自然穿墙，并应用其结果证明广义 Losev-Manin 空间相关爆破的 Q-因子分解在超平面数足够多时不是 Mori dream 空间，同时给出平面直线情形的精确描述。
Let $(({\mathbb B} / Γ_1)', T(1))$ be á torsion free toroidal compactification with abelian minimal model $(A_1, D(1))$. An arbitrary principal isogeny $μ_a : A_2 \rightarrow A_1$, $a \in {\mathbb C}$ pulls-back $(A_1, D(1))$ to the abelian minimal model $(A_2, D(2))$ of a torsion free toroidal compactification $(({\mathbb B} / Γ_2)', D(2))$. The present work makes use of the isogeny pull-backs of abelian ball quotient models, towards the construction of infinite series of mutually non-birational co-abelian torsion free Galois covers $(({\mathbb B} / Γ_n)', T(n))$ of a ball quotient compactifi…
We characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space. It follows then that the GIT compactification of the moduli space of cubic fourfolds is isomorphic to the Looijenga's compactification associated to this arrangement. This work builds on and is a natural continuation of our previous paper on the GIT compactification of the moduli space of cubic fourfolds.
Curvature properties and Shafarevich conjecture for toroidal compactifications of ball quotients
本文研究有限体积复双曲流形的环面紧化，探讨了其上满足非正曲率性质的Kähler度量的存在性，并通过对足够深的非均匀算术格群商证明万有覆盖为Stein，验证了Shafarevich猜想。
The first part of this paper is a survey on algebro-geometric aspects of sheaves of logarithmic vector fields of hyperplane arrangements. In the second part we prove that the relative de Rham cohomology (of degree two) of ADE-type adjoint quotient map is naturally isomorphic to the module of certain multiderivations. The isomorphism is obtained by the Gauss-Manin derivative of the Kostant-Kirillov form.
Moduli spaces of arrangements of 12 projective lines with a sextic point
本文研究了具有六重点的12条射影直线构型的模空间分类问题，证明了该模空间可包含两个以上连通分支，并通过复共轭取商及提供潜在Zariski对给出了模空间不可约的构型定义方程。
For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H^{<=2}(X), to the second nilpotent quotient, G/G_3. We define invariants of G/G_3 by counting normal subgroups of a fixed prime index p, according to their abelianization. We show how to compute this distribution from the resonance varieties of the Orlik-Solomon algebra mod p. As an application, we establish the cohomology classification of 2-arrangements of n<=6 planes in R^4.
We consider Kapranov's Chow quotient compactification of the moduli space of ordered n-tuples of hyperplanes in P^{r-1} in linear general position. For r=2 this is canonically identified with the Grothendieck-Knudsen compactification of M_{0,n} which has among others the nice properties 1) Modular meaning: stable pointed rational curves 2) Canonical description of limits of one parameter degenerations 3) Natural Mori theoretic meaning: log canonical compactification. We prove (1-2) generalize naturally to all (r,n), but that (3), which we view as the deepest, fails except possibly in the cases…
Let F be a global field, and let S be a finite set of places of F containing all archimedean places. Consider the product X of the symmetric spaces and Bruhat-Tits buildings for PGL_d of the completions of F at archimedean and non-archimedean places in S, respectively. We construct compactifications of the quotient of X by S-arithmetic subgroups of PGL_d(F). The constructions make delicate use of reductive Borel-Serre spaces for archimedean places and polyhedral and seminorm compactifications at nonarchimedean places. We also briefly discuss a few potential applications of our compacifications…
We study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non-archimedean amoebas. These compactifications have many nice properties, for example any k boundary divisors intersect in codimension k. We consider some examples including $M_{0,n}\subset\bar M_{0,n}$ (and more generally log canonical models of complements of hyperplane arrangements) and compact quotients of Grassmannians by a maximal torus.
We study the GIT compactifications of pairs formed by a hypersurface and a hyperplane. We provide a general setting to characterize all polarizations which give rise to different GIT quotients. Furthermore, we describe a finite set of one-parameter subgroups sufficient to determine the stability of any GIT quotient. We characterize all maximal orbits of non stable and strictly semistable pairs, as well as minimal closed orbits of strictly semistable pairs. Our construction gives natural compactifications of the space of log smooth pairs for Fano and Calabi-Yau hypersurfaces.
The present work obtains the fundamental group of a toroidal compactification X' of a non-compact quotient X of a Hermitian symmetric space D of non-compact type by a lattice L in the isometry group G of D. As a consequence it derives the equality of the ranks of the first homology groups of X' and X with integral coefficients. The paper provides also a sufficient condition on a torsion free non-uniform lattice L, under which the fundamental group of X' is residually finite. Articles of Hummel-Schroeder, Hummel and Di Cerbo imply that the toroidal compactifications X' of generic non-compact to…
Non-isomorphic smooth compactifications of the moduli space of cubic surfaces
针对复三次曲面模空间的Kirwan爆破与环面紧化因上同调相同是否同构的问题，证明二者在Grothendieck环中等价但非同构、非$K$-等价，并建立相关奇点与典范类的研究技术。
These lecture notes are based on lectures given by the author at the summer school "Arrangements in Pyrénées" in June 2012. We survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gel'fand-MacPherson construction, Kapranov's visible contours compactification, and De Concini and Procesi's wonderful compactification. We explain how these constructions are unified by some ideas from the modern origins of tropical geometry. The paper contains a few new arguments intended to make the presentation as self-contained as possible.
Let $Ω$ be the complement of a connected, essential hyperplane arrangement. We prove that every dominant endomorphism of $Ω$ extends to an endomorphism of the tropical compactification $X$ of $Ω$ associated to the Bergman fan structure on the tropicalization of $Ω$. This generalizes a previous result by Rémy, Thuillier and the second author which states that every automorphism of Drinfeld's half-space over a finite field $\mathbb{F}_q$ extends to an automorphism of the successive blow-up of projective space at all $\mathbb{F}_q$-rational linear subspaces. This successive blow-up is in fact the…
A characterization of ball quotient stacks
该研究刻画了作为球商栈紧化的光滑真Deligne-Mumford栈，证明了球商紧化的边界除子结构，推广了Deng-Cadorel的结果，所用方法结合了非阿贝尔霍奇对应等理论。
We prove a result on the singularities of ball quotients $Γ\backslash\CC H^n$. More precisely, we show that a ball quotient has canonical singularities under certain restrictions on the dimension $n$ and the underlying lattice. We also extend this result to the toroidal compactification $(Γ\backslash\CC H^n)^*$.
We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of $\frac{8}{3}π^{2}$, i.e., they attain all possible volumes of complex hyperbolic $2$-manifolds. The surfaces in one of the two families have all $2$-cusps, so that we can saturate the entire volume spectrum with $2$-cusped manifolds. Finally, we show that the associated neat lattices have infinite abelianization and finitely generated commut…
A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion of the quotient away from the singularities. New compactifications of classical spaces arise naturally: a nonsingular and irreducible compactification of the moduli of maps from genus 1 curves to projective space is obtained. Localization on the moduli of stable quotients leads to new relations in the tautological ring generalizing Brill-Noether constructions…
In this paper we show a general method to compactify certain open varieties by adding normal crossing divisors. This is done by proving that {\it blowing up along an arrangement of subvarieties} can be carried out. Important examples such as Ulyanov's configuration spaces, spaces of holomorphic maps, etc., are covered. Intersection ring and (non-recursive) Hodge polynomails are computed. Further general structures arising from the blowup process are described and studied.
We introduce complete quotients over the projective line and prove that they form smooth projective varieties. The resulting parameter spaces coincide with the varieties constructed in [HLS11] and [Shao11]. Hence they provide modular smooth compactifications with normal crossing boundaries of the spaces of algebraic maps from the projective line to Grassmannian varieties.
We construct the Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves. The Hilbert compactification is the GIT quotient of some open part of an appropriate Hilbert scheme of curves in a Grassmannian. It has all the properties asked for by Teixidor.
We define a period map for classical Campedelli surfaces, using a covering trick as in the case of Enriques surfaces: the period map is shown to come from a family of Enriques surfaces, obtained as quotients of the Campedelli surface by an involution. The period map realises an isomorphism between a projective variety obtained by invariant theory, and the Baily-Borel compactification of an arithmetic quotient, in the same fashion as in the work of Matsumoto, Sasaki and Yoshida. The result is proved from scratch using traditional methods. As another consequence we determine properties of the mo…
We study the geometry of the stratification induced by an affine hyperplane arrangement H on the quotient of a complex affine space by the action of a discrete group preserving H. We give conditions ensuring normality or normality in codimension 1 of strata. As an application, we provide the list of those categorical quotients of closures of Jordan classes and of sheets in all complex simple algebraic groups that are normal. In the simply connected case, we show that normality of such a quotient is equivalent to its smoothness.
In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmueller space by the action of the mapping class group gives a compactification of the moduli space. We put an analytic structure on this compact quotient and prove that with respect to this structure, it is canonically isomorphic (as an analytic space) to the Deligne-Mumford compactification.
We calculate Chow quotients of some families of symmetric \(T\)-varieties. In complexity two we obtain new examples of Kähler-Einstein metrics by bounding the symmetric alpha invariant of their orbifold quotients. As an additional application we determine the homeomorpism class of the orbit space of the compact torus action.
We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. Moreover, several results concerning the Riemannian and complex algebraic geometry of these spaces are given. In particular we show that there are compact complex surfaces which admit Riemannian metrics of nonpositive curvature, but which do not admit Kähler metrics of nonpositive curvature. An infinite class of such examples arise as smooth toroidal compactifications of ball quotients.
It is known that some GIT compactifications associated to moduli spaces of either points in the projective line or cubic surfaces are isomorphic to Baily-Borel compactifications of appropriate ball quotients. In this paper, we show that their respective toroidal compactifications are isomorphic to moduli spaces of stable pairs as defined in the context of the MMP. Moreover, we give a precise mixed-Hodge-theoretic interpretation of this isomorphism for the case of eight labeled points in the projective line.
We study the Chow groups and the Chow motives of the wonderful compactifications $Y_{\mathcal{G}}$ of arrangements of subvarieties. We prove a natural decomposition of the Chow motive of $Y_\mathcal{G}$, in particular of the Fulton-MacPherson configuration space $X[n]$. As a consequence, we prove a decomposition of the Chow motive of $X[n]/\frak{S}_n$. A generating function for the Chow groups and for the Chow motive of $X[n]$ is given.
We describe the GIT compactification for the moduli space of smooth quintic surfaces in projective space. In particular, we show that a normal quintic surface with at worst an isolated double point or a minimal elliptic singularity is stable. We also describe the boundary of the GIT quotient, and we discuss the stability of the non-normal surfaces.
We prove that any toroidal compactification of arithmetic quotient of Hermitian symmetric domain has no boundary branch divisor, in the setting where the algebraic group is of adjoint type.
Chow Quotient Moduli Space of Dynamical Systems on the Projective Line and Rescaling Limits
本文研究射影直线上次数为d的有理自映射模空间的Chow商紧化，计算三维轨道环度数，引入带标记Chow商记录重标度极限，并扩展不动点乘子映射。
Galois lines for the quotient curve of the Hermitian curve by an involution
研究involution作用下Hermitian曲线商曲线在有限域上的Galois线排列，确定三种平面模型的Galois点。
On the graded singularity category of Abelian quotient singularities, I. Smooth categorical compactification
针对有限 Abel 子群作用于仿射空间所得商奇点，本文利用非交换 crepant 消解构造其非交换射影空间导出范畴与分次奇点范畴的 Efimov 光滑紧化，并给出核中典范经典生成元。
Stable maps to quotient stacks with a properly stable point
本文研究具有射影好模空间的商栈上曲线映射的模空间紧化，提出扩展加权吹胀的双有理操作，将拟映射理论紧化扩展至多类映射与纤维化模空间，并在附录中模块化证明Hassett猜想。
Groupes de Brauer algébriques modulo les constants d'espaces homogènes et leurs compactifications
本文证明连通线性代数群齐性空间的代数 Brauer 群对常数 Brauer 群的商到 $\mathrm H^1(K,\operatorname{Pic}\bar X)$ 的自然包含未必为同构，并将结论推广到光滑紧化。
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