arXiv subjects
Publications and source records attributed to 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 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.
The symmetric space $X_n={\rm SL}(n,\Rb)/{\rm SO}(n)$ can be interpreted as the Teichmüller space of marked, unit volume, flat $n$-dimensional tori. It comes with a unique (up to scale) ${\rm SL}(n,\Rb)$-invariant metric $d_{X_n}$. In 1939 Teichmüller gave a modular interpretation of $d_{X_2}$ (the hyperbolic metric) in terms of an extremal mapping problem for quasiconformal dilatation. Such a modular interpretation for $d_{X_n}$ for $n\geq 3$ has remained unaddressed: the natural candidates - minimal quasiconformal dilatation, Lipschitz constant, or total energy - do not work. In this paper we give such a modular interpretation, two in fact. We introduce the {\em total expansion} $\TE(f)\in [0,\infty]$ of a Lipschitz map $f:M\to N$ between Riemannian manifolds, a notion related to the notion of ``$k$-dilatation'' developed by Gromov, Guth and others. For volume-preserving Lipschitz maps $f:\Tc_0\to\Tc_1$ between $n$-dimensional, flat, unit-volume tori, we prove that $\TE(f)$ is minimized in the homotopy class of $f$ precisely by the affine maps in that class and takes on these the value $d_{X_n}$. We prove similar results for the \emph{Hilbert-Schmidt expansion} $\HE(f)$, which is a simple integral over $M$ and has more of an $L^2$ flavor.
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.
Let $Z'\subset \mathbb{P}^{n}$ be a smooth projective hypersurface of degree $d>1$ and let $Z\to \mathbb{P}^n$ be the $μ_d$-cover totally ramified along $Z'$. We relate full level $d$ structures on the primitive cohomology $Z'$ with full level $d$ structures on the primitive cohomology of $Z$. In the special case, $d=n=3$ this makes a marking of a smooth cubic surface determine a level $3$-structure on the associated cubic threefold, thereby answering a question by Beauville. We expect many more such applications.
Let $C$ be a smooth projective curve over an algebraically closed field $k$ of characteristic zero. We prove that a Lagrangian supplement of $H^0(C, Ω_C)$ in the de Rham cohomology group $H^1_{dR}(C)$ determines and is determined by a particular type of symmetric bidifferential on $C^2$ (its polar divisor must be twice the diagonal and have biresidue one along it). When $k$ is the complex field, a natural choice of such supplement is $H^{0,1}(C)$ and we show that this corresponds with the bidifferential that after a twist is the rational $2$-form on $C^2$ found by Biswas-Colombo-Frediani-Pirola. We determine the cohomology class carried by that $2$-form and define an analogue of this form as rational $n$-form on $C^n$ that is regular on the $n$-point configuration space of $C$. The proof relies on a local version of the above correspondence, which can be stated in terms of a complete discrete valuation ring. We use this local version also to construct in a natural manner the Virasoro extension of the Lie algebra of derivations of a local field.
This is a paper in smooth $4$-manifold topology, inspired by the Néron-Lang Theorem in number theory. More precisely, we prove that a smooth version $\MW(π)$ of Mordell-Weil group of an elliptic fibration $π:M\to\Pb^1$ is finitely generated. We compute $\MW(π_d)$ explicitly for elliptic fibrations $π_d:M_d\to\Pb^1$, where $M_d$ is a simply-connected complex surfaces $M_d$ of arithmetic genus $d\geq 1$ and all fibers of $π_d$ are nodal. We prove in this case that the fibered structure is unique up topological isotopy. By combining this with a result of Donaldson, we obtain the following remarkable consequence: any diffeomorphism of $M_d$ with $d\geq 3$ is topologically isotopic to a diffeomorphism taking fibers to fibers.
Let $U$ be a space whose one point compactification $U^*$ is a CW-complex for which the added point $*$ is the only $0$-cell. We observe that the configuration space $Conf_n(U)$ of $n$ numbered distinct points in $U$ has no closed support homology in degree $<n$ and prove that Borel-Moore homology group $H^{cl}_n(Conf_n(U))$ depends only on the fundamental group $π_1(U^*,*)$. We describe this homology group in terms of a presentation of $π_1(U^*,*)$. A case of interest is when $U$ is a connected closed oriented surface of positive genus minus a finite nonempty set. Then the mapping class group $Mod(U)$ of $U$ acts on both $π_1(U^*,*)$ and ${H^k}(Conf_n(U){)}\cong H^{cl}_ {2n-k}(Conf_n(U))$ and we prove that its action on the latter is through its action on the nilpotent quotient $π_1(U^*,*)/ π_1(U^*,*)^{(k+1)}$. Furthermore, we give an example of a mapping class of a once punctured closed surface $U$ which acts trivially on ${H^n}(Conf_n(U))$, but not on the nilpotent quotient $π_1(U^*,*)/ π_1(U^*,*)^{(n+1)}$. The former generalizes a theorem of Bianchi-Miller-Wilson and the latter disproves a conjecture of theirs.
We construct diffeomorphisms of ``pseudo-Anosov type'' on K3 surfaces M. In particular we obtain infinitely many examples of such diffeomorphisms that minimize entropy in their homotopy class, and for which neither the diffeomorphism nor any diffeomorphism homotopic to it preserves any complex structure on M.
In this paper we construct various moduli spaces of K3 surfaces $M$ equipped with a surjective holomorphic map $π:M\to\Pb^1$ 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 $M$.
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.
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 $\mathbb{C}[\mathfrak{g}]^G\to \bigoplus_{n\ge 0} H^0(S,\,Ω^n_{S,cl}\otimes R^nf_*\mathbb{C})$, where $Ω^n_{S,cl}$ 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 $S$.
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 and a weaker condition for this to have no finite nonzero orbit (the Putman-Wieland property).
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 $Γ_1(3)\subset \SL_2(\Zds)$ 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.
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. 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 $g^1_3$.
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 $p\in \{2,3,5,7\}$).
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.
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.
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.