Mapping Class Groups and Moduli Spaces of Curves
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.
alg-geom↗
arXiv subjects
Eduard Looijenga
Publications and source records attributed to Eduard Looijenga.
A Lie algebra attached to a projective variety
Each choice of a Kähler class on a compact complex manifold defines an action of the Lie algebra $\slt$ on its total complex cohomology. If a nonempty set of such Kähler classes is given, then we prove that the corresponding $\slt$-copies generate a semisimple Lie algebra. We investigate the formal properties of the resulting representation and we work things out explicitly in the case of complex tori, hyperkähler manifolds and flag varieties. We pay special attention to the cases where this leads to a Jordan algebra structure or a graded Frobenius algebra.
alg-geom↗
Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map
This replacement corrects statement and proof of the main result. Also, a section on the universal Abel-Jacobi map has been added.
alg-geom↗
On the tautological ring of $\M _g$
We prove among other things that any product of tautological classes of $\M_g$ of degree $d$ (in the Chow ring of $\M _g$ with rational coefficients) vanishes for $d>g-2$ and is proportional to the class of the hyperelliptic locus in degree $g-2$.
alg-geom↗
Cellular decompositions of compactified moduli spaces of pointed curves
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 r\^ ole 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.
alg-geom↗