(1) A note on polynomial isolated singularities, Indag. Math. 33, 418-421 (1971).
In this paper are constructed differentiable map-germs (R4,0) → (R2,0) having the origin as an isolated singular point that are not topologically equivalent to complex map-germs (C2,0) → (C,0), thus answering a question Milnor posed in his book on isolated hypersurface singularities. This was part of my Masters thesis. Its Math. Review is supplemented in an interesting manner by a paragraph in Lee Rudolph's review MR 0643562 (84d:57005) of B. Perron's paper Le noeud `huit' est algébrique reel.
(2) Structural Stability of smooth families of C∞-functions, Thesis, Amsterdam (1974).
This was never published, but C.T.C. Wall reported extensively on it in his survey paper Geometric properties of generic differentiable manifolds, Geometry and topology, pp. 707–774. L. N. in Math., Vol. 597, Springer, Berlin, 1977. The main result might be stated as follows. Fix a manifold M and observe that for every embedding j : M → EN in euclidean N-space, we get a family of real valued functions on M parameterized by EN: to u ∈ EN is associated the function p ∈ M → F(p, u) := ∥ j(p) − u ∥2. The claim is that among the proper embeddings, those with the property that the associated family of functions is topologically stable (in the sense that the topological type of M × EN → R × EN → EN as a diagram of continuous mappings is insensitive to small deformations of F) are dense. This was conjectured by R. Thom and provides the ultimate justification for his approach to extrinsic differential geometry.
(3) The complement of the bifurcation variety of a simple singularity, Invent. Math. 23, 105–116 (1974).
One of the main results of this paper is what Russian colleagues (because of O.V. Ljaško: The geometry of bifurcation diagrams, Uspekhi Mat. Nauk 34 (1979), no. 3(207), 205–206) have baptized the Lyashko-Looijenga theorem. Conjecture 3.5 was proved later that year by Deligne with the help of Tits and Zagier.
(4) A period mapping for certain semi-universal deformations, Comp. Math. 30, 299-316 (1974).
This paper contained the first (written) proof that the discriminant of a rational double point equals the discriminant of its associated Weyl group. The Grothendieck programme suggested a different approach, and this was, after the initial steps by Brieskorn, later brought to completion by Peter Slodowy (see his Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, 815. Springer, Berlin, 1980). My original motivation for introducing this period map was to give a proof that would lend itself to extending this result to other unimodal singularities. This was accomplished for the unimodal singularities in some of my subsequent papers.
(5) (with C.G. Gibson, K. Wirthmüller, and A. du Plessis) Topological Stability of Smooth Mappings, Springer Lecture Note 552 (1976).
This is an account of a seminar run in Liverpool in 1974-75. It presents the first complete proof of the theorem that says that in the space of proper smooth maps between two given manifolds, the topological stable ones are dense. The theorem is due to J. Mather (and rooted in the work of R. Thom), but the proof presented here is somewhat different than his and is modeled after one (of a similar result) in my thesis. The first chapter, by C.G. Gibson, is based on a lecture of mine.
(6) Root Systems and Elliptic Curves, Invent. Math. 38 17–32 (1976).
My investigation of the simple elliptic singularities led me to discover a new invariant theory for finite root systems (namely one in terms of formal theta expansions). Henry Pinkham (who reviewed this paper) observed that we thus have for every connected commutative algebraic group of dimension one—Ga , Gm or an elliptic curve—an associated invariant theory for root systems, namely the polynomial, the exponential, and the theta invariants: in all cases we have a polynomial algebra. The geometric content is that a finite Weyl group acts on an affine cone over a product of elliptic curves in such a manner that the orbit space is an affine space. The paper also contains a new proof (of a geometric nature) of a famous formula of I. Macdonald. The orbit space in question has several geometric interpretations, one of which was later exploited by Friedman-Morgan-Witten. (The reader be warned that the proof of Theorem (3.4) is only given for the generic elliptic curve, or rather for the Tate curve, for the argument in (4.4) based on the fact that all elliptic curves have the same underlying manifold is incomplete.)
(7) On the semi-universal deformation of a simple elliptic singularity I, Topology 16, 257–262 (1977).
This established topological equisingularity of the semi-universal deformation of a simple elliptic singularity along its modular stratum and was the first result of that kind. The technique introduced here has since been further refined and amplified, mostly by J. Damon.
(8) On the semi-universal deformation of a simple elliptic singularity II, Topology 17, 23–40 (1978).
We find a description of the discriminant of a semi-universal deformation of a simple elliptic singularity in terms of a finite Weyl group acting on the affine cone over a product of elliptic curves. It makes full use of the period map introduced in (4) and the invariant theory developed in (6). This made it more manageable to verify a conjecture of R. Thom which states that the complement of such a discriminant has contractible universal covering. I put this as a thesis problem to my first student H. van der Lek, but this turned out to be a very hard (and is still open). He gave however a neat presentation of the fundamental group appearing here. That presentation suggested to Cherednik and others the type of Hecke algebra that is now called a Cherednik algebra. Since Van der Lek’s thesis was never published, much of his work on generalized braid groups was later duplicated.
(9) The discriminant of a real simple singularity, Comp. Math. 37, 51–62 (1978).
We enumerate the connected components of the complement of the real discriminant and prove that each such component is contractible, in agreement with a conjecture of R. Thom.
(10) Homogeneous spaces associated to certain semi-universal deformations, Proc. Intern. Congr. Math. Helsinki, vol. 2, 529–536 (1978).
After reviewing earlier work we outline a programme for describing the discriminants of all the unimodal singularities. This has subsequently been carried out in (12), (13), (17), (19) and was finally brought to a completion a quarter of a century later in (43).
(11) On quartic surfaces in projective 3-space, Nieuw Arch. Wisk. 27, 98–103 (1979)
(12) Invariant theory for generalized root systems, Invent. Math. 61, 1–32 (1980).
The generalized root system of the title is one that is given by a generalized Cartan matrix. We construct a Stein manifold such that the highest weight characters of the associated (Kac-Moody) group define holomorphic function on it that separate the points. Although this is certainly relevant in the theory of Kac-Moody groups, our motivation was in applying this to hyperbolic singularities. This was later taken shown to fit in a ‘Grothendieck package’ for such groups by Peter Slodowy.
(13) Rational Surfaces with an anticanonical cycle, Ann. of Math. 114, (1981).
The main goal is generalize our results on the discriminant of a simple elliptic singularity to the hyperbolic case. For this we develop the theory of rational surfaces as in the title and show that such surfaces can degenerate into certain Inoue surfaces with (socalled) cusp singularities. (Some prominent algebraic geometers found it at the time hard to believe that such a degeneration was possible.) This implies that if a cusp (surface) singularity is smoothable, then there exists a smooth projective rational surface with an anticanonical cycle that is the minimal resolution of its `dual cusp’. We state a conjecture that essentially says that the converse also holds: all smoothable cusp singularities are thus accounted for. Partial results were obtained by Friedman-Miranda and Friedman-Pinkham, but a remarkable proof of the full conjecture has been recently obtained by Gross-Hacking-Keel (Mirror symmetry for log Calabi–Yau surfaces I, Publ. math. de l'IHÉS, 122 (2015), 65–168). It is based on mirror symmetry considerations. An independent (and somewhat more direct) proof was obtained by Philip Engel (Looijenga's conjecture via integral-affine geometry, J. Diff. Geom. 109 (2018), 467–495).
(14) Moduli spaces of marked Del Pezzo surfaces, Appendix to a paper by I. Naruki, Proc. London Math. Soc. 45, 24–30 (1982).
We observe that if X is a Del Pezzo surface and D is an effective anticanonical divisor on X, then the group homomorphism from Pic(X) → Pic(D) is a complete invariant of the pair (X, D). It is pointed out that this is quite helpful for constructing moduli spaces of such pairs (X, D) and can serve to better understand Naruki’s paper. We exploited this fact further in (27) and (51); others have too.
(15) (with C. Peters) Torelli theorems for Kähler K3-surfaces, Comp. Math. 42, 145-186 (1981).
This is mostly an account of the fundamental paper by Piatetski-Shapiro and Shafarevic, but with the gaps filled in.
(16) A Torelli Theorem for Kähler–Einstein K3-surfaces, Proc. Geometry Symposium Utrecht 1980, Springer Lecture Note 894, 107–112.
This paper and one by Y.T. Siu (Manuscripta Math. 35 (1981), no. 3, 311–321) give the first proofs of the surjectivity of the period map for Kähler K3-surfaces that do not rely on the hard proof of the polarized case, due to Kulikov, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 5, 1008–1042 and Persson-Pinkham, Ann. of Math. (2) 113 (1981), no. 1, 45–66. As Huybrechts later showed, this proof generalizes in a straightforward manner to hyperkähler manifolds.
(17) The Smoothing Components of a Triangle Singularity I, Proc. Symp. Pure Math. 40, vol. 2, 173–184 (1983).
A triangle singularity is a normal surface singularity associated to one of Klein’s hyperbolic triangle groups. We here enumerate the smoothing components in terms of combinatorial (lattice) data. Shortly afterwards Pinkham (Smoothings of the Dpqr singularities, p + q + r = 22. Singularities, Part 2, 373–377, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983) used this to determine when the set in question is nonempty , i.e., when such a singularity is smoothable. We also announce a description of the structure of these smoothing components.
(18) Isolated singular points on complete intersections, 200 p.+xi, L.M.S. Lecture Note Series, 77 (1984), Cambridge University Press.
This book originates in notes that I wrote for an intercity seminar that we ran on the subject. It was the first systematic treatment of isolated complete intersection singularities in book form and included some new material as well. A second (TeXed) edition with minor changes has been published in 2013 by International Press.
(19) The smoothing components of a triangle singularity II, Math. Ann. 269, 357–387 (1984).
We determine the rational double point locus in terms of lattice data, thus proving most of the results announced in Part I.
(20) (with G.M. Greuel) The dimension of smoothing components, Duke Math. J. 52, 263-272 (1985).
We prove here a conjecture of J. Wahl in the general framework he anticipated.
(21) (with J. Steenbrink) A formula for µ − τ, Math. Ann. 271, 121-124 (1985).
This paper does what the title promises (the formula being a sum of nonnegative sum of singularity invariants). It is the shortest paper I ever co-authored. In retrospect, I think it was too short.
(22) Riemann-Roch and smoothings of singularities, Topology 25, 293–302 (1986).
(23) (with J.M. Wahl) Quadratic functions and smoothings of singularities, Topology 25, 261–291 (1986).
(24) New Compactifications of Locally Symmetric Varieties, Proc. Vancouver Conf. in Alg. Geom., CMS Conference Proceedings 6, 341-364 (1986).
Outlines a construction that existed essentially in preprint form (1985). The final version appeared 20 years later.
(25) L2-Cohomology of Locally Symmetric Varieties, Comp. Math. 6, 3–20 (1988).
This gives a proof of the Zucker Conjecture. Competition was stiff (and included the late A. Borel); a different proof was at about the same time given by L. Saper and M. Stern (Ann. of Math. (2) 132 (1990), no. 1, 1–69). Our main tool is a local Hecke operator that we introduce here. That operator was later used by Goresky-Harder-MacPherson in their work on weighted cohomology.
(26) (with M. Rapoport) Weights in the local cohomology of a Baily–Borel compactification, Complex geometry and Lie theory, Proc. Symp. Pure Math. 53, 223–260 (1991).
We show among other things that various possible notions of weight in this local cohomology all coincide here. The local Hecke operator of (25) is used here and we obtain another proof of the Zucker conjecture.
(27) Cohomology of M3 and M3,1, in Mapping class groups and moduli spaces of Riemann surfaces, Contemp. Math. 150, 205–228 (1993).
We describe the rational cohomology in question with its Hodge structure (it is all Tate, but the weight filtration is nontrivial). We thus find a new cohomology class in M3 in degree 6 (whose weight is 12). For our main result we needed a description of the cohomology of a toric arrangement complement (as the toric counterpart of the work of Arnol’d-Brieskorn and Orlik-Solomon). That such a description is included in this paper is of course not at all clear from the title and that explains why that part has been duplicated a number of times since. (This experience taught me that is not always a good idea to squeeze everything you need in a single paper.) Ezra Getzler noticed that my computation of the cohomology of M3,1 could not be correct as stated. The two of us wrote a joint note in which the correction was made (The Hodge polynomial of M3,1 , 4 p. (1999), math.AG/9910174).
(28) Intersection theory on Deligne–Mumford compactifications, Séminaire Bourbaki, Exposé 768, Astérisque 216, 187–212 (1993).
An account of Kontsevich’s proof of the Witten conjecture.
(29) Smooth Deligne–Mumford compactifications by means of Prym level structures, J. Algebraic Geometry 3, 283–293 (1994).
We construct here finite orbifold covers of Mg whose normalization over Mg is smooth. I fortunately did not know that many experts (among them, Mumford, or so I have been told) doubted that this be possible. In any case, this result allows us to bypass Mumford’s setting for Chow theory on Mg . This was later generalized to Mg,n by my students Boggi and Pikaart and to positive characteristic by De Jong-Pikaart.
(30) On the tautological ring of Mg, Invent. Math. 121, 411–419 (1995).
This short paper verifies one of the conjectures of C. Faber concerning the tautological ring of Mg : it proves that this ring vanishes in degree ≥ g−2 and is of dimension ≤ 1 in degree g − 2. Faber himself proved shortly afterwards that summand in degree g − 2 is of dimension exactly one.
(31) Cellular decompositions of compactified moduli spaces of pointed curves, The moduli space of curves (Texel Island, 1994), 369–400; Progr. Math., 129, Birkhaüser Boston, Boston, MA (1995).
(32) The stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map, J. Algebraic Geometry 5, 135–150 (1996).
We prove that the cohomology of the mapping class group of genus g with values in an algebraic representation of the symplectic group Sp(2g,Q) stabilizes with g (there is an obvious way to stabilize such representations). This stable twisted cohomology is universally (and quite concretely) expressed as a module over the stable cohomology of the mapping class group (which, thanks to Madsen and Weiss, is now known to be the polynomial algebra in the Miller-Morita-Mumford classes).
(33) Cohomology and intersection homology of algebraic varieties, Complex Algebraic Geometry at Park City, IAS/Park City mathematical Series 3, AMS. 221–263 (1997).
During my visiting professorship in Salt Lake City (Spring 1991) I gave a course on the Lefschetz approach to (intersection) cohomology of algebraic varieties. Herb Clemens took very carefully notes of that course and I used his notes for a (much more concise) lecture series given in Park City. This is an edited version of the notes that accompanied these lectures.
(34) Prym representations of mapping class groups, Geometriae Dedicata 64, 69–83 (1997).
For every genus g≥ 2 we produce a family of representations (unitary or symplectic over a ring of integers) of the corresponding mapping class group (which is essentially given by letting act on the cohomology of a finite cyclic cover of the surface). Our main result is that these representations are independent and almost surjective. So for every r ≥ 0, this mapping class group has an arithmetic group of rank ≥ r as quotient.
(35) (with V. Lunts) A Lie algebra attached to a projective variety, Invent. Math. 129, 361–412 (1997).
This paper could have been written half a century earlier. According to Weil, if M is a compact connected Kaehler manifold, then cupping with its Kaehler class k∈ H1,1(M;R) turns H∗(M;R) into a sl(2,R)-module. This is however an open property for any k ∈ H1,1(M;R) to have. We here investigate among other things the graded Lie algebra generated by all such copies of sl(2,R) (and show that is always semisimple). Of particular interest is the case when we get a Jordan algebra. The reader is challenged to find a Calabi-Yau 3-fold with Neron-Severi group of rank 27 so that the above Lie algebra acts on the Tate part of the even dimensional cohomology as the irreducible representation of E7 of degree 56(=1 + 27 + 27 + 1). The paper got a Featured Review in Math. Reviews.
(36) (with R. Hain) Mapping class groups and moduli spaces of curves, Proc. Symp. Pure Math. 62.2, 97–142 (1997).
Both Hain and I gave talks at the AMS conference at Santa Cruz (1995). We decided to join forces in order to survey recent developments in the area of the title. In this overview we also posed some questions we felt were important and were not able to answer. We included some new results as well, among them a version of the Mumford conjecture for a certain completion of the mapping class group as a pro-arithmetic group, using Pikaart’s purity theorem and work of Kontsevich (which was obtained at the same time by Morita and Kawazumi, whose based themselves on the same ideas). But the interest of that result is somewhat diminished since Madsen and Weiss proved the original Mumford conjecture.
(37) Arrangements, KZ systems and Lie algebra homology, in Singularity Theory, B. Bruce and D. Mond eds., London. Math. Soc. Lecture Note Series, CUP 263, 109–130 (1999).
This first part of this paper surveys Dunkl connections and the KZ equation. The second part explains a construction due to Varchenko-Schechtman of a map from a KZ local system to a local system of Gauss-Manin type. As in (27) we here take (and propagate) an approach to the cohomology of local systems on arrangement complements via the derived category of constructible sheaves, which we feel is tailor made for this question. This leads quite naturally to modifying the Varchenko-Schechtman map as to make it always injective (it is not even a priori clear whether the original map is nonzero).
(38) Correspondences between moduli spaces of curves, in Moduli of Curves and Abelian Varieties, C. Faber and E. Looijenga eds., Aspects of Math. E33, 131–150 (1999).
We here take a motivic approach to the stability theorem and to (what was then still) Mumford’s conjecture, the stability maps being replaced by correspondences. We obtain among other things motivic Adams operators.
(39) A minicourse on Moduli of Curves, in ICTP Lecture Note series 1, 267–291 (2000).
Notes accompanying a series of lectures given at a summer school at the ICTP.
(40) Motivic measures, Séminaire Bourbaki, Exposé 874, Astérisque 276, 267–297 (2002).
This is mostly an account of the work of Denef-Loeser. We added a few things of our own, though (among them a simple presentation of the Grothendieck ring of varieties over a field of characteristic zero and a convolution product).
(41) (with G. Heckman) The Moduli Space of Rational Elliptic Surfaces, Algebraic geometry 2000, Azumino (Hotaka), 185–248, Adv. Stud. Pure Math. 36, Math. Soc. Japan, Tokyo (2002).
The moduli space of rational elliptic surfaces admitting a section is the same as the moduli space of Del Pezzo surfaces of degree 1. We identify this moduli space with a ball quotient of dimension 8 and we compare various compactifications of that moduli space.
(42) Compactifications defined by arrangements I: The ball quotient case, Duke Math. J. 118, 151–187 (2003).
(43) Compactifications defined by arrangements II: Locally symmetric varieties of type IV, Duke Math. J. 119, 527–588 (2003).
This paper and its predecessor are the fruit of a sort of internal deliberation that took two decades to arrive at a conclusion: I did not want the kind of generality I had no use for but I also wanted to make sure that this theory would cover all the examples I had encountered (a set that increased over time). In order to explain what these papers are about, let us first observe that there are only two types of irreducible symmetric domains that contain symmetric subdomains of complex codimension one: complex balls and domains of type IV (these are attached to a real orthogonal group of signature (2,n)). If D is such a domain and Γ is an arithmetic group, then the Baily-Borel theory furnishes a canonical projective compactification of the orbit space Γ \D. This theory is in fact a whole package with automorphic, geometric and topological aspects. These papers extend that package to the case where D has been replaced by a Γ -invariant open subset D′ of D whose complement is a locally finite union of symmetric subdomains of complex codimension one. Our justification is that varieties of the type Γ \D′ and their Baily-Borel compactification (as defined here) occur remarkably often in algebraic geometry.
(44) (with W. Couwenberg and G. Heckman) Geometric structures on the complement of a projective arrangement, Publ. Math. de l’IHES 101, 69–161 (2005).
Couwenberg was a Ph.D. student of Heckman who took his degree eleven years earlier. His thesis was a beautiful piece of work, but it failed to accomplish the goal Heckman and he had set for themselves, namely to generalize the Deligne-Mostow theory to Weyl group arrangements (from this point of view Deligne-Mostow treated the An -case). Around 2002 I noticed that the view point and the accompanying techniques developed in (42) (plus a Stein factorization argument) could help to bring this work to a completion. This approach worked in even greater generality then was originally envisaged and the present paper is the result. Writing it was not easy, though, in part because it was often unclear what the appropriate setting should be. After the work was done, we felt justified in beginning the introduction with the sentence: This article wants to be the child of two publications which saw the light of day in almost the same year, referring to the work of Deligne-Mostow (1986) and of Barthel-Hirzebruch-Hoefer (1987).
(45) (with W. Couwenberg and G. Heckman) On the geometry of the Calogero-Moser system, Indagationes Math. 16, 443–459 (2005).
Here we describe a generalization of (44) to toric arrangements. (The Calogero-Moser system can be regarded as the toric An-case.) The details of this construction have yet to be published.
(46) (with T. Springer) Peter Slodowy, Jahresber. Dtsch. Math.-Ver. 108, No. 2, 105-117 (2006).
This paper in German consists of a brief biography and a description of the main mathematical accomplishments of Slodowy.
(47) Uniformization by Lauricella functions—an overview of the theory of Deligne-Mostow, Lectures Notes for the CIMPA Summer School Arithmetic and Geometry Around Hypergeometric Functions 207–244, Progr. Math. 260, Birkhaüser Verlag Basel (2007).
Just what the title promises. There are no original results here, but I hope the paper is helpful for anyone wanting to study the original papers.
(48) Invariants of quartic plane curves as automorphic forms, Algebraic Geometry (Dolgachev Festschrift), Contemp. Math. 422, 107–120 (2007).
We show that the GIT for quartic plane curves produces a Baily-Borel compactification of the type introduced in (42). This yields an identification of the algebra of invariants of quartic forms in 3 variables with an algebra of meromorphic automorphic forms on a complex 6-ball.
(49) (with Rogier Swierstra) On period maps that are open embeddings, J. f. d. reine u. angew. Math. (Crelle) 617, 169-192 (2008).
(50) (with Rogier Swierstra) The period map for cubic threefolds, Compositio Math. 143, 1037–1049 (2007).
The GIT for cubic threefolds in P4 was carried out by D. Allcock in J. Algebraic Geom. 12 (2003), 201–223 and independently by M. Yokoyama (unpublished). We show that this GIT orbit variety is a Baily-Borel compactification of the type introduced in (42). This includes a determination of the image of the period map for these varieties, a result that was independently obtained by Allcock-Carlson-Toledo (Mem. Amer. Math. Soc. 209 (2011), no. 985 and arXiv:math/0608287 [math.AG]). We also obtain an identification of the algebra of invariants of cubic forms in 5 variables with an algebra of meromorphic automorphic forms on a complex 10-ball.
(51) Affine Artin groups and the fundamental groups of some moduli spaces, J. of Topology 1, 187–216 (2008).
In this paper, of which a first version was written ten years earlier, we define for every Artin group associated to an affine Coxeter group a natural quotient, called the reduced Artin group, which still maps to the associated finite Coxeter group modulo its center. We prove that the fundamental group of some moduli stacks are of this type. For instance, for the moduli stack of smooth cubic surfaces we get the group attached to the affine system of type E6. A similar result for the moduli stack of smooth quartic curves in relation to E7 leads to a new presentation for the mapping class group of genus three surfaces with a boundary component. This produces all the exotic relations for the Humphries generators of a mapping class group of any genus at least 3.
(52) (with Claudio Fontanari) A perfect stratification of Mg for g at most 5, Geometriae Dedicata 136, 133-143 (2008).
The stratification in question is of the form Mg=Z0 ⊃ Z1 ⊃ .. ⊃ Zg-2 such that each Zi is irreducible, the classes [Zi] generate the Chow group of Mg additively (over Q) and each successive difference Zi -Zi+1 is affine. Here Mg stands for the moduli space of curves of genus g. Conjecturally a stratification enjoying the last property exists also for g>5.
(53) The period map for cubic fourfolds, Invent. Math. 177, 213-233 (2009).
Since Voisin proved the Torelli theorem for cubic fourfolds the question remained to characterize the image is of the period map (it was known not to be surjective). A precise conjecture regarding this image was proposed by Brendan Hassett. In this paper we prove his conjecture. An independent proof was given by Radu Laza (Ann. of. Math. 172, 673-711, 2010). We also give a new proof of Voisin’s Torelli theorem.
(54) Unitarity of SL(2) conformal blocks in genus zero, J. of Geometry and Physics, vol. 59, 654-662 (2009).
The (projectively flat) WZW-connection is supposed to be unitary, but an actual inner product has not been exhibited except in the first nontrivial case of the title. This is due to T.R. Ramadas (Ann. of Math. 169, 1-39, 2009). We offer here a somewhat simpler proof, which also yields a more precise result, as it characterizes the conformal block in question (together with its inner product) in terms of Hodge theory. This suggests that a flat unitary inner product relative the WZW-connection can always be defined via such an interpretation.
(55) (with Alex Boer) On the unitary nature of abelian conformal blocks, J. of Geometry and Physics, vol. 60, 205-218 (2010).
It is since long known that theta functions satisfy the heat equation and that this fact can be understood as the existence of a projectively flat connection on the vector bundle of theta functions over the moduli space of polarized abelian varieties. In case the abelian varieties are Jacobians it makes sense to ask whether this `heat’ connection can be identified with the WZW connection for the case of the multiplicative group. We show that the answer is yes (as was perhaps expected), and we also write down the flat inner product (that is conjectured to exist in general).
(56) (with Elisabetta Colombo and Bert van Geemen) Del Pezzo moduli via root systems, in: Algebra, Arithmetic, and Geometry: Volume I: In Honor of Y.I. Manin, Y. Tschinkel, Y. Zarhin eds., 289-335, Progr. Math. 269, Birkhaüser Verlag Basel (2010).
This paper can be somewhat cryptically summarized by saying that we embed the invariant theory of Coble in his treatise Algebraic Geometry and Theta functions (1929) in the theory of root systems (so that for instance the invariants of cubic surfaces become invariants of the E6 root system).
(57) Fermat varieties and the periods of some hypersurfaces, Advanced Studies in Pure Mathematics (Proceedings of Moduli 2007).
If one wants to set up a period map for smooth projective hypersurfaces of given dimension n and given degree d (at least 3), then it is convenient to have a base point. Especially if one wants to make the comparison between the one in dimension n and the next dimension n+1 (such a passage is obtained by taking the d-fold cover of the ambient projective (n+1)-space which ramifies along the hypersurface). With this in mind, the most natural choice of base point is given by a Fermat hypersurface. The first part of this paper determines the primitive cohomology of such a hypersurface with integral coefficients as a module over the integral group ring of its automorphism group (this had been done before over Q only). We use this to show that starting with the results of Laza and myself (53) on the period maps of cubic fourfolds, one can quickly descend to the period map for cubic threefolds and cubic surfaces and thus recover the results of Allcock-Carlson-Toledo and Looijenga-Swierstra (we also reproduce the period map for cubic one-folds).
(58) (with Wilberd van der Kallen) Spherical complexes attached symplectic lattices, Geom. Dedicata, 152, 197-211 (2011). The main result here is that if Ag denotes the stack of principally polarized abelian varieties of genus g and Ag,dec the substack which parametrizes the genuinely decomposable ones, then the pair (Ag, Ag,dec) is (g-2)-connected. This is a consequence of a certain simplicial complex being Cohen-Macaulay, but the latter property was not covered in a simple manner by the existing literature. We offer here a general result of the needed type.
(59) (with Gert Heckman) Hyperbolic structures and root systems, in: Casimir Force, Casimir Operators and the Riemann Hypothesis, G. van Dijk, M. Wakayama (eds.), 211-228, W. de Gruyter (2010).
Follow up of (45).
(60) The KZ-system via polydifferentials, in: Arrangements of hyperplanes (Sapporo 2009), 189-231, Advanced Studies for Pure mathematics of the JMS, Vol. 62.
We give a complete topological characterization of every KZ-system as a Gauss-Manin system. In order to make the exposition self-contained and transparent, we review and reprove some of the results of a paper of Schechtman-Varchenko.
(61) From WZW models to modular functors, in: Handbook of Moduli, vol II, G. Farkas, I. Morrison (eds.), 427-166, International Press (2013).
We develop in a concise and coordinate free manner the main properties of the WZW model and make the connection with topological quantum field theory.
(62) Connectivity of complexes of separating curves, Groups Geom. Dyn. 7 (2013), 443–450.
The complex in question is associated to an oriented connected surface S of genus g with n punctures such that it has negative Euler characteristic. A separating curve is an embedded circle in S whose complement is disconnected. The isotopy classes of such curves are the vertices of a simplicial complex that is helpful for analysing the homological properties of the Torelli groups. Farb and Ivanov announced in 2005 that this complex is connected for g ≥ 3, Putman improved this in 2008 that this is simply connected and Hatcher and Vogtmann (unpublised) have later shown that is about (g-3)/2-connected. We prove here among other things that this complex is (g-3)-connected (and even (g-2)-connected when n>1).
(63) The fine structure of the moduli space of abelian differentials in genus 3 (with G. Mondello), Geom. Dedicata 169 (2014), 109-128.
The space of holomorphic differentials of the universal curve of genus g ≥ 2 is naturally stratified (by looking at the type of the zero divisor of such a differential). Kontsevich and Zorich initiated the investigation of this stratified space and made a number of conjectures about them. We give here a complete description in genus 3. It is of a sufficiently topological character to make it feasable for a homotopy theorist to verify Kontsevich’s conjecture that theses strata are classifying spaces for their fundamental group. But for us it is not at all obvious that they have this property.
(64) Discrete automorphism groups of convex cones of finite type, Compositio Math. 150 (2014), 1939-1962.
Perhaps the main result of this paper is the following: let V be a real vector space of finite dimension and an open nondegenerate convex cone C in V. Then the convex hull of the intersection of C with a lattice L in V is locally polyhedral: its intersection with any bounded polyhedron is a polyhedron. Our main applications concern the siutuation where we are given a subgroup G of GL(V) which stabilizes C and L such that G has on C a fundamental domain that is spanned by a finite subset of L that lies in the closure of C. This is a situation that one encounters in the theory of (real) reflection groups and in reduction theory. It also often occurs when V is the Neron-Severi group tensored with R of a complex projective manifold M, C is the ample cone and G is the group of automorphisms of M.
(65) Moduli spaces and locally symmetric varieties, in: Development of Moduli Theory—Kyoto 2013, 33-75, Adv. studies in pure math, Math. Soc. Japan (2016).
We review the various compactifications of locally symmetric varieties: Satake-Baily-Borel, Mumford’s toric compactifications and the intermediate semitoric compactifications.
(66) (with Jiaming Chen) The stable cohomology of the Satake compactification of Ag, Geom. Topol. 21 (2017), 2231–2241.
(67) Goresky-Pardon lifts of Chern classes and associated Tate extensions, Compos. Math. 153 (2017), no. 7, 1349–1371.
(68) Moduli spaces of Riemann surfaces, 1–87, Surv. Mod. Math., 14, Int. Press, Somerville, MA, 2017.
(69) (with Igor Dolgachev and Benson Farb) Geometry of the Wiman-Edge pencil, I: algebro-geometric aspects. Eur. J. Math. 4 (2018), 879–930.
(70) (with Marco Boggi) Deforming a Canonical Curve Inside a Quadric, Int. Math. Research Notices, https://doi.org/10.1093/imrn/rny027
(As yet) unpublished preprints.
(a) Semi-toric partial compactifications I, Preprint series KUN Nijmegen, 72 p. (1985).
(b) (with E. Getzler) The Hodge polynomial of Mbar3,1, 4 p. (1999), available at math.AG/9910174