Old and new perspectives on higher dimensional classification ==================== MSRI farewell performance A better title might be A 1980s view of 21st century classification of varieties Moral: don't ask old generals to fight new wars. ``The mission needs new thinking.'' Contents 1. History 2. Change of generation and Lazi{\'c}'s work 3. Lower bound on canonical volume 4. Plurigenus formula 5. A path to abundance? 6. Graded rings for KX and D 7. What's in store for Fano or CY 4-folds? 8. Toric SL(4) quotients, crepant resolutions, A-Hilb. ------------ 1. History If you write out the history of 19th cent alg geom, you might get 1831 Pluecker's book on PP^2 and its dual and the Pluecker formulas 1849 Cayley and Salmon discover the 45 tritangent planes to the cubic surface 1857 first form of Riemann-Roch 1845-1890s Explicit invariant theory (many contributors) 1885-1890 Hilbert on invariants ``Gordan sprach davon, daß dies keine Mathematik, sondern Theologie sei.'' (A Google snippet, not any attempt at serious history.) Hilbert proved general theorems about finite generation with theoretical methods that do not involve finding explicit generators. In other words, you can say something about all invariants without actually knowing how to calculate a single one. From the late 1970s Mori, Reid and others do MMP for 3-folds with a mass of explicit methods and examples; e.g., explicit classification of terminal singularities, hundreds of examples of QQ-Fano 3-folds and birational maps between them, and much more. In 2005--2006 (following Shokurov and Siu) Hacon and McKernan, then Birkar, Cascini, Hacon and McKernan prove MMP in all dimensions by abstract methods, without doing any examples or explicit calculations. ``not mathematics but theology'' I guess that Paul Gordan (1837-1912, the ``King of Invariant Theory'') understood perfectly well the significance of Hilbert's results and methods, and was not disputing its status as a magnificent breakthrough (although Google says he criticised the presentation quite fiercely and Hilbert was obliged to stand his ground). As far as I am concerned, there is no criticism of the new regime, just natural initial disappointment that what is a theoretical solution to a vast array of long-standing problems is primarily only an existence theorem, and does not (as it stands) tell us how to construct anything explicitly, or help out with the calculations I am currently stuck on. There are of course many difference between 1890 and 2009, starting off with the global scale of our current endeavour and the ease of communication. ------------ 2. Change of generation and Lazi{\'c}'s work At Bowdoin in 1985 I gave my ``dinosaur'' talk, and outlined some aspects of the MMP as a series of conjectures (this was also in [Pagoda], Section 4). At the time, this was a generational change from the Iitaka program to Mori theory; the Iitaka program worked primarily with holomorphic differentials, such as 1-forms or pluricanonical forms (as did Castelnuovo and Enriques), whereas Mori theory works in the first instance with numerical properties of the canonical class. A main item of MMP that has been the main orthodoxy since 1980 is that the MMP proceeds recursively -- e.g. on a surface of general type, you contract -1-curves one at a time to get a model with K nef and big. Given a rational curves with C^2 <= -2, it has KC >= 0, and whether or not it will be contracted on passing to a minimal model can't be predicted in advance, but is determined in the course of running the MMP. You can say Zariski decomposition if you prefer, but this is also proved recursively. Corti and Vlad Lazi{\'c}'s idea overturns this ``main orthodoxy''; this is probably the next generational change. Lazi{\'c}'s claim [Adjoint rings are finitely generated, arXiv:0905.2707] is that you can prove f.g. of general adjoint rings directly without the MMP. If correct, this is a major simplification, and makes a fraction of the MMP literature redundant. I sketch the idea in colloquial style for those of you who don't have time to download the paper. An adjoint ring is a multigraded ring R(X, Si), where Si is a f.g. semigroup made up of adjoint divisors, i.e., Si = , where each D_i is the adjoint divisor D_i = KX + De_i + A of a Klt divisor De_i+A (here Si = Sigma and De = Delta; each De_i is an effective QQ-divisor and A is a small general ample QQ-divisor). The main claim is that any such R(X, Si) is a finitely generated multigraded ring. The idea of the proof is to think of Si as a rational polyhedral cone, and cut it up into a fan of subcones, each of which admits a nice restriction to a log centre. For each subcone, we write down a restriction sequences, having a surjective restriction map (by Siu's extension theorem, as reworked by Hacon and McKernan) to an adjoint ring in smaller dimension (so finitely generated by induction), and kernel a principal ideal generated by the equation of the restriction divisor, which is an element of our multigraded ring by construction. Assuming all that works, there is still a small barrage of lemmas involved in passing between a fan of finitely generated cones and the whole of Si, (basically comparable to Gordan's lemma). There are many technical details in the above argument that require study, but Corti and Lazi{\'c} believe that it works. ------------ 3. Lower bound on canonical volume Hacon and McKernan have proved a theoretical lower bound on the ``canonical volume'' of an n-fold of general type for any n (depending only on the dimension). However, their result is completely ineffective and likely to remain so. In dimensions 1, 2, 3 we know the bound and the champions, namely C6 in PP(1,1,3), a canonical curve of genus 2 S(10) in PP(1,1,2,5), a canonical surface with pg = 2, K^2 = 1 V(46) in PP(4,5,6,7,23), a canonical 3-fold with pg = P2 = P3 = 0, KX^3 = 1/420, and fie(26K) not birational. (By the way, I believe that this example is not just world champion, but beats its nearest competitor by a factor of 2.) I hope that with a few years' computation, one can write out explicit lower bounds for 4-folds of the order of KX^4 >= 1/(million), and that toric complete intersections (more specifically, weighted hypersurfaces, superficially similar examples to those in dimension 1, 2, 3) will be candidates for ``world champion'', with canonical volume of the same order. My expected bound comes from a simplistic analogy with the traditional question of sums of reciprocals: 1-1/2-1/3 = 1/6 1-1/2-1/3-1/7 = 1/42 1-1/2-1/3-1/7-1/43 = 1/1806 1-1/2-1/3-1/7-1/43-1/1807 = 1/3263442 1-1/2-1/3-1/7-1/43-1/1807-1/3263443 = 1/10650056950806 This numerology appear in classification problems all over geometry, starting from Felix Klein's discovery of the (2,3,7) triangle group around 1870. For example, the orbifold X = PP^4 with 6 general orbifold hyperplanes of degree 2, 3, 7, 43, 1807, 3263443 has K_{X,orb} numerically (1/N)H, where N = 10650056950806 approx 10^13, so K^4 = 1/N^4 approx 10^{-52}. This very low bound comes because we allow codimension 1 orbifold behaviour, whereas canonical varieties only have quotient stuff going on in codimension >= 3. I give some arguments (admittedly feeble) for this guesswork. First, note that if H^0(Om^1) <> 0 then we get an Albanese variety and a positive dimensional Abelian variety Pic^0 X, either of which provides simple, practical and technically powerful methods of attacking X -- either the Albanese fibration or the paracanonical systems will give methods of increasing the canonical volume, probably by some large amount (e.g. a positive integer). We don't really know how to exploit nonzero i-forms in H^0(Om^i) for 2 <= i < n in the same convenient way, but my guess is that these should again give some quite coarse increase in the canonical volume. Therefore, my tentative conclusion is that the bound should be smallest for X with H^0(Om^i) = 0, that is, having Gorenstein canonical ring. The simplest of these, and those with smallest growth are likely to be weighted hypersurfaces. As a very rough indication of what to expect, choose (more or less at random) the case of the general hypersurface X(112) in PP(9,10,11,12,13,56) with K = Oh(1) and K^4 = 1/77220. The toric and Newton polygon methods of [YPG], together with some fairly large computational effort, will determine whether or not this X has canonical singularities. (I'm mostly worried about the point (0,0,0,1,0,0), which seems not to have very many monomials of low degree.) If this guy doesn't work, some similar ones will. ------------ 4. Plurigenus formula Compare [YPG], Chap. III for the background to the problem and its solution for surfaces and 3-folds, and see my question in the MSRI problem session, online at http://www.msri.org/calendar/attachments/programs/251/Questions2.pdf A 4-fold Mori minimal model has curves of 3-fold terminal singularities, with the orbifold contributions from curves of 1/r(1,a,r-a) points presumably typical; the tricky point here is that the normal bundle to such a curve is a direct sum of three isotypical line bundles, and the degrees of the three factors will appear in their plurigenus contributions (not just their symmetric combinations). If X has only orbifold points (i.e., is nonsingular as a stack), these curves can meet transversally in embedded dissident points, up to 4 at a time. The harder point is to deal with more general QQ-Gorenstein singular points; the standard wisdom of the 1980s was that 4-fold terminal singularities are ``intractable''. I hope that the main case is when the index 1 cover is locally complete intersection. After jiggling the coefficients a bit, assume that the singularities are nondegenerate for their Newton polyhedra. The condition to be terminal is toric (although at present not especially easy to calculate with), and it seems to me quite likely that the main point is to sort out the group action on the ambient space: more non-isolated Dedekind sums as in [YPG], then passing to the quotient by a regular sequence. These calculations should at least be well suited to finding canonical 4-folds as weighted hypersurface, which is where I hope the champions live. ------------ 5. A path to abundance? This section is definitely an ``old perspective'': for my view of the proof of abundance for surfaces and its difficulties, see my Park City [Chapters], E.9.6. The strategy of trying to construct the K numerically trivial fibration geometrically before studying the Kodaira dimension was discussed in [Pagoda], Section 4. The proofs of abundance in the surface and 3-fold cases are logically many-layered, and depend on low-dimensional features at several points. For example, the surface case uses that K^2 = c_2 = 0 implies that q>0, so a nontrivial Pic^0 and Albanese map. It seems rather implausible that anyone will ever succeed in doing anything similar in n dimensions, or even for 4-folds. It is slightly risky to talk about abundance, since I may give the impression of trying to claim credit for intuition about problems that are completely outside my experience. For example, it is doubtless pretentious to talk about abundance in dimension >= 4 when one has not spent any time analysing the colourful terms in the Todd class: Td4 = 1/720*(-c1^4 + 4*c1^2*c2 + c1*c3 + 3*c2^2 - c4), Td5 = 1/1440*(c1^3*c2 - c1^2*c3 - 3*c1*c2^2 + c1*c4), Td6 = 1/60480*(2*c1^6 - 12*c1^4*c2 + 5*c1^3*c3 + 11*c1^2*c2^2 - 5*c1^2*c4 + 11*c1*c2*c3 - 2*c1*c5 + 10*c2^3 - 9*c2*c4 - c3^2 + 2*c6), etc. The problem here is not just that there are many terms, most of which have no clear-cut geometric interpretation. But even if the sign of Td were given (say chi(OhX) = Td4 >= 100), we wouldn't immediately get a usable conclusion for H^0(KX): it might still happen that H^2(OhX) and H^0(Om^2) account for all of it, so that we get a large number of holomorphic 2-forms that we just don't have any idea how to use. Abundance will probably be solved at some future point, but not by this path. I propose instead to use what we are given: if X is a Mori minimal model (projective, QQ-factorial terminal singularities, and K nef), the numerical dimension nu(X) = nu(KX) is essentially by definition the number nu such that for some ample divisor H, H^0(H + lK) grows as c x l^nu as l -> infinity with c > 0 (this uses Kodaira vanishing). If nu = n then K is big and there is nothing to prove. If nu = 0 (that is, K numerically 0), abundance is also known by Kawamata's argument on the irregularity: if q > 0 there is an Albanese fibration, and we use some form of additivity. Otherwise Pic is finitely generated, so numerically zero implies some multiple is linearly zero. Assume that nu is between 1 and n-1. If abundance is true, the Iitaka fibre space X -> Y maps to a nu-fold Y and contracts exactly the curves with KX.C = 0, or more generally, the i-dimensional subvarieties with K|V numerically trivial. My proposal is to look directly for this K numerically trivial fibration. Write fie_l for the map defined by |H+lK|. I want a statement saying that if these maps for all l >> 0 are not correlated then necessarily K is ample. Say that a subscheme S in X is special for fie_l if the restriction map to H^0(Oh_S(H+lK)) has small rank; under appropriate numerical assumptions, lots of these exist for fixed l. One possibility is to ask for subschemes S of X (starting with 0-dimensional ones) that are special for several l >> 0, so candidates for being on a subvariety with K|V numerically trivial. Of course, I can't make this work. Another possible strategy, in the style of Shokurov, would be to force a divisor in |H+lK| to have a singularity of large multiplicity at some point P. The abundance question in dimension n-1 is a necessary step for log canonical models in dimension n. A key example that bears repetition is Zariski's famous counterexample to finite generation: PP^2 blown up in 10 or more general points on an elliptic curve. The elliptic curve E on the blown up surface has negative self-intersection, but cannot be contracted algebraically because the surface has no nonzero divisor class that restricts to zero on E. In this case, we know that KS+E is analytically trivial near E because we know abundance in dimension one. ------------ 6. Graded rings for KX and D Studying the canonical ring of a variety of general type as a graded ring applies (esp. when it is Gorenstein) to give lots of nice examples of varieties and their moduli, so why not the log canonical ring and log pairs? To my knowledge, it is an open problem to find any substantial cases in which this can be used effectively. The point of the question is that knowing good properties of KX and D, there should be methods in appropriate cases of studying X and D (including questions such as existence and moduli) by generators and relations for their graded rings. If X is a variety, D a smooth divisor, and both D and KX + D ample (say), then R(X, KX+D) is not Gorenstein, but it is closely related to Gorenstein rings in several ways. E.g. the orbifold canonical ring R(X, KX + (r-1)/r D) (for any r) is likely to be Gorenstein, as is the canonical ring of two copies of X (or two different varieties X, X' containing D) glued along D, that restricts to R(X, KX+D). I've been accused quite recently of advocating that everything should be embedded into weighted projective space. However, if a variety X is given with several different Cartier divisors, it might be even better to construct a multigraded ring (as Lazi{\'c}'s adjoint rings in Section 2 above). For example, there might in good cases be a multigraded ring involving nKX+mD for an appropriate subcone of that is Gorenstein (or better still, a hypersurface) and includes R(X, KX+D). ------------ 7. What's in store for Fano or CY 4-folds? My point: alongside developments in the MMP, governed by the sign of KX, and more precisely by the Mori cone and canonical models, the last 25 years have seen astonishing parallel developments in mirror symmetry and geometry of CY 3-folds, that amount to ``special geometry'' or ``CY3 magic''. This is now coming into focus with symmetric deformation theories (the obstruction space T^2 is dual to first order deformations T^1, so that the moduli spaces are virtually 0-dimensional), the comparison of Gromov-Witten versus Donaldson-Thomas invariants, mirror symmetry and so on). CY3 magic has leaked out even to pure algebra, with CY3 algebras defined as noncommutative rings having generating relations dual to the generators. One must expect Fano 4-folds and CY 4-folds to display new and quite different phenomena, and remain on the alert for indications of these. ------------ 8. Toric SL(4) quotients, crepant resolutions, A-Hilb. Crepant resolutions exist for orbifolds by finite subgroups of SL(3); this was proved case-by-case by Ito, Roan, Markushevich and others, and without case division by derived category tricks in [BKR]. The phenomenon can presumably be explained away as part of the CY3 magic. The 4-fold case is nothing like that simple and consistent. I now have some preliminary experience of trying to resolve orbifolds CC^4/A by a diagonal cyclic group A = 1/r(a1,a2,a3,a4) in SL(4) (that is, 4-fold cyclic Gorenstein orbifolds). Some cases have a crepant resolution, and some don't; a necessary condition is that the group has sufficiently many junior elements (so that every age 2 element is a sum of two juniors). That condition is not sufficient in a very few cases -- Sarah Davis found the first counterexample 1/39(1,5,25,8), and there are about 10 more up to r = 125. It is possible that a crepant resolution exists if and only if there exists a crepant resolution obtained as a chain of barycentric subdivisions in junior points. Following Nakamura and [BKR], one could also ask about A-Hilb CC^4. I can compute its affine pieces and their discrepancy by computer algebra. However, the answer is messy -- in some cases it is a crepant resolution; in others it is nonsingular and with quite small discrepancy (so a reasonably simple blowup of a crepant resolution); in others it has a few pretty easy singularities and quite small discrepancy. However, there seem to be exuberant cases when A-Hilb CC^4 is very complicated -- for example 1/30(1,6,10,13) has about 150 affine pieces, and many irreducible components. I have some preliminary notes and Magma routines for anyone who is interested. P.S. Dmitrios Dais points out that the counterexamples such as 1/39(1,5,25,8) are already contained in the Diplomarbeit of Robert Firla: R. Firla: Algorithms for Hilbert-cover and Hilbert-partition problems (in German), Diplomarbeit, TU-Berlin, 1997. R. Firla and G. Ziegler: Hilbert bases, unimodular triangulations, and binary covers of rational polyhedral cones, Discrete and Computational Geometry 21 (1999) 205-216. ------------ Magma code for Todd classes dd :=8; Td := [Coefficient(S!(s/(1-Exp(-s))),i) : i in [0..dd]]; // Bernouilli numbers RR := PolynomialRing(Q,dd); Tda := [&+[Td[i]*a^(i-1) : i in [1..7]] : a in [RR.i : i in [1..dd]]]; T := &*Tda; SS := PolynomialRing(Q,dd); // this is the ring containing the Chern classes Todd := [SS!1]; for i in [1..dd] do Tdi := &+[m : m in Terms(T) | WeightedDegree(m) eq i]; x,y := IsSymmetric(Tdi,SS); Append(~Todd,y); end for; Todd; [ 1, 1/2*c1, 1/12*c1^2 + 1/12*c2, 1/24*c1*c2, -1/720*c1^4 + 1/180*c1^2*c2 + 1/720*c1*c3 + 1/240*c2^2 - 1/720*c4, -1/1440*c1^3*c2 + 1/1440*c1^2*c3 + 1/480*c1*c2^2 - 1/1440*c1*c4, 1/30240*c1^6 - 1/5040*c1^4*c2 + 1/12096*c1^3*c3 + 11/60480*c1^2*c2^2 - 1/12096*c1^2*c4 + 11/60480*c1*c2*c3 - 1/30240*c1*c5 + 1/6048*c2^3 - 1/6720*c2*c4 - 1/60480*c3^2 + 1/30240*c6, 1/60480*c1^5*c2 - 1/60480*c1^4*c3 - 1/12096*c1^3*c2^2 + 1/60480*c1^3*c4 + 11/120960*c1^2*c2*c3 - 1/60480*c1^2*c5 + 1/12096*c1*c2^3 - 1/13440*c1*c2*c4 - 1/120960*c1*c3^2 + 1/60480*c1*c6, 1/362880*c1^5*c3 + 1/362880*c1^4*c2^2 - 1/362880*c1^4*c4 - 1/51840*c1^3*c2*c3 + 17/3628800*c1^3*c5 - 1/90720*c1^2*c2^3 + 53/3628800*c1^2*c2*c4 + 13/1209600*c1^2*c3^2 - 17/3628800*c1^2*c6 + 61/1814400*c1*c2^2*c3 - 1/56700*c1*c2*c5 - 61/3628800*c1*c3*c4 + 1/134400*c1*c7 + 1/134400*c2^4 - 29/1814400*c2^2*c4 - 1/113400*c2*c3^2 + 37/3628800*c2*c6 + 1/134400*c3*c5 + 17/3628800*c4^2 - 1/134400*c8 ]