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
]