Brent Doran

Summary of Papers:

B. Doran and F. Kirwan, 2007, Towards non-reductive geometric invariant theory. Pure and applied mathematics quarterly 3, 61-105.

We develop the background, including a brief discussion of some motivating problems, for a Geometric Invariant Theory of non-reductive algebraic group actions. Most of the seemingly critical results about reductive groups fail badly for non-reductive groups, but we show that, even so, many ideas of GIT still hold true. Issues like non-finite generation of invariants are shown to be irrelevant as far as a theory of quotients is concerned, but nevertheless can still be addressed geometrically through a study of compactifications. We include a detailed discussion of stability in particular, which we show admits an intrinsic characterization (given the choice of a G-linearization, just as in reductive GIT).

A. Asok, B. Doran and F. Kirwan, Yang-Mills and Tamagawa numbers Bull. Lon. Math.Soc., to appear.

We apply our theory of motivic cohomology of quotients to the problem of moduli of vector bundles (of coprime rank and degree) on curves. By this approach to understanding the motive of BunG (for G = SLn) we try to give a geometric understanding of a result that number theorists find rather deep, namely that the Tamagawa number of SLn is one. We suggest how to extend the analysis to BunG for other reductive groups G, but do not investigate because of difficulties with singularities.

A. Asok and B. Doran , 2007, On unipotent quotients and some A1-contractible smooth schemes. Int. Math. Res. Papers, 1-51.

We characterize when the quotient of an affine variety by a unipotent action is a quasi-affine but not affine variety. Applying this to the ``simplest case" of additive group actions on affine space, we conclude that ``essentially any" variety can arise as the complement of the image of a quotient of such an action in Spec of the ring of invariant regular functions. In particular, this is an interesting statement about the size and complexity of Aut(A^n). It is instructive to compare with the smooth category, where Diff(R^n) contains enough distinct additive R^+ actions to characterize any contractible manifold as the quotient of R^n by an R^+ action (using results of Stallings, h-cobordism, and the Poincare conjecture).

We construct arbitrarily large dimensional families of non-isomorphic smooth exotic (or "fake") affine spaces in every dimension greater than or equal to six, and countably many nonisomorphic exotic affines in dimensions four and five. These are all A1-contractible, so no algebraic cohomology theory (motivic cohomology, algebraic K-theory, higher Chow groups, etc.) can distinguish among them. Indeed, we can show that when these are affine (as opposed to quasiaffine), then up to isomorphism, all vector bundles are isomorphic to trivial ones (the analogy with topology is that A1-homotopy classes of maps from X into BGLn should be in bijection with isomorphism classes of algebraic vector bundles on X ).

The basic conclusion: even the most sophisticated known invariants miss an enormous amount of structure in algebraic geometry. Optimistic corollary: we should look to methods of topology, adapted to algebraic geometry perhaps via A1-homotopy theory, for our classification theorems.

We conjecture, in well-motivated analogy with results of Stallings in classical smooth topology, that all A1-contractible schemes can be constructed in the manner we describe, as certain generalized "unipotent quotients" of affine space.

In producing these examples, we derive a canonical decomposition of Ga-invariants into sums of two parts; this decomposition admits a nice geometric interpretation in terms of boundaries of quotients, and seems to be new (although perhaps it is buried in disguise in the classical literature on covariants).

A. Asok and B. Doran, Vector bundles on contractible smooth schemes. Duke Math. J., to appear.

We show that for A1-contractible smooth quasi-affine schemes which are not affine, under very mild (and probably removable) conditions, there always exists a non-trivial vector bundle in every sufficiently large rank. Indeed, we show that for every m and n, there exist n dimensional families of A1-contractible varieties over any field, each variety in the family equipped with at least m-dimensional moduli of non-isomorphic vector bundles in every sufficiently large rank. In particular, A1-invariance of isomorphism classes of vector bundles breaks down arbitrarily badly if one leaves the category of affine schemes. None of these vector bundles can be distinguished from the trivial bundle by any of the standard invariants, e.g., Chern classes, algebraic K -theory, algebraic cycles, etc. This is another sign that enormous amount of algebro-geometric structure is missed by our usual tools, and even by A1-homotopy theory, and is one way in which smooth algebraic geometry in the motivic homotopy category behaves genuinely differently from topological manifolds in the ordinary homotopy category.

We also prove triviality of vector bundles for certain smooth topologically contractible but not A1-contractible varieties, and indeed show there are moduli of these.

A. Asok, and B. Doran, A1-homotopy groups, excision, and solvable quotients. Submitted.

We study some properties of A1-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of geometric quotients by solvable groups in terms of covering spaces in the sense of A1-homotopy theory. These concepts and results are well-suited to the study of certain quotients via geometric invariant theory. As a case study in geometry of solvable group quotients, we focus on A1-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree A1-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases we can actually compute the "next" non-vanishing A1-homotopy group (beyond p1A1) of a smooth toric variety. From this point of view, A1-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost "as tractable" (in low degrees) as ordinary homotopy for large classes of interesting varieties.

A. Asok and B. Doran. On unipotent quotients, Zariski cancellation and affine A1-contractible smooth schemes. In preparation.

If X × A1 = An, is X isomorphic to An-1? This famous question in affine geometry is known as the Zariski Cancellation Problem. Any such X must be A1-contractible. We prove that X is a member of an explicitly computable family of hypersurfaces in an affine space, thus reducing the problem to determining whether or not a given type of hypersurface is isomorphic to affine space. The case where X is dimension 3 is completely addressed. We are in the process of computing the Makar-Limanov invariant for the higher dimensional hypersurfaces to try to distinguish some from affine space and thus disprove the conjecture in higher dimensions. The problem is viewed as a special application of effective geometric and algebraic characterizations we give for when the quotient under a unipotent action of an affine variety is itself an affine variety.

A. Asok and B. Doran, On A1-homotopy types of affine contractible schemes. In preparation.

We analyze in detail the application of Zariski's affine modifications (an affine version of blowups) to produce topologically contractible complex varieties which are not A1-contractible. We know such exist and give explicit examples, but the optimistic goal is to use A1-homotopy groups to distinguish among a large class of topologically contractible varieties. Conjecturally, all A1-contractible smooth affine schemes can be produced by unipotent quotient techniques, and so in particular admit presentations as explicit hypersurfaces in affine space.

B. Doran, Free Ga actions and invariant theory. In preparation.

We show that the study of invariants for affine algebraic group actions on affine varieties can always be replaced by the study of everywhere stable (in the sense of non-reductive GIT) Ga actions on affine varieties. We furthermore characterize (and make computationally effective) when the geometric quotient by a free action is a variety and when it is not even a scheme but rather a separated algebraic space (some fairly simple quotients of An are shown to be examples). We show Zariski's generalization of Hilbert's 14th Problem is also of this type, and we analyze the question of finite generation of rings from this perspective.

B. Doran and F. Kirwan, Effective non-reductive geometric invariant theory. In preparation.

We make the techniques of non-reductive GIT fully effective, so that it can be used with roughly comparable ease as reductive GIT. Functorial properties are discussed. A generalized Hilbert-Mumford criterion is given. A canonical (given a G-linearization) compactification of the quotient is introduced, which from one point of view is a generalization of the theory of symplectic implosion of Guillemin, Jeffrey, and Sjamaar. A canonical finitely generated subring of the ring of invariants for any action of an affine algebraic group on an affine variety is likewise given.

A. Asok, B. Doran and F. Kirwan, Equivariant motivic cohomology and quotients, (first paper in a sequence, to be submitted after some reworking of the structure/exposition)

We show that the procedure for computing ordinary cohomology of quotients in the stable equals semistable setting holds for an enormous class of cohomology theories, including algebraic K-theory, motivic cohomology, etc. Many disparate results in the literature, including some from just the past couple of years, are subsumed. Much of the work is done in the A1-homotopy (or "motivic homotopy") category, both for its strength and generality, and also in an attempt to capture the topological flavour of the subject. In this spirit we conjecture that there is a (rational) splitting of the motive of the equivariant Borel model XG induced by the Kempf-Hesselink-Kirwan instability stratification.

A. Asok, B. Doran and F. Kirwan, Moduli of torsion-free coherent sheaves on smooth algebraic varieties. In preparation.

We derive a generating function for the ordinary Poincare polynomial of the moduli space of matrix divisors, in the sense of Weil-Grothendieck and following a construction of Bifet-Ghione-Letizia. We hope to derive from this a generating function for the Poincare polynomial of the moduli space of torsion-free coherent sheaves on any algebraic surface. This is a natural GIT compactification of the moduli space of vector bundles on a surface, and may be of interest in higher dimensional Geometric Langlands.

A. Asok and B. Doran, Motivic homotopy of smooth affine quadrics. In preparation.

By generalizing the unipotent quotient construction of affine space, we show that smooth affine quadrics can be built out of unions of A1-contractible quasi-affine varieties and in particular have the A1-homotopy type of motivic spheres. We also explore analogs of the J homomorphism in this setting, and suggest that this is the proper setting for realizing the hopes of Atiyah and Baum relating stable homotopy groups of spheres to polynomial maps between quadrics and spaces of quadratic forms.

A. Asok and B. Doran, Constructing highly "rationally" connected smooth hypersurfaces. In preparation.

We show how, by careful use of excision results we have proved and explicit Ak-bundles realizing the Jouanolou-Thomason homotopy lemma, one can construct large families of A1-k-connected affine hypersurfaces for any k . We suggest that, in analogy with results of Kollar et al. for rational connectivity and de Jong and Starr for simple rational connectivity, that, for d the degree of a smooth hypersurface X in An, if dm is less than or equal to n then X is A1-(m-1)-connected.

A. Asok and B. Doran, Hodge conservation conjecture and A1-homotopy types of contractible smooth varieties. In preparation.

It is expected, using a form of the conservation conjecture for the Hodge realization functor, that the motive of a topologically contractible smooth complex variety is that of a point, at least rationally. For contractible surfaces this is here established (indeed it is true integrally). However, the A1-homotopy type is far stronger: e.g., the unique A1-contractible surface is A2. We explore the extent to which A1-homotopy, and in particular pi0A1, distinguishes among contractible varieties, and establish for certain families of examples that the motive is indeed nonetheless that of a point.

B. Doran, Moduli space of cubic surfaces as ball quotient via hypergeometric functions, (math.AG/0404062)

We use generalized hypergeometric functions, in the sense of Deligne and Mostow, to show that the moduli space of cubic surfaces is a complex hyperbolic manifold, that is, a quotient of the complex ball by an explicitly presented arithmetic group. This result had been established originally by Allcock, Carlson, and Toledo (without recourse to hypergeometric functions), and through K3 surface moduli by Dolgachev, van Geeman, and Kondo. However, this approach is quick, clean, and relies only on simple classical geometry.

B. Doran, Hurwitz spaces and moduli spaces as ball quotients via pull-back, submitted. (math.AG/0404062)

We approach the theory of hypergeometric functions due to Deligne and Mostow via the Hodge theory of intersection cohomology valued in a local system. We observe that the original tables of Deligne and Mostow make errors of omission (which William Thurston had independently and much earlier, in a circulating but unpublished paper, observed and corrected from a different point of view). We apply their theory to inherit uniformizations for certain subvarieties of Hurwitz spaces, and thence some moduli spaces like the moduli space of rational elliptic surfaces (proved originally from a different perspective in a paper of Looijenga and Heckman). All of the discrete groups involved are explicitly presentable as monodromy groups, and most (but not all) are arithmetic.

B. Doran , Cohomology and resolutions of algebraic quotients. Solicited for subseries "Algebraic Transformation Groups and Invariant Theory", Encyclopaedia of Mathematical Sciences.