We study the isospectral Hilbert scheme , defined as the reduced fiber product of with the Hilbert scheme of points in the plane , over the symmetric power . By a theorem of Fogarty, is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over . We derive two important consequences.
(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients . This establishes the Macdonald positivity conjecture, namely that .
(2) We show that the Hilbert scheme is isomorphic to the -Hilbert scheme of Nakamura, in such a way that is identified with the universal family over . From this point of view, describes the fiber of a character sheaf at a torus-fixed point of corresponding to .
The proofs rely on a study of certain subspace arrangements , called polygraphs, whose coordinate rings carry geometric information about . The key result is that is a free module over the polynomial ring in one set of coordinates on . This is proven by an intricate inductive argument based on elementary commutative algebra.
Averaging lemmas deduce smoothness of velocity averages, such as
from properties of . A canonical example is that is in the Sobolev space whenever and are in . The present paper shows how techniques from Harmonic Analysis such as maximal functions, wavelet decompositions, and interpolation can be used to prove versions of the averaging lemma. For example, it is shown that implies that is in the Besov space , . Examples are constructed using wavelet decompositions to show that these averaging lemmas are sharp. A deeper analysis of the averaging lemma is made near the endpoint .
Our results are statistical and asymptotic in the degree of the polynomials. We equip the space of polynomials of degree in complex variables with its usual SU-invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope . We then determine the asymptotics of the conditional expectation of simultaneous zeros of polynomials with Newton polytope as . When , the unit simplex, it is clear that the expected zero distributions are uniform relative to the Fubini-Study form. For a convex polytope , we show that there is an allowed region on which is asymptotically uniform as the scaling factor . However, the zeros have an exotic distribution in the complementary forbidden region and when (the case of the Bernstein-Kouchnirenko Theorem), the expected percentage of simultaneous zeros in the forbidden region approaches 0 as .
Let be either or the one point blow-up of . In both cases carries a family of symplectic forms , where -1$"> determines the cohomology class . This paper calculates the rational (co)homology of the group of symplectomorphisms of as well as the rational homotopy type of its classifying space . It turns out that each group contains a finite collection , of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups ``asymptotically commute", i.e. all the higher Whitehead products that they generate vanish as . However, for each fixed there is essentially one nonvanishing product that gives rise to a ``jumping generator" in and to a single relation in the rational cohomology ring . An analog of this generator was also seen by Kronheimer in his study of families of symplectic forms on -manifolds using Seiberg-Witten theory. Our methods involve a close study of the space of -compatible almost complex structures on .
I will also show that there is a function such that if are uncountable and , then there are in and respectively with . Previously it was unknown whether such a function existed even if was replaced by . Finally, I will prove that there is no basis for the uncountable regular Hausdorff spaces of cardinality .
The results all stem from the analysis of oscillations of coherent sequences of finite-to-one functions. I expect that the methods presented will have other applications as well.