Siegel coordinates and moduli spaces for morphisms of Abelian varieties

We describe the moduli spaces of morphisms between polarized complex abelian varieties. The discrete invariants, derived from a Poincaré decomposition of morphisms, are the types of polarizations and of lattice homomorphisms occurring in the decomposition. For a given type of morphisms the moduli variety is irreducible, and is obtained from a product of Siegel spaces modulo the action of a discrete group. Mathematics Subject Classification (2000) 14K20     

A compactification of open varieties

In this paper we prove a general method to compactify certain open varieties by adding normal crossing divisors. This is done by showing that blowing up along an arrangement of subvarieties can be carried out. Important examples such as Ulyanov's configuration spaces and complements of arrangements of linear subspaces in projective spaces, etc., are covered. Intersection ring and (nonrecursive) Hodge polynomials are computed. Furthermore, some general structures arising from the blowup process are also described and studied.

     

A compactification of Igusa varieties

We investigate the notion of Igusa level structure for a one-dimensional Barsotti–Tate group over a scheme X of positive characteristic and compare it to Drinfeld’s notion of level structure. In particular, we show how the geometry of the Igusa covers of X is useful for studying the geometry of its Drinfeld covers (e.g. connected and smooth components, singularities). Our results apply in particular to the study of the Shimura varieties considered in Harris and Taylor (On the geometry and cohomology of some simple Shimura varieties. Princeton University Press, Princeton, 2001). In this context, they are higher dimensional analogues of the classical work of Igusa for modular curves and of the work of Carayol for Shimura curves. In the case when the Barsotti–Tate group has constant p-rank, this approach was carried-out by Harris and Taylor (On the geometry and cohomology of some simple Shimura varieties. Princeton University Press, Princeton, 2001).     

A compactification of the moduli space of polynomials

In this paper, we introduce a compactification of the moduli space of polynomial maps with a fixed degree such that the map from it to defined by using the elementary symmetric functions of multipliers at fixed points is a continuous surjection.

     

A compactification of the moduli space of twisted holomorphic maps

Given compact symplectic manifold X with a compatible almost complex structure and a Hamiltonian action of S¹ with moment map , and a real number K?0, we compactify the moduli space of twisted holomorphic maps to X with energy ?K. This moduli space parameterizes equivalence classes of tuples (C,P,A,?), where C is a smooth compact complex curve of fixed genus g, P is a principal S¹ bundle over C, A is a connection on P and ? is a section of PS¹_×X satisfying

     

Abelian varieties and the general Hodge conjecture

We investigate the relationship between the usual and general Hodgeconjectures for abelian varieties. For certain abelian varieties A, weshow that the usual Hodge conjecture for all powers of A implies thegeneral Hodge conjecture for A.     

Abelian and Hamiltonian varieties of groupoids

We study certain groupoids generating Abelian, strongly Abelian, and Hamiltonian varieties. An algebra is Abelian if t( a,[`(c)] ) = t( a,[`(d)] ) ? t( b,[`(c)] ) = t( b,[`(d)] ) t\left( {a,\bar{c}} \right) = t\left( {a,\bar{d}} \right) \to t\left( {b,\bar{c}} \right) = t\left( {b,\bar{d}} \right) for any polynomial operation on the algebra and for all elements a, b, [`(c)] \bar{c} , [`(d)] \bar{d} . An algebra is strongly Abelian if t( a,[`(c)] ) = t( b,[`(d)] ) ? t( e,[`(c)] ) = t( e,[`(d)] ) t\left( {a,\bar{c}} \right) = t\left( {b,\bar{d}} \right) \to t\left( {e,\bar{c}} \right) = t\left( {e,\bar{d}} \right) for any polynomial operation on the algebra and for arbitrary elements a, b, e, [`(c)] \bar{c} , [`(d)] \bar{d} . An algebra is Hamiltonian if any subalgebra of the algebra is a congruence class. A variety is Abelian (strongly Abelian, Hamiltonian) if all algebras in a respective class are Abelian (strongly Abelian, Hamiltonian). We describe semigroups, groupoids with unity, and quasigroups generating Abelian, strongly Abelian, and Hamiltonian varieties.     

On the Natanzon-Turaev compactification of the Hurwitz space

Natanzon and Turaev have constructed by topological methods a compactification of the Hurwitz space, that is, the space of simple branched covers of the two-sphere. Here we show that this compactification is homeomorphic to a compactification mentioned by Diaz and Edidin (in 1996) that was constructed by algebraic methods. Using this we are able to show by example that the Natanzon-Turaev compactification can be singular, that is, not a manifold.

     

A Hilbert cube compactification of the function space with the compact-open topology

Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)^ℕ of the Hilbert cube Q = [−1, 1]^ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $ \bar C $ \bar C (X) of C(X) such that the pair ($ \bar C $ \bar C (X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification $ \bar C $ \bar C (X) coincides with the space USCC _F (X,      

Abelian fourfold of Mumford-type and Kuga-Satake varieties

Let X be a complex abelian fourfold of Mumford-type and let V = H¹(X, ). The complex Mumford-Tate group of X is isogenous to SL(2)³. We recover information about the Hodge structure of X using representations of the Lie algebras (2)³ and (8) acting on V . Using these techniques we show that there is a Kuga-Satake variety A associated to X in such a way that A is isogenous to X³².     

A note on characterizations of Abelian varieties by topological invariants

In this note, for any given n3 and 2mn (when m=n, we assume n divides 3 and n6), we construct examples of smooth projective varieties X of dimension n with p_g(X)=1, ₁(X)²ⁿ and the Kodaira dimension (X)=m.Mathematics Subject Classification (2000):14H45, 14H99     

A smooth compactification of the space of transfer functions with fixed McMillan degree

It is a classical result of Clark that the space of all proper or strictly properp ×m transfer functions of a fixed McMillan degreed has, in a natural way, the structure of a noncompact, smooth manifold. There is a natural embedding of this space into the set of allp × (m+p) autoregressive systems of degree at mostd. Extending the topology in a natural way we will show that this enlarged topological space is compact. Finally we describe a homogenization process which produces a smooth compactification.This author was supported in part by NSF grant DMS-9201263.     

Torsion of Abelian varieties of finite characteristic

The finiteness of the torsion of Abelian varieties with a complete real field of endomorphisms in the maximal Abelian extension of the field of definition is proven. This assertion is formally deduced from the finiteness hypothesis for isogenic Abelian varieties, proven for characteristic p > 2. The structure is studied of the Lie algebra of Galois groups acting in a Tate module; in particular, for fields of characteristic greater than two there is proven one-dimensionality of the center of the Lie algebra.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 3–11, July, 1977.     

Abelian varieties over fields of finite characteristic

The aim of this paper is to extend our previous results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.