The Hodge conjecture for certain moduli varieties

For smooth projective varietiesX over ℂ, the Hodge Conjecture states that every rational Cohomology class of type (p, p) comes from an algebraic cycle. In this paper, we prove the Hodge conjecture for some moduli spaces of vector bundles on compact Riemann surfaces of genus 2 and 3.     

Direct images of bundles under Frobenius morphism

Let X be a smooth projective variety of dimension n over an algebraically closed field k with char(k)=p>0 and F:X→X ₁ be the relative Frobenius morphism. For any vector bundle W on X, we prove that instability of F _* W is bounded by instability of W⊗T^ℓ(Ω¹ _X ) (0≤ℓ≤n(p-1)) (Corollary 4.9). When X is a smooth projective curve of genus g≥2, it implies F _* W being stable whenever W is stable. Dedicated to Professor Zhexian Wan on the occasion of his 80th birthday.     

The Gersten conjecture for Milnor K-theory

We prove that the n-th Milnor K-group of an essentially smooth local ring over an infinite field coincides with the (n,n)-motivic cohomology of the ring. This implies Levine’s generalized Bloch–Kato conjecture.     

Transcendence degree of division algebras

We define a transcendence degree for division algebras by modifying the lower transcendence degree construction of Zhang. We show that this invariant has many of the desirable properties one would expect a noncommutative analogue of the ordinary transcendence degree for fields to have. Using this invariant, we prove the following conjecture of Small. Let k be a field, let A be a finitely generated k-algebra that is an Ore domain, and let D denote the quotient division algebra of A. If A does not satisfy a polynomial identity, then GKdim(K) ≤ GKdim(A) − 1 for every commutative subalgebra K of D.     

Weak approximation over function fields of curves over large or finite fields

Let K = k(C) be the function field of a curve over a field k and let X be a smooth, projective, separably rationally connected K-variety with ${X(K)\neq\emptyset}Let K = k(C) be the function field of a curve over a field k and let X be a smooth, projective, separably rationally connected K-variety with X(K) 1 ?{X(K)\neq\emptyset}. Under the assumption that X admits a smooth projective model p: X? C{\pi: \mathcal{X}\to C}, we prove the following weak approximation results: (1) if k is a large field, then X(K) is Zariski dense; (2) if k is an infinite algebraic extension of a finite field, then X satisfies weak approximation at places of good reduction; (3) if k is a nonarchimedean local field and R-equivalence is trivial on one of the fibers X_p{\mathcal{X}_p} over points of good reduction, then there is a Zariski dense subset W í C(k){W\subseteq C(k)} such that X satisfies weak approximation at places in W. As applications of the methods, we also obtain the following results over a finite field k: (4) if |k| > 10, then for a smooth cubic hypersurface X/K, the specialization map X(K)? ?_{p ? P}X_p(k(p)){X(K)\longrightarrow \prod_{p\in P}\mathcal{X}_p(\kappa(p))} at finitely many points of good reduction is surjective; (5) if char k 1 2, 3{\mathrm{char}\,k\neq 2,\,3} and |k| > 47, then a smooth cubic surface X over K satisfies weak approximation at any given place of good reduction.     

On degrees of maps between Grassmannians

Let G_n,k denote the oriented grassmann manifold of orientedk-planes in ℝⁿ. It is shown that for any continuous mapf: G_n,k → G_n,k, dim G_n,k = dim G_m,l = l(m −l), the Brouwer’s degree is zero, providedl > 1,n ≠ m. Similar results for continuous mapsg: ℂG_m,l → ℂG_n,k,h: ℍG_m,l → ℍG_n,k, 1 ≤ l < k ≤ n/2, k(n — k) = l(m — l) are also obtained.     

The K-theory of fields in characteristic p

We show that for a field k of characteristic p, H ⁱ (k,ℤ(n)) is uniquely p-divisible for i≠n (we use higher Chow groups as our definition of motivic cohomology). This implies that the natural map K _n ^M (k)?K _n (k) from Milnor K-theory to Quillen K-theory is an isomorphism up to uniquely p-divisible groups, and that K _n ^M (k) and K _n (k) are p-torsion free. As a consequence, one can calculate the K-theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example K _n (X,ℤ/p ^r )=0 for n>dimX. Another consequence is Gersten’s conjecture with finite coefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch’s cycle complexes localized at p satisfy all Beilinson-Lichtenbaum-Milne axioms for motivic complexes, except possibly the vanishing conjecture. Oblatum 21-I-1998 & 26-VII-1999 / Published online: 18 October 1999     

On Isotopies and Homologies of Subvarieties of Toric Varieties

In ℂⁿ we consider an algebraic surface Y and a finite collection of hypersurfaces S_i. Froissart’s theorem states that if Y and S_i are in general position in the projective compactification of ℂⁿ together with the hyperplane at infinity then for the homologies of Y \∪ S_i we have a special decomposition in terms of the homology of Y and all possible intersections of S_i in Y. We prove the validity of this homological decomposition on assuming a weaker condition: there exists a smooth toric compactification of ℂn in which Y and S_i are in general position with all divisors at infinity. One of the key steps of the proof is the construction of an isotopy in Y leaving invariant all hypersurfaces Y ∩ S_k with the exception of one Y ∩ S_i, which is shifted away from a given compact set. Moreover, we consider a purely toric version of the decomposition theorem, taking instead of an affine surface Y the complement of a surface in a compact toric variety to a collection of hypersurfaces in it.     

Stein extensions of Riemannian symmetric spaces and dualities of orbits on flag manifolds

It is known [M4] that K_ℂ-orbits S and G_ℝ-orbits S' on a complex flag manifold are in one-to-one correspondence by the condition that S ∩ S' is nonempty and compact. It is possible to replace K_ℂ by some conjugate xK_ℂx⁻¹ so that the correspondence is preserved. We investigate the sets C(S) of such x for various orbits S and their relations with each other. We prove that for classical groups the intersection C = ∩_S C(S) equals D₀Z where D₀ = D₀/K_ℂ is the universal domain in G_ℂ/K_ℂ introduced in [AG] and Z is the center of G. As a corollary we prove that D₀ is Stein for classical groups. Moreover we conjecture that C(S)₀ = D₀ for generic S where C(S)₀ is the connected component of C(S) containing the identity.     

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface X _k obtained by blowing up ℂℙ² at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W _k :M _k →ℂ with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of X _k , and give an explicit correspondence between the deformation parameters for X _k and the cohomology class [B+iω]∈H ²(M _k ,ℂ).     

Congruences between derivatives of abelian <Emphasis Type="Italic">L</Emphasis>-functions at <Emphasis Type="Italic">s</Emphasis>=0

Let K/k be a finite abelian extension of global fields. We prove that a natural equivariant leading term conjecture implies a family of explicit congruence relations between the values at s=0 of derivatives of the Dirichlet L-functions associated to K/k. We also show that these congruences provide a universal approach to the ‘refined abelian Stark conjectures’ formulated by, inter alia, Stark, Gross, Rubin, Popescu and Tate. We thereby obtain the first proofs of, amongst other things, the Rubin–Stark conjecture and the ‘refined class number formulas’ of both Gross and Tate for all extensions K/k in which K is either an abelian extension of ℚ or is a function field. Mathematics Subject Classification (1991)  Primary 11G40; Secondary 11R65; 19A31; 19B28     

On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

The Grothendieck ring of varieties and piecewise isomorphisms

Let K ₀(Var _k ) be the Grothendieck ring of algebraic varieties over a field k. Let X, Y be two algebraic varieties over k which are piecewise isomorphic (i.e. X and Y admit finite partitions X ₁, ..., X _n , Y ₁, ..., Y _n into locally closed subvarieties such that X _i is isomorphic to Y _i for all i ≤ n), then [X] = [Y] in K ₀(Var _k ). Larsen and Lunts ask whether the converse is true. For characteristic zero and algebraically closed field k, we answer positively this question when dim X ≤ 1 or X is a smooth connected projective surface or if X contains only finitely many rational curves.     

A minimal triangulation of complex projective plane admitting a chess colouring of four-dimensional simplices

We construct and study a new 15-vertex triangulation X of the complex projective plane ℂP². The automorphism group of X is isomorphic to S ₄ × S ₃. We prove that the triangulation X is the minimal (with respect to the number of vertices) triangulation of ℂP² admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for the simplices of X and show that the automorphism group of X can be realized as a group of isometries of the Fubini-Study metric. We find a 33-vertex subdivision $ \bar X $ \bar X of the triangulation X such that the classical moment mapping μ: ℂP² → Δ² is a simplicial mapping of the triangulation $ \bar X $ \bar X onto the barycentric subdivision of the triangle Δ². We study the relationship of the triangulation X with complex crystallographic groups.     

Mader’s Conjecture On Extremely Critical Graphs

A non-complete graph G is called an (n,k)-graph if it is n-connected but G—X is not (n−|X|+1)-connected for any X ⊂V (G) with |X|≤k. Mader conjectured that for k≥3 the graph K_2k+2−(1−factor) is the unique (2k,k)-graph(up to isomorphism). Here we prove this conjecture.     

Principal bundles whose restriction to curves are trivial

Let k be an algebraically closed field and X a smooth projective variety defined over k. Let E_G be a principal G–bundle over X, where G is an algebraic group defined over k, with the property that for every smooth curve C in X the restriction of E_G to C is the trivial G–bundle. We prove that the principal G–bundle E_G over X is trivial. We also give examples of nontrivial principal bundle over a quasi-projective variety Y whose restriction to every smooth curve in Y is trivial.     

The proof of Ushio’s conjecture concerning path factorization of complete bipartite graphs

Let K _m,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A P _v -factorization of K _m,n is a set of edge-disjoint P _v -factors of K _m,n which partition the set of edges of K _m,n . When v is an even number, Wang and Ushio gave a necessary and sufficient condition for existence of P _v -factorization of K _m,n . When k is an odd number, Ushio in 1993 proposed a conjecture. Very recently, we have proved that Ushio’s conjecture is true when v = 4k − 1. In this paper we shall show that Ushio Conjecture is true when v = 4k − 1, and then Ushio’s conjecture is true. That is, we will prove that a necessary and sufficient condition for the existence of a P _4k+1-factorization of K _m,n is (i) 2km≤(2k+1)n, (ii) 2kn≤(2k+1)m, (iii) m+n≡0 (mod 4k+1), (iv) (4k+1)mn/[4k(m+n)] is an integer.     

Effective algebraic degeneracy

We show that for every smooth projective hypersurface X⊂ℙ ⁿ⁺¹ of degree d and of arbitrary dimension n ≥2, if X is generic, then there exists a proper algebraic subvariety Y ⊊ X such that every nonconstant entire holomorphic curve f :ℂ→X has image f(ℂ) which lies in Y, as soon as its degree satisfies the effective lower bound d\geqslant 2ⁿ⁵d\geqslant 2^{n^{5}} .     

On structure of small contractions of odd dimensional projective varieties

Let X be a smooth projective variety of dimension 2k-1 (k≥3) over the complex number field. Assume that fR: X→Y is a small contraction such that every irreducible component Ei of the exceptional locus of fR is a smooth subvariety of dimension k. It is shown that each Ei is isomorphic to the k-dimensional projective space Pk, the k-dimensional hyperquadric surface Qk in Pk 1, or a linear Pk-1-bundle over a smooth curve.     

Smoothness and Euler characteristic of the variety of complete pairsX 23 of zero-dimensional subschemes of length 2 and 3 of algebraic surfaces

A. S. Tikhomirov’s conjecture about the smoothness of the variety of complete pairsX ₂₃ of zero-dimensional subschemes of an irreducible projective algebraic surface is verified. In the caseS=P², the topological Euler characteristic of this variety is computed. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 276–287, February, 2000.