Homological symbols and the Quillen conjecture

We formulate a “correct” version of the Quillen conjecture on linear group homology for certain arithmetic rings and provide evidence for the new conjecture. In this way we predict that the linear group homology has a direct summand looking like an unstable form of Milnor K-theory and we call this new theory “homological symbols algebra”. As a byproduct, we prove the Quillen conjecture in homological degree two for the rank two and the prime 5.     

ON THE FUNDAMENTAL GROUP OF OPEN MANIFOLDS WITH NONNEGATIVE RICCI CURVATURE＊＊＊＊

The authors establish some uniform estimates for the distance to halfway points of minimal geodesics in terms of the distantce to end points on some types of Riemannian manifolds, and then prove some theorems about the finite generation of fundamental group of Riemannian manifold with nonnegative Ricci curvature, which support the famous Milnor conjecture.     

具有非负Ricci曲率的开流形的基本群

我们对某些类型的Riemannian流形,通过点到极小测地圈端点的距离建立了它到极小测地圈中点的距离的一致估计,然后利用这种一致估计证明了具有非负Ricci 曲率Riemannian流形的基本群有限生成的一个定理,对著名的Milnor猜测起到更强的支持作用．     

The Witt ring kernel for a fourth degree field extension and related problems

In the first part of this paper we compute the Witt ring kernel for an arbitrary field extension of degree 4 and characteristic different from 2 in terms of the coefficients of a polynomial determining the extension. In the case where the lower field is not formally real we prove that the intersection of any power n of its fundamental ideal and the Witt ring kernel is generated by n-fold Pfister forms.In the second part as an application of the main result we give a criterion for the tensor product of quaternion and biquaternion algebras to have zero divisors. Also we solve the similar problem for three quaternion algebras.In the last part we obtain certain exact Witt group sequences concerning dihedral Galois field extensions. These results heavily depend on some similar cohomological results of Positselski, as well as on the Milnor conjecture, and the Bloch-Kato conjecture for exponent 2, which was proven by Voevodsky.     

3-Manifolds with nonnegative Ricci curvature

For a complete noncompact 3-manifold with nonnegative Ricci curvature, we prove that either it is diffeomorphic to ?³ or the universal cover splits. This confirms Milnor’s conjecture in dimension 3.     

Omega-Limit Sets of Generic Points of Partially Hyperbolic Diffeomorphisms

We prove that, for any E^u ⊕ E^cs partially hyperbolic C² diffeomorphism, the ω-limit set of a generic (with respect to the Lebesgue measure) point is a union of unstable leaves. As a corollary, we prove a conjecture made by Ilyashenko in his 2011 paper that the Milnor attractor is a union of unstable leaves. In the paper mentioned above, Ilyashenko reduced the local generecity of the existence of a “thick” Milnor attractor in the class of boundary-preserving diffeomorphisms of the product of the interval and the 2-torus to this conjecture.     

On the fundamental group of hyperelliptic fibrations and some applications

We prove that for a hyperelliptic fibration on a surface of general type with irreducible fibers over a (possibly) non-complete curve, the image of the fundamental group of a general fiber in the fundamental group of the surface is finite. Examples show that the result is optimal. As a corollary of this result we prove two conjectures; the Shafarevich conjecture on holomorphic convexity for the universal cover of these surfaces, and a conjecture of Nori on the finiteness of the fundamental groups of some surfaces. We also prove a striking general result about the multiplicities of multiple fibers of a hyperelliptic fibration on a smooth, projective surface.     

The Monodromy Conjecture for hyperplane arrangements

The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2πic) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture asserts that every such c is a root of the Bernstein-Sato polynomial of the hypersurface. In this note we prove the weak version of the conjecture for hyperplane arrangements. Furthermore, we reduce the strong version to the following conjecture: −n/d is always a root of the Bernstein-Sato polynomial of an indecomposable essential central hyperplane arrangement of d hyperplanes in C ⁿ .     

Durfee-type bound for some non-degenerate complete intersection singularities

The Milnor number, \(\mu (X,0)\), and the singularity genus, \(p_g(X,0)\), are fundamental invariants of isolated hypersurface singularities (more generally, of local complete intersections). The long standing Durfee conjecture (and its generalization) predicted the inequality \(\mu (X,0)\ge (n+1)!p_g(X,0)\), here \(n=\dim (X,0)\). Recently we have constructed counterexamples, proposed a corrected bound and verified it for the homogeneous complete intersections. In the current paper we treat the case of germs with Newton-non-degenerate principal part when the Newton diagrams are “large enough”, i.e. they are large multiples of some other diagrams. In the case of local complete intersections we prove the corrected inequality, while in the hypersurface case we prove an even stronger inequality.     

Vanishing structure set of Haken 3-manifolds

In this article we prove that the surgery groups of the fundamental group of a certain class of Haken 3-manifolds can be computed in terms of a generalized homology theory even if the manifolds do not support any nonpositively curved Riemannian metric. A consequence of this result is that the integral Novikov conjecture is true for the fundamental group of this class of manifolds. Received October 2, 1998 / in revised form February 10, 2000 / Published online July 20, 2000     

Some phases of oscillatory integrals

The first example of a phase is presented for which Arhold’s conjecture on the validity of uniform estimates for oscillatory integrals with maximal singularity index is true, while his conjecture on the semicontinuity of the singularity index is false. A rough upper bound for the Milnor number such that the latter conjecture fails is obtained. The corresponding counterexample is simpler than Varchenko’s well-known counterexample to Arnold’s conjecture on the semicontinuity of the singularity index. This gives hope to decrease codimension and the Milnor number for which the conjecture on the semicontinuity of the singularity index fails.     

On the Shafarevich conjecture for genus-2 fibrations

We prove a result about fundamental group of a smooth projective surface with ample canonical divisor which admits a genus two fibration. As a corollary we prove that the universal cover of such a surface is holomorphically convex. This proves the conjecture of Shafarevich for such surfaces. This article is dedicated to Madhav V. Nori.     

Deformations of Functions and F-Manifolds

We study deformations of functions on isolated singularities.A unified proof of the equality of Milnor and Tjurina numbersfor functions on isolated complete intersections singularitiesand space curves is given. As a consequence, the base spaceof their miniversal deformations is endowed with the structureof an F-manifold, and we can prove a conjecture of V. Goryunov,stating that the critical values of the miniversal unfoldingof a function on a space curve are generically local coordinateson the base space of the deformation. 2000 Mathematics SubjectClassification 32S05.     

The generalized Chern conjecture for manifolds that are locally a product of surfaces

We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old conjecture proved by Milnor in dimension 2. In particular, the Chern conjecture follows in these cases. The proof goes via a new sharp Milnor–Wood inequality for Riemannian manifolds that are locally a product of hyperbolic planes. Furthermore, we analyze the possible flat vector bundles over such manifolds. Over closed Hilbert–Blumenthal modular varieties, we show that there are finitely many flat structures with nonzero Euler number and none of them corresponds to the tangent bundle. Some of the main results were announced in [M. Bucher, T. Gelander, Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes, C. R. Acad. Sci. Paris Ser. I 346 (2008) 661–666].     

Curvatures of homogeneous Randers spaces

We study curvatures of homogeneous Randers spaces. After deducing the coordinate-free formulas of the flag curvature and Ricci scalar of homogeneous Randers spaces, we give several applications. We first present a direct proof of the fact that a homogeneous Randers space is Ricci quadratic if and only if it is a Berwald space. We then prove that any left invariant Randers metric on a non-commutative nilpotent Lie group must have three flags whose flag curvature is positive, negative and zero, respectively. This generalizes a result of J.A. Wolf on Riemannian metrics. We prove a conjecture of J. Milnor on the characterization of central elements of a real Lie algebra, in a more generalized sense. Finally, we study homogeneous Finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected Lie group admitting a left invariant Finsler metric with positive flag curvature is SU(2)$\text{SU}\left(2\right)$.     

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.     

Complete affine locally flat manifolds with a free fundamental group

Examples of free noncommutative subgroups of the affine group A(3), which act properly discontinuously on ³, are constructed in the paper. These examples refute a conjecture of Milnor to the effect that the fundamental group of any complete affine locally flat manifold contains a sòlvable subgroup of finite index.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 134, pp. 190–205, 1984.     

Oka's conjecture on irreducible plane sextics

We partially prove and partially disprove Oka's conjecture onthe fundamental group/Alexander polynomial of an irreducibleplane sextic. Among other results, we enumerate all irreduciblesextics with simple singularities admitting dihedral coveringsand find examples of Alexander equivalent Zariski pairs of irreduciblesextics.     

An analogue of the Novikov Conjecture in complex algebraic geometry

We introduce an analogue of the Novikov Conjecture on higher signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain ``higher Todd genera' are birational invariants. This implies birational invariance of certain extra combinations of Chern classes (beyond just the classical Todd genus) in the case of varieties with large fundamental group (in the topological sense). We prove the conjecture under the assumption of the ``strong Novikov Conjecture' for the fundamental group, which is known to be correct for many groups of geometric interest. We also show that, in a certain sense, our conjecture is best possible.