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We prove a mild strengthening of a theorem of C̆esnavic̆ius which gives a criterion for a vector bundle on a smooth complete intersection of dimension at least 3 3 to split into a sum of line bundles. We also prove an analogous statement for bundles on a general complete intersection surface.

Let $X$ be a compact Riemann surface of genus $g \geq 3$. Let $\cat{M}_{Hod}$ denote the moduli space of stable $\lambda$-connections over $X $ and $\cat{M}'_{Hod} \subset \cat{M}_{Hod}$ denote the subvariety whose underlying vector bundle is stable. Fix a line bundle $L$ of degree zero. Let $\cat{M}_{Hod}(L)$ denote the moduli space of stable $\lambda$-connections with fixed determinant $L$ and $\cat{M}'_{Hod}(L) \subset \cat{M}_{Hod}(L)$ be the subvariety whose underlying vector bundle is stable. We show that there is a natural compactification of $\cat{M}'_{Hod}$ and $\cat{M}'_{Hod} (L)$, and study their Picard groups. Let $\M_{Hod}(L)$ denote the moduli space of polystable $\lambda$-connections. We investigate the nature of algebraic functions on $\cat{M}_{Hod}(L)$ and $\M_{Hod}(L)$. We also study the automorphism group of $\cat{M}'_{Hod}(L)$.

We give examples of surfaces which are Ulrich-wild, i.e. that support families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$., Comment: 15 pages. Several typos have been fixed. The proof of Theorem 1.2 has been simplified. The final version will appear in Proceedings of the A.M.S

This note provides certain computations with transfer associated with projective bundles of Spin vector bundles. One aspect is to revise the proof of the main result of \cite{B} which says that all fourfold products of the Ray classes are zero in symplectic cobordism., 7 pages

In this paper we study the Brill Noether locus of rank 2, (semi)stable vector bundles with at least two sections and of suitable degrees on a general ν-gonal curve. We classify its irreducible components having at least expected dimension. We moreover describe the general member F of such components just in terms of extensions of line bundles with suitable "minimality properties", providing information on the birational geometry of such components as well as on the very-ampleness of F.

We generalize a construction of Hitchin to prove that, given any compact Kähler manifold M M with positive holomorphic sectional curvature and any holomorphic vector bundle E E over M M , the projectivized vector bundle P ( E ) {\mathbb P}(E) admits a Kähler metric with positive holomorphic sectional curvature.

Inspired by recent papers on twisted $K$-theory, we consider in this article the question of when a twist $\mathcal{R}$ over a locally compact Hausdorff groupoid $\mathcal{G}$ (with unit space a CW-complex) admits a twisted vector bundle, and we relate this question to the Brauer group of $\mathcal{G}$. We show that the twists which admit twisted vector bundles give rise to a subgroup of the Brauer group of $\mathcal{G}$. When $\mathcal{G}$ is an \'etale groupoid, we establish conditions (involving the classifying space $B\mathcal{G}$ of $\mathcal{G}$) which imply that a torsion twist $\mathcal{R}$ over $\mathcal{G}$ admits a twisted vector bundle., Comment: Theorem 4.2 in v1 was incorrect as stated. Consequently v2 represents a substantial revision. This version (v2) to be published in Proceedings of the AMS

We give a formula for pushing forward the classes of Hall-Littlewood polynomials in Grassmann bundles, generalizing Gysin formulas for Schur S- and Q-functions., Comment: 7 pages; the version corresponding to the published one and its corrigenda, and corrigenda published electronically on March 16, 2016

This paper is devoted to the study of a cohomological criterion for the splitting of a vector bundle on multiprojective space. The criterion extends a result of Ballico-Malaspina towards a generalization of the Horrocks criterion on multiprojective space.

We find a full strongly exceptional collection for the Cayley plane OP2, the simplest rational homogeneous space of the exceptional group E6. This collection, closely related to the one given by the second author in [J. Algebra, 330:177-187, 2011], consists of 27 vector bundles which are homogeneous for the group E6, and is a Lefschetz collection with respect to the minimal equivariant embedding of OP2.

A basic question for any property of quasi–coherent sheaves on a scheme X X is whether the property is local, that is, it can be defined using any open affine covering of X X . Locality follows from the descent of the corresponding module property: for (infinite dimensional) vector bundles and Drinfeld vector bundles, it was previously proved by Kaplansky’s technique of dévissage. Since vector bundles coincide with ℵ 0 \aleph _0 –restricted Drinfeld vector bundles, a question arose as to whether locality holds for κ \kappa –restricted Drinfeld vector bundles for each infinite cardinal κ \kappa . We give a positive answer here by replacing the dévissage with its recent refinement involving C \mathcal C –filtrations and the Hill Lemma.

Let M M be a Riemannian manifold and let E → M E\to M be a Hermitian vector bundle with a Hermitian covariant derivative ∇ \nabla . Furthermore, let H ( 0 ) H(0) denote the Friedrichs extension of ∇ ∗ ∇ / 2 \nabla ^*\nabla /2 and let V : M → E n d ( E ) V:M\to \mathrm {End}(E) be a potential. We prove that if V V has a decomposition of the form V = V 1 − V 2 V=V_1-V_2 with V j ≥ 0 V_j\geq 0 , V 1 V_1 locally integrable and | V 2 | \left | V_2 \right | in the Kato class of M M , then one can define the form sum H ( V ) := H ( 0 ) ∔ V H(V):=H(0)\dotplus V in Γ L 2 ( M , E ) \Gamma _{\mathsf {L}^2}(M,E) without any further assumptions on M M . Applications to quantum physics are discussed.

In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. We generalized Morrison's result to higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group. In this article, we give a simple proof for polarizations OPE�(d) � � L k , where d is a positive integer, k � 0 and the base manifold is a compact Riemann surface of genus g � 2.

Let X X be a geometrically connected smooth projective curve of genus g X ≥ 2 g_X \geq 2 over R \mathbb {R} . Let M ( r , ξ ) M(r, \xi ) be the coarse moduli space of geometrically stable vector bundles E E over X X of rank r r and determinant ξ \xi , where ξ \xi is a real point of the Picard variety P i c _ d ( X ) \underline {\mathrm {Pic}}^d( X) . If g X = r = 2 g_X = r = 2 , then let d d be odd. We compute the Brauer group of M ( r , ξ ) M(r,\xi ) .

Let X X be a smooth projective variety, defined over an algebraically closed field of positive characteristic, such that the tangent bundle T X TX is trivial. Let F X : X ⟶ X F_X\, :\, X\,\longrightarrow \, X be the absolute Frobenius morphism of X X . We prove that for any n ≥ 1 n\, \geq \,1 , the n n –fold composition F X n F^n_X is a torsor over X X for a finite group–scheme that depends on n n . For any vector bundle E ⟶ X E\,\longrightarrow \, X , we show that the direct image ( F X n ) ∗ E (F^n_X)_*E is essentially finite (respectively, F F –trivial) if and only if E E is essentially finite (respectively, F F –trivial).

We consider maps between different spaces of vector bundles on curves obtained by taking wedge powers, elementary transformations or kernels of evaluation maps and studying their respective fibers. We apply the results to construct large dimensional sets in the base locus of the generalized theta divisor.

'. Let Cg be a smooth projective irreducible curve over C of genus g ≥ 1 and let {p 1 ,..., p s } be a set of distinct points on Cg. We fix a nonnegative integer l and denote by M g (p, λ) the moduli space of parabolic semistable vector bundles of rank r on Cg with trivial determinant and fixed parabolic structure of type A = (λ 1 ,..., λ s ) at p = (p 1 ,..., p s ), where each weight λ i is in P l (SL(r)). On M g (p, λ) there is a canonical line bundle L(λ, l), whose global sections are called generalized parabolic SL(r)-theta functions of order l. In this paper we prove the existence of such nonzero nonabelian theta functions, thus establishing a part of higher genus generalizations of the celebrated saturation conjectures.

When B B is a biquotient, we show that there exist vector bundles over B B with metrics of nonnegative curvature whose normal holonomy groups have arbitrarily large dimension.

Suppose that E → B E\to B is a vector bundle with a linear periodic map of period p p ; the map is assumed free on the outside of the 0 0 -section. A polynomial c E ( y ) c_{E}(y) , called a mod p p Chern polynomial of E E , is defined. It is analogous to the Stiefel-Whitney polynomial defined by Dold for real vector bundles with the antipodal involution. The mod p p Chern polynomial can be used to measure the size of the periodic coincidence set for fibre preserving maps of the unit sphere bundle of E E into another vector bundle.

We classify C 0 ( X ) C_0(X) -endomorphisms of stable continuous-trace C ∗ C^\ast -algebras up to inner automorphism by a surjective multiplicative invariant taking values in finite-dimensional vector bundles over the spectrum. Specializing to automorphisms, this gives a different approach to results of Lance, Smith, Phillips and Raeburn.

We consider the complex 0 → Λ 0 (M; E) →∂ ω Λ 1 (M; E) →∂ ω ... →∂ ω Λ m (M; E), where E is a finite-dimensional vector bundle over a suitable differential manifold M, A q (M;E) denotes the space of all smooth or real analytic or holomorphic sections of the q-exterior product of E and ∂ ω (η):= w A η for ω ∈ Λ 1 (M;E). We give sufficient and necessary conditions for the above complex to be exact and, in smooth and holomorphic cases, we give sufficient conditions for its splitting, i.e., for existence of linear continuous right inverse operators for ∂ w : Λ q (M;E) → Im∂ ω ⊆ Λ q+1 (M; E). Analogous results are obtained whenever M is replaced by a suitable closed subset X or ∂ ω are replaced by the interior product operators ∂ Z , ∂ Z (η):= Z?η for a given section Z of the dual bundle E*.

A vanishing theorem is proved for families over higher dimensional bases and applied to generalize some Shafarevich type statements to that setting.

Let us consider a riemannian vector bundle E E with compact basis ( M , g ) (M,g) and the rough laplacian Δ ¯ \bar {\Delta } associated to a connection D D on E E . We prove that the eigenvalues of Δ ¯ \bar {\Delta } are bounded above by a function of the first eigenvalue and of the geometry of ( M , g ) (M,g) , but independently of the choice of the connection D D .

The first section of this paper provides an improvement upon known finiteness theorems for Riemannian submersions; that is, theorems which conclude that there are only finitely many isomorphism types of fiber bundles among Riemannian submersions whose total spaces and base spaces both satisfy certain geometric bounds. The second section of this paper provides a sharpening of some recent theorems which conclude that, for an open manifold of nonnegative curvature satisfying certain geometric bounds near its soul, there are only finitely many possibilities for the isomorphism class of a normal bundle of the soul. A common theme to both sections is a reliance on basic facts about Riemannian submersions whose A A and T T tensors are both bounded in norm.

We characterize the triples (X, L, H), consisting of line bundles L and H on a complex projective manifold X, such that for some positive integer k, the k-th holomorphic jet bundle of L, J(k) (X, L), is isomorphic to a direct sum H + . . . + H.

We prove that a holomorphic vector bundle E E over a compact connected Kähler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials f f and g g , with nonnegative integral coefficients, such that the vector bundle f ( E ) f(E) is isomorphic to g ( E ) g(E) . An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group. When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.

Given a certain kind of linear operator A (possibly a differential operator or a properly supported pseudodifferential operator) between sections of Hermitian vector bundles over a Riemannian manifold, a necessary and sufficient condition is obtained for the operator A to be solvable in a class of nonstandard sections in a generalized sense of weak solutions. The existence of a fundamental-solution-like internal section is established in the solvable case.

Fix a smooth projective 3-fold X X , c 1 c_1 , H ∈ P i c ( X ) H\in \mathrm {Pic}(X) with H H ample, and d ∈ Z d\in \mathbf {Z} . Assume the existence of integers a , b a,b with a ≠ 0 a\not =0 such that a c 1 ac_1 is numerically equivalent to b H bH . Let M ( X , 2 , c 1 , d , H ) M(X,2,c_1,d,H) be the moduli scheme of H H -stable rank 2 vector bundles, E E , on X X with c 1 ( E ) = c 1 c_1(E)=c_1 and c 2 ( E ) ⋅ H = d c_2(E)\cdot H=d . Let m ( X , 2 , c 1 , d , H ) m(X,2,c_1,d,H) be the number of its irreducible components. Then lim sup d → ∞ m ( X , 2 , c 1 , d , H ) = + ∞ \limsup _{d\rightarrow \infty }m(X,2,c_1,d,H)= +\infty .

The purpose of this note is to extend the results of V. Guillemin on elliptic self-adjoint pseudodifferential operators of order one, from operators defined on smooth functions on a closed manifold to operators defined on smooth sections in a vector bundle of Hilbert modules of finite type over a finite von Neumann algebra., Comment: 11 pages, AMS-Tex, Minor Corrections

We prove a series of rigidity theorems. Examples of higher rank ASD vector bundles are given on quaternion-Kähler manifolds, which admit the one and only ASD connections modulo the gauge equivalence.

The purpose of this paper is to classify ample and spanned vector bundles of top Chern number two on smooth projective varieties of arbitrary dimension defined over an algebraically closed field of characteristic zero.

Here we give an upper bound for the number of irreducible components of the moduli scheme of stable rank r torsion-free sheaves of fixed degree on the integral curve X. This bound depends only on r, Sing(X),pa(X) and the corresponding number for the rank 1 case. Let X be an integral complete curve of arithmetic genus Pa and geometric genus g. For all integers r, d with r > 1 let M(X, r, d) (or just M(r, d)) be the moduli scheme of stable rank r torsion-free sheaves on X with degree d. For general background on stable vector bundles and sheaves on curves and their moduli schemes, see [N] or [S]. The number of irreducible components of M (r, d) does not depend on d, but only on X and r (see Theorem 1.2). Let n(r, X) (or just n(r)) be the number of irreducible components of M(r, d). Here we want to give upper bounds on n(r) depending only on the singularities of X, the integer r and n(l). This is given by Theorem 0.1 (see the discussion in the last part of Remark 1.4). To state Theorem 0.1 (our main result) we need to introduce a few concepts (e.g. measures for the singularities of X) and fix some notation. Let 7r: C -* X be the normalization (hence g = pa(C)). Set 6 := Pa g. Set o:= Ox. For any x C Sing(X), set Ox := Ox,x and let mx be the maximal ideal of Ox; set kx := Ox/mx. Set fx := dimK(Ext1(kx,Ox)) and f := max{fx}xEsirlg(x). For example, we have fx = 1 if and only if Ox is Gorenstein (see e.g. [Co], Lemma 2.1.4). Let ex be the multiplicity of X at x; set e := max(ex) and e" := Ex ex. Let F be a rank r torsion-free sheaf on X; the set of all formal isomorphism classes of the germs {FX}xEsirlg(x) will be called the formal isomorphism class of F. The set W of all possible isomorphism classes for rank r torsion-free sheaves around Sing(X) has in a natural way a scheme structure; for any reduced subscheme Q of W, let M(r, d, Q) be the reduced subscheme of M(r, d)r.eci parametrizing sheaves with formal isomorphism type in Q. Let n(r, Q) (or n(r, d, Q)) be the number of irreducible components of M(r, d, Q). By Theorem 1.2 this integer does not depend on d. Our main interest is when Q is an irreducible component of Wr.ec. Here is our main result. Received by the editors November 28, 1994. 1991 Mathematics Subject Classification. Primary 14H60, 14D20, 14B99. This research was partially supported by MURST and GNSAGA of CNR (Italy). The author is a member of Europroj (and its group "Vector bundles on curves"). ?)1997 Arnericari Mathernatical Society

We show the stable triviality of all the elements in VEC ⁡ ( R 2 , R n ) \operatorname {VEC}(R_{2},R_{n}) concretely, and describe VEC ⁡ ( R 2 , R n ) \operatorname {VEC}(R_{2},R_{n}) as surjection classes from a trivial bundle to another. The results also contain the explicit description of non-linearizable S L 2 SL_{2} actions on C n \mathbb {C}^{n} .

Swan’s theorem verifies the equivalence between finitely generated projective modules over function algebras and smooth vector bundles. We define A ( r ) {A^{(r)}} -maps that correspond to usual non-linear differential operators of degree r under the equivalence of Swan’s theorem and thus generalize Swan’s theorem to include non-linear differential operators as morphisms. An A ( r ) {A^{(r)}} -manifold structure is introduced on the space of sections of a fiber bundle through charts with A ( r ) {A^{(r)}} -maps as transition homeomorphisms. A characterization for all the smooth maps between the spaces of sections of vector bundles, whose kth derivatives are linear differential operators of degree r in each variable, is given in terms of A ( r ) {A^{(r)}} -maps.

A basis is presented for the R P n {\mathbf {R}}{{\mathbf {P}}^n} classes in N n ( B O ) {\mathfrak {N}_n}(BO) . This basis is used to prove that every smooth involution ( M , T ) (M,T) on a closed manifold bounds if its fixed point set is a disjoint union of odd-dimensional projective spaces of constant dimension.

Let E E be an ample and spanned vector bundle of rank r r over a complex projective surface S S . It is shown that the sectional genus g ( S , det E ) g(S,\det E) is bounded from below by the number b = max { q ( S ) , 2 − r χ ( O S ) } b = \max \{ q(S),2 - r\chi ({\mathcal {O}_S})\} and pairs ( S , E ) (S,E) satisfying b ⩽ g ( S , det E ) ⩽ b + 1 b \leqslant g(S,\det E) \leqslant b + 1 are characterized.

Let E 0 , J 0 {E_0},\;{J_0} , and L 0 {L_0} be the symplectic 2 n 2n -vector bundle, the compatible complex operator, and the Lagrangian subbundle that are determined by the U ( n ) U(n) -extension of the principal O ( n ) O(n) -bundle U ( n ) → U ( n ) / O ( n ) U(n) \to U(n)/O(n) . We compute the Chern-Simons-Maslov class μ 1 ( E 0 , J 0 , L 0 ) {\mu ^1}({E_0},{J_0},{L_0}) . Then for a trivial symplectic 2 n 2n -bundle E E , a compatible complex operator J J , and a Lagrangian subbundle L L , we compute Chern-Simons-Maslov classes μ h ( E , J , L ) {\mu ^h}(E,J,L) under some condition on the base space of E E .

The main result of this paper is an equation representing the value of the Dirichlet series L ( f , A , s ) L(f,A,s) at s = k s = k in terms of the height pairings of special points on a vector bundle V V associated with the quaternion algebra over Q {\mathbf {Q}} ramified at N N and ∞ \infty .

Conformal invariance of middle-dimensional harmonic forms is used to improve Kato's inequality for four-manifolds. An application to positively curved four-manifolds is given. 0. INTRODUCTION The purpose of this paper is to prove the following: Theorem 1. Let (M4, g) be a four-dimensional Riemannian manifold. Let w be a harmonic two-form on (M, g). Then w satisfies the pointwise inequality: (0. 1 ) ~ ~~~~1'7(ol' > 3ld l ,l,l ' Kato's inequality [1, p. 130], states that if E is a Riemannian vector bundle with connection V over a Riemannian manifold M, then any smooth section s of E, satisfies the pointwise inequality: (0.2) IVS-2 > ldlsl 12 . Now by definition, if s(w) vanishes at p E M, then dlsl(dlwl) = 0 at p. Thus, (0.1) and (0.2) are automatically valid at such a point. At points where w does not vanish (0.1) can be thought of as a quantitative improvement of (0.2), for the case of harmonic two-forms on four-dimensional manifolds. As an application of the above theorem, we prove: Theorem 2. Let (M4, g) be a compact, connectedfour-dimensional Riemannian manifold whose sectional curvature K(g) satisfies 1 > K(g) > 3. If (0.3) 3 > 1/(3(1 + 3.2 14/51/2)1/2 + 1) . 1714 then M is definite. This theorem represents an improvement of results starting with [2] followed by [4, 7, 6]. The relevance of Theorems 1 and 2 stems from the following facts Received by the editors November 6, 1989 and, in revised form, July 27, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C20; Secondary 57N1 3.

Using vector bundle method, we study the tricanonical system on a minimal surface of general type defined over an algebraically closed field of positive characteristic. Under some conditions, it is proved that it has no fixed component. INTRODUCTION Pluricanonical systems on minimal surfaces of general type have been studied by many authors after the fundamental work of Bombieri [B]. Recently, Ekedahl extended many of the classical results of Bombieri to the positive characteristic case [E]. In particular, he proved that the mth pluricanonical system ImKI is base point free if m > 4 or m > 3 and K2 > 2. Some of the remaining cases are treated in [SB]. In this short note we shall consider the fixed part of the tricanonical system 13KI under certain conditions. Our purpose is to show the following Theorem. Let X be a minimal surface of general type defined over an algebraically closed field k. Assume char(k) = p > 2 and K2 = X(6,x) = 1. Then 13I'1 has no fixed component if it is not composed with a pencil. PROOF OF THE THEOREM Proposition 1.1. Let X be a minimal surface of general type defined over an algebraically closed field of characteristic p > 2 such that K2 = 1. Then (-2) curves cannot be contained in the fixed part of 13K . Proof. Suppose that a (-2) curve C is a fixed component of 13K I. We have the exact sequence 0 -+ 6x(3K C) -+ 6x(K) -+ 6'C(3K) -O0, which induces a sequence H0(6x(3K)) -+ H0(6c(3K)) -* H1(6x(3K -C)) H1(x(3K)). Received by the editors March 13, 1991 and, in revised form, May 17, 1991. 1991 Mathematics Subject Classification. Primary 14J29; Secondary 14J60. (@31993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page

Let L L be an ample line bundle on an abelian surface A A . We prove that the four conditions: (1) L L is base point free, (2) L L is fixed component free, (3) L ⊗ 2 {L^{ \otimes 2}} is very ample, (4) L ⊗ 2 {L^{ \otimes 2}} is normally generated, are equivalent if ( L 2 ) > 4 ({L^2}) > 4 . Moreover we prove that L ⊗ 2 {L^{ \otimes 2}} is not normally generated if ( L 2 ) = 4 ({L^2}) = 4 .

We give a new proof of the existence of the Green bundles. Let (M, g) be a compact Riemannian manifold, and denote by 9t its geodesic flow on the tangent bundle TM. Let -r : TM -* M be the canonical projection and for all 0 E TM let V(0) be the kernel of d-ro. Two points 01, 02 are said to be conjugate if 02 g9tO and dgtV(01) n V(02) & o0 It was proved by Hopf [5] that a two-dimensional torus without conjugate points is flat. Afterwards, Green [8] proved that the integral of the scalar curvature of a manifold without conjugate points is nonpositive and it vanishes if the metric is locally flat. A main ingredient was the existence, under the condition of no conjugate points, of the following bundles: (1) Es(0) lim dg_tV(gt(0)), t-> oo (2) E'(0) = lim dgtV(g_t(0)). t-* oo Hopf's result was generalized to higher dimensions in [2], but there are still new rigidity type results using these bundles; see for example [1]. These bundles have other applications: among other ideas they where used by Freire and Mane' [7] to obtain estimates of the topological entropy. Foulon [6] generalized this result to the case of Finsler metrics. The bundles were also used by Eberlain [4] who proved that these are transverse if and only if the geodesic flow is Anosov. This result was also generalized to the case of convex Hamiltonians without conjugate points; see [31. The purpose of this note is to give a new proof of the following Theorem. If the geodesic flow 9t of a compact manifold does not have conjugate points, then for every 0 in TM the limits (1) and (2) exist. We recall from [4] the definition of the connection map K: T0TM -> T7r(o)M. For ( on T0TM let Z: (-c, e) -* TM be a curve with initial velocity (. Define K(s) = Z'(O) to be the covariant derivative of Z along the curve ir o Z. The definition does not depend on the curve Z. Received by the editors February 8, 2001. 2000 Mathematics Subject Classification. Primary 37D40. The author was partially supported by CONACYT-Mexico grant #28489-E and EPSRCUnited Kingdom GR/M5610. ?)2002 American Mathematical Society

Let k be an algebraically closed field. For every affine variety X with dim X > 7 we construct a smooth affine variety Y which is birationally equivalent to X and which possesses a stably trivial but not trivial algebraic vector bundle. We give some application of this fact to the cancellation problem.

We show that certain line bundles on the cotangent bundle of a Grassmannian arising from an anti-dominant character λ have cohomology groups isomorphic to those of a line bundle on the cotangent bundle of the dual Grassmannian arising from the dominant character wo(λ), where ω 0 is the longest element of the Weyl group of SL l+1 (k).

There are many affine subalgebras of polynomial rings with highly non-trivial projective modules, whose initial algebras (toric degenerations) are still finitely generated and have all projective modules free. Let k[X1, . . . , Xn] be a polynomial algebra (k a field, n ∈ N) and A an affine k-subalgebra. Let ≺ denote a term order on the multiplicative semigroup of monomials in the Xi and let in≺(A) denote the monomial subalgebra of k[X1, . . . , Xn], generated by the leading monomials of elements f ∈ A with respect to ≺. In case the initial algebra in≺(A) is finitely generated, one can obtain many properties of A by checking them for in≺(A) (called sometimes a toric degeneration of A) (see [CHV], [RS]). However, this is not the case for the property ‘all projective modules are free’ – thanks to Bernd Sturmfels for asking me this question. Theorem 1. Let A = k[X, Y, Z, Z − XY Z] and ≺ be the lexicographic term order corresponding to Z ≺ Y ≺ X. Then SK0(A) = Ker(K0(A) det → Pic(A)) is not trivial (equivalently, there are projective A-modules which are not even stably of type free⊕rank 1), while in≺(A) is finitely generated and all projective in≺(A)-modules are free. (Here ‘projective’ includes ‘finitely generated’.) Proof. Since (Z2−XY )k[X, Y, Z] ⊂ A, the following diagram with the upper horizontal identity embedding is a pull-back diagram (all the letters refer to variables) A −−−−→ k[X, Y, Z] y y k[U, V ] −−−−→ k[S, ST, T ] , where X 7→ S, Y 7→ T , Z 7→ ST , U 7→ S, V 7→ T . It is similarly easy to show that in≺(A) = k[X, Y, Z, XY Z] – a seminormal monomial algebra (as a kvector space it is generated by the normal monomial subalgebras k[X, Y ], k[X, Z], k[Y, Z] and k[{XaY bZc|a > 0, b > 0, c > 0}]). So by [Gu1] projective modules over in≺(A) are free, while the Mayer-Vietoris sequence (see [Bass], p.490), applied to the diagram above, implies SK0(A) = SK1(k[S, ST, T ]). Hence, by [Gu2] SK0(A) 6= 0. Received by the editors February 20, 1998. 1991 Mathematics Subject Classification. Primary 13D15, 19A49. This research was supported in part by the Alexander von Humboldt Foundation and CRDF grant #GM1-115. c ©1999 American Mathematical Society