In the 1960’s and 1970’s, the Adams Conjecture figured prominently both in homotopy theory and in geometric topology. Quillen sketched one way to attack the conjecture and then proved it with an entirely different line of argument. Both of his approaches led to spectacular and beautiful new mathematics. 1. Background on the Adams Conjecture For a finite CW -complex X, let KO(X) be the Grothendi...

Let Mr denote the moduli space of semi-stable rank-r vector bundles with trivial determinant over a smooth projective curve C of genus g. In this paper we study the base locus Br ⊂ Mr of the linear system of the determinant line bundle L over Mr, i.e., the set of semi-stable rank-r vector bundles without theta divisor. We construct base points in Bg+2 over any curve C, and base points in B4 ove...

Let X be a smooth projective curve of genus g ≥ 2 defined over an arbitrary algebraically closed field K. For any vector bundle E on X, call μ(E) := deg(E)/rank(E) the slope of E. A vector bundle E on X is said to be spanned if the natural map H(X,E)⊗OX → E is surjective. Such bundles are important tools for the projective geometry ofX because they are exactly the vector bundles associated to a...

We prove that, for every r ≥ 2, the moduli space M X (r; c1, c2) of rank r stable vector bundles with Chern classes c1 = rH and c2 = 1 2 (3r 2 − r) on a nonsingular cubic surface X ⊂ P3 contains a nonempty smooth open subset formed by ACM bundles, i.e. vector bundles with no intermediate cohomology. The bundles we consider for this study are extremal for the number of generators of the correspo...

We relate Berezin–Toeplitz quantization of higher rank vector bundles to quantum-classical hybrid systems and in stages symplectic fibrations. apply this picture the analysis geometry bundles, including spectral gap Berezin transform convergence rate Donaldson’s iterations toward balanced metrics on stable bundles. also establish refined estimates scalar case compute Kähler manifolds with const...

The Lie derivation of multivector fields along multivector fields has been introduced by Schouten (see cite{Sc, S}), and studdied for example in cite{M} and cite{I}. In the present paper we define the Lie derivation of differential forms along multivector fields, and we extend this concept to covariant derivation on tangent bundles and vector bundles, and find natural relations between them and...

Mochizuki’s work on torally crys-stable bundles [9] has extensive implications for the theory of logarithmic connections on vector bundles of rank 2 on curves, once the language is translated appropriately. We describe how to carry out this translation, and give two classes of applications: first, one can conclude almost immediately certain results classifying Frobenius-unstable vector bundles ...