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Let X be a curve in positive characteristic. A Hasse--Witt matrix for X is a matrix that represents the action of the Frobenius operator on the cohomology group H^1(X,O_X) with respect to some basis. A Cartier--Manin matrix for X is a matrix that represents the action of the Cartier operator on the space of holomorphic differentials of X with respect to some basis. The operators that these matrices represent are adjoint to one another, so Hasse--Witt matrices and the Cartier--Manin matrices are related to one another, but there seems to be a fair amount of confusion in the literature about the exact nature of this relationship. This confusion arises from differences in terminology, from differing conventions about whether matrices act on the left or on the right, and from misunderstandings about the proper formulae for iterating semilinear operators. Unfortunately, this confusion has led to the publication of incorrect results. In this paper we present the issues involved as clearly as we can, and we look through the literature to see where there may be problems. We encourage future authors to clearly distinguish between Hasse--Witt and Cartier--Manin matrices, in the hope that further errors can be avoided., Corrected version; an error in Section 2.5 has been fixed

A moduli space of sheaves satisfies weak Brill-Noether if the general sheaf in the moduli space has no cohomology. Goettsche and Hirschowitz prove that on the projective plane every moduli space of Gieseker semistable sheaves of rank at least two and Euler characteristic zero satisfies weak Brill-Noether. In this paper, we give sufficient conditions for weak Brill-Noether to hold on rational surfaces. We completely characterize Chern characters on Hirzebruch surfaces for which weak Brill-Noether holds. We also prove that on a del Pezzo surface of degree at least 4 weak Brill-Noether holds if the first Chern class is nef., 21 pages, comments welcome! Dedicated to Lawrence Ein on the occassion of his sixtieth birthday

This paper realises the Khovanov homology of a link in the 3-sphere as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k of characteristic zero. Here we prove the symplectic cup and cap bimodules which relate different symplectic arc algebras are themselves formal over k, and construct a long exact triangle for symplectic Khovanov cohomology. We then prove the symplectic and combinatorial arc algebras are isomorphic over the integers in a manner compatible with the cup bimodules. It follows that Khovanov homology and symplectic Khovanov cohomology co-incide in characteristic zero., Comment: 77 pages, 16 figures. v2 (which dates from Dec 2017) corrects one topological hypothesis in the construction of the nc vector field (cf. Remark 3.2) and implements numerous other minor clarifications. This version to appear in JAMS

We present a holomorphic version of the bosonic string in the formalism of quantum field theory developed by Costello and collaborators. In this paper we focus on the case in which space-time is flat and construct a one-loop exact quantization. Starting from first principles, we arrive at the critical dimension as an obstruction to satisfying the quantum master equation. Moreover, we show how the factorization algebra recovers the BRST cohomology of the string and give another construction of the Gerstenhaber structure. Finally, we show how the factorization homology along closed manifolds encodes the determinant line bundle over the moduli space of Riemann surfaces., Fixed typos and clarified exposition. Modified Section 5

In this paper we conjecture the stability and vanishing of a large piece of the unstable rational cohomology of SL_n Z, of mapping class groups, and of Aut(F_n).

These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the second cohomology group of the level p congruence subgroup of the mapping class group, following my papers "The second rational homology group of the moduli space of curves with level structures" and "The Picard group of the moduli space of curves with level structures".

Preface. Conference Programme. Conference Picture. List of participants. Authors' addresses. Cohomological Methods. Lectures on algebro-geometric Chern-Weil and Cheeger-Chern-Simons theory for vector brundles Bloch, et al. Deligne cohomology and the geometric (co-)bar constructions P. Gajer. Kuga-Satake varieties and the Hodge conjecture B. van Geemen.Hodge and Weil classes on abelian varieties K.V. Murty. Bloch-Kato conjecture and motivic cohomology with finite coefficients A. Suslin, V. Voevodsky. Chow Groups and Motives. Indecomposable higher Chow cycles B.B. Gordon, J.D. Lewis. Equivalence relations on algebraic cycles U. Jannsen. Letter to Dick Gross on higher Abel-Jacobi maps U. Jannsen. Finiteness of torsion in the codimension-two Chow group: An Axiomatic Approach A. Langer. Algebraic cycle complexes Basic Properties S. Muller-Stach. Algebraic cycles on abelian varieties Application of abstract Fourier theory J.P. Murre. Motives and filtrations on Chow groups, II S. Saito. Zero cycles on singular varietis V. Srinivas. Arithmetic Methods. Prepotentials of Yukawa couplings of certain Calabi-Yau 3-folds and mirror symmetry M. Saito. Weight-monodromy conjecture for l-adic representations associated to modular forms: A supplement to the paper 'IO' T. Saito. Cohomology computations related to the l-adic Abel-Jacobi map module l C. Schoen. Integral elements in K-theory and products of modular curves A.J. Scholl. Appendix to Scholl's article: A counterexample to a conjecture of Beilinson R. de Jeu. Reduction of abelian varieties A. Silverberg, Y. Zahrin. The arithmetic of certain Calabi-You varieties over number fields N. Yui. Classical and elliptic polylogarithms and special values of L-series D. Zagier, H.Gangl.

The goal of this paper is to give a cohomological characterization of F n , t {F_{n,t}} , where F n , t := Ker ⁡ ( ( n + t ; n ) O P n ( − t ) → O P n ) {F_{n,t}}: = \operatorname {Ker} ((n + t;n){\mathcal {O}_{{{\mathbf {P}}^n}}}( - t) \to {\mathcal {O}_{{{\mathbf {P}}^n}}}) .

The main result of this paper is a version of the Atiyah-Singer index theorem for Dirac-type operators on (noncompact) complete Riemannian manifolds. The statement of the theorem involves a novel "cohomology" theory for such manifolds. This theory, called exotic cohomology, depends on the structure at infinity of a space; more precisely, it depends on the way that large bounded sets fit together. For each cohomology class in this theory, we define a "higher index" of a Dirac-type operator, enjoying the stability and vanishing properties of the usual Atiyah-Singer index; these higher indices are analogous to the Novikov higher signatures. Our main theorem will compute these higher indices in terms of standard topological invariants. Applications of this index theorem include a different approach to some of the results of Gromov and Lawson [10] on topological obstructions to positive scalar curvature. The concept of index that we will use involves the A^-theory functors K0 and Kx for operator algebras [3, 14]. Suppose that B is an ideal in a unital algebra C, and let T e C be invertible modulo B. (In the classical Atiyah-Singer index theorem, one takes C to be the bounded operators on the L space of some compact manifold, B the compact operators, and T an elliptic pseudodifferential operator of order zero.) Then T has an "index" in the ^-theory group KQ(B) (in the classical case this is just Z , and one recovers the usual Fredholm index). Now let M be a complete Riemannian manifold, possibly noncompact. In [18] I introduced an algebra %?(M) which is defined as follows: Sf(M) consists of all bounded operators A on L(M) that have a kernel representation

This paper sketches the proofs that (1) an automorphism of a complete filtered ring is a limit of successive approximations, (2) given an nth order approximate automorphism, there is an obstruction to prolonging it to an (n + l)st order approximation, the obstruction lying in a certain 2nd cohomology group, and (3) the mapping which sends an nth order approximate automorphism to its obstruction is a crossed homomorphism from the multiplicative group of nth order approximate automorphisms to the (additive) 2nd cohomology group containing the obstructions. The rings in question need not be associative: we tacitly assume that there is given a "category of interest," 6, in the sense of [l ] (which may be, in particular, the category of associative, Lie, or commutative associative rings), and "ring" and "morphism" are meant relatively to