Projective modules over smooth real affine varieties

Witten's top Chern class is a particular cohomology class on the moduli space of Riemann surfaces endowed with r-spin structures. It plays a key role in Witten's conjecture relating to the intersection theory on these moduli spaces. Our first goal is to compute the integral of Witten's class over the so-called double ramification cycles in genus 1. We obtain a simple closed formula for these integrals. This allows us, using the methods of [15], to find an algorithm for computing the intersection numbers of the Witten class with powers of the \psi-classes (or tautological classes) over any moduli space of r-spin structures, in short, all numbers involved in Witten's conjecture.

We discuss secondary (and higher) characteristic classes for algebraic vector bundles with trivial top Chern class. We then show that if $X$ is a smooth affine scheme of dimension $d$ over a field $k$ of finite 2-cohomological dimension (with $\mathrm{char}(k)\neq 2)$ and $E$ is a rank $d$ vector bundle over $X$, vanishing of the Chow-Witt theoretic Euler class of $E$ is equivalent to vanishing of its top Chern class and these higher classes. We then derive some consequences of our main theorem when $k$ is of small 2-cohomological dimension.

On any smooth affine surface over an algebraically closed field, it is easy to construct rank two projective modules with given determinant as the first Chern class and an arbitrary rational equivalence class of a zero cycle as the second Chern class. We will briefly indicate this method in Section 1. Here we construct projective modules over smooth affine 3-folds with prescribed Chern classes and of the right rank. That is to say, we construct rank two projective modules with prescribed first and second Chem classes and rank three projective modules with prescribed first, second and third Chern classes (Theorem 2.1). As a corollary, using Roitman's theorem on torsion in zero cycles and Suslin's cancellation theorem, one gets that a rank three projective module over a smooth affine threefold with top Chem class zero has a free direct summand of rank one (Cor. 2.4). For example, over a rational threefold any projective module splits into a free module and a rank two projective module. Thus we obtain necessary and sufficient conditions in terms of appropriate Chern classes for modules to be efficiently generated. For example, on an affine rational 3-fold any line bundle is 3-generated and any rank two projective module is 4-generated. We also prove that any maximal ideal over a smooth affine variety of dimension d is the zero of a section of a rank d projective module (Theorem 3.1). For an affine 3-fold, if A3(X) = 0, we prove the validity of the Eisenbud-Evans estimate for finitely generated modules. We do not know the answers for most of these questions in higher dimensions. Sometimes we assume that the characteristic of the ground field is different from 2, 3 and 5 due to technical reasons.**

Del Pezzo surfaces with irregularity and intersection numbers on quotients in geometric invariant theory Zachary Alexander Maddock This thesis comprises two parts covering distinct topics in algebraic geometry. In Part I, we construct the first examples of regular del Pezzo surfaces for which the first cohomology group of the structure sheaf is nonzero. Such surfaces, which only exist over imperfect fields, arise as generic fibres of fibrations of singular del Pezzo surfaces in positive characteristic whose total spaces are smooth, and their study is motivated by the minimal model program. We also find a restriction on the integer pairs that are possible as the irregularity (that is, the dimension of the first cohomology group of the structure sheaf) and anti-canonical degree of regular del Pezzo surfaces with positive irregularity. In Part II, we consider a connected reductive group acting linearly on a projective variety over an arbitrary field. We prove a formula that compares intersection numbers on the geometric invariant theory quotient of the variety by the reductive group with intersection numbers on the geometric invariant theory quotient of the variety by a maximal torus, in the case where all semi-stable points are properly stable. These latter intersection numbers involve the top equivariant Chern class of the maximal torus representation given by the quotient of the adjoint representation on the Lie algebra of the reductive group by that of the maximal torus. We provide a purely algebraic proof of the formula when the root system decomposes into irreducible root systems of type A. We are able to remove this restriction on root systems by applying a related result of Shaun Martin from symplectic geometry.

In this article, we study various kinds of indices of a vector field on a singular variety and as an application, we prove, for a compact strong local complete intersection V with isolated singularities, a formula expressing the Euler-Poincare characteristic x (^ ) 0 I ^ i n terms of the top Chern class of the tangent bundle of V and the Milnor numbers of the singularities (Theorem 2.4). For a vector field t i o n a singular variety V, we consider the index, the GSV-index and the virtual at the singularity of v. All these reduce to the usual Poincare-Hopf index when the singularity of v is in the regular part of V, so we compare them when it is in the singular part of V. M.-H. Schwartz defined an index for vector fields on a singular variety V, see [21, 4]. When the singularities of V are isolated, as they are in this article, this definition can be easily extended to vector fields which are not radial. We do this in Sec. 1 below and we call the corresponding index the Schwartz index of a vector field. We show that, for a global vector field with isolated singularities on a compact variety V, the sum of the Schwartz indices gives x(V) (Theorem 1.2). In [12] there is a definition of a local index for stratified vector fields on singular varieties, extending Schwartz' definition for radial vector fields. Presumably our definition of the Schwartz index coincides with that in [12]. We then recall the GSV-index, which is defined in [22, 9, 23]. It is defined for a vector field on a local complete intersection V in a complex manifold M and it

0.1. Hodge integrals. Let Mg,n be the Deligne-Mumford moduli stack of stable curves of genus g with n marked points. Let A ∗ (Mg,n) denote the Chow ring with Q-coefficients. We will consider two types of tautological

We study higher-genus Fan–Jarvis–Ruan–Witten theory of any chain polynomial with any group of symmetries. Precisely, we give an explicit way to compute the cup product of Polishchuk and Vaintrob’s virtual class with the top Chern class of the Hodge bundle. Our formula for this product holds in any genus and without any assumption on the semi-simplicity of the underlying cohomological field theory.

Let T be a maximal torus of a connected reductive group G that acts linearly on a projective variety X so that all semi-stable points are stable. This paper compares the integration on the geometric invariant theory quotient X//G of Chow classes σ to the integration on the geometric invariant theory quotient X//T of certain lifts of σ twisted by c top(g/t), the top Chern class of the T-equivariant vector bundle induced by the quotient of the adjoint representation on the Lie algebra of G by that of T. We provide a purely algebraic proof that the ratio between any two such integrals is an invariant of the group G and that it equals the order of the Weyl group whenever the root system of G decomposes into irreducible root systems of type A n , for various \( n\in \mathbb{N} \). As a corollary, we are able to remove this restriction on root systems by applying a related result of Martin from symplectic geometry.

We show how the “finite Quot scheme method” applied to Le Potier’s strange duality on del Pezzo surfaces leads to conjectures (valid for all smooth complex projective surfaces) relating two sets of universal power series on Hilbert schemes of points on surfaces: those for top Chern classes of tautological sheaves and those for Euler characteristics of line bundles. We have verified these predictions computationally for low order. We then give an analysis of these conjectures in small ranks. We also give a combinatorial proof of a formula predicted by our conjectures: the top Chern class of the tautological sheaf on $S^{[n]}$ associated to the structure sheaf of a point is equal to $(-1)^n$ times the nth Catalan number.

In this paper we study projective modules over subrings of k [ X , Y ] k[X,Y] . Conditions are given for projective modules to decompose into free ⊕ \oplus rank 1 modules. Our main result is that if k is an algebraically closed field and A a subring of B = k [ X , Y ] B = k[X,Y] with A ⊂ B A \subset B integral and sing ( A ) {\text {sing}}(A) finite, then all f.g. projective A-modules have the form free ⊕ \oplus rank 1. We also give several examples of subrings of k [ X , Y ] k[X,Y] which have indecomposable projective modules of rank 2.

We investigate compact projective generators in the category of equivariant -modules on a smooth affine variety. For a reductive group G acting on a smooth affine variety X, there is a natural countable set of compact projective generators indexed by finite dimensional representations of G. We show that only finitely many of these objects are required to generate; thus the category has a single compact projective generator. The proof goes via an analogous statement about compact generators in the equivariant derived category, which holds in much greater generality and may be of independent interest.

This paper deals with the unit groups of the endomor- phism rings of projective modules over polynomial rings and further over formal power series rings. A normal subgroup of the unit group is dened and discussed. The local splitting properties of elements of endomorphism rings of projective modules over polynomial rings are given. So, if P is a projective module over a ring A, then (id+ xEndA(x)(P(x))) is constructed. This is a normal subgroup of the unit group of endomorphism rings of a projective module over a poly- nomial ring. If (A,m) is a local ring with dim(A) ≥ µ(m), then we show that (id+ xEndA(x)(P(x))) is a normal subgroup of SLA(x)(P(x)). Also, we will show that under these conditions the similar conclusion can be drawn for the ring of formal power series. In section 3, let P be a projective A-module. Let s1, s2 ∈ A be such that As1 + As2 = A. Then we use the splitting lemma of Quillen to show that every element of (idPs1s2 + xEnd As1s2(x) (Ps1s2 (x))) has two decompositions. Finally, we generalize the result to Theorem 3.4.

We show that for every $n\ge 7$ there is a smooth rational affine variety with exotic model. Moreover, we show that for every $n\ge 6$ there are Zariski open subsets $U_1, U_2$ of $\mathbb {C}^n$, such that they are holomorphically but not algebraically equivalent.

We describe maximal commutative unipotent subgroups of the automorphism group \mathrm{Aut}(X) of an irreducible affine variety X . Furthermore, we show that a group isomorphism \mathrm{Aut}(X) \allowbreak\to \mathrm{Aut}(Y) maps unipotent elements to unipotent elements, where Y is irreducible and affine. Using this result, we show that the automorphism group detects sphericity and the weight monoid.As an application, we show that an affine toric variety different from an algebraic torus is determined by its automorphism group among normal irreducible affine varieties, and we show that a smooth affine spherical variety different from an algebraic torus is determined by its automorphism group (up to an automorphism of the base field) among smooth irreducible affine varieties. This generalizes results obtained by Cantat, Kraft, Liendo, Urech, Xie and the authors.

Let $W$ be an irreducible subvariety of codimension $r$ in a smooth affine variety $X$ of dimension $n$ defined over the complex field $C$. Suppose that $W$ is left pointwise fixed by an automorphism of $X$ of infinite order or by a one-dimensional algebraic torus action on $X$. In the present article, we consider whether or not $X$ is then an affine space bundle over $W$ of fiber dimension $n-r$. Our results concern the case $r=1$ or the case $r=2$ and $n\\leq3$. As by-products, we obtain algebro-topological characterizations of the affine 3-space.