Kähler–de Rham Cohomology and Chern Classes

There is a surprising occurrence of some minus signs in the isomorphisms produced in the well-known technique of dimension shifting in calculating derived functors in homological algebra. We explicitly determine these signs. Getting these signs right is important in order to avoid basic contradictions. We illustrate the result − which we call as the sign lemma for dimension shifting − by some de Rham cohomology and Chern class considerations for compact Riemann surfaces.

These notes offer a short presentation of de Rham cohomology on differentiable manifolds with a proof of the isomorphism between de Rham cohomology and singular cohomology. We also recall some duality results, show how Betti numbers relate to Morse theory and shortly present the relation between de Rham and Cech cohomologies with coefficients in the sheaf of real locally constant functions for de Rham cohomology.

This chapter describes a general group-theoretical property enjoyed by fundamental groups of compact Kahler manifolds. In general, the condition that an infinitesimal deformation be tangent to an analytic path is equivalent to the existence of a formal power series deformation with the infinitesimal deformation as its leading term. If a k-jet of a deformation exists, the condition that this k-jet of a deformation extend to a -jet of a deformation is given by the vanishing of a certain cohomology class, which depends on the previous k terms of the series. The chapter discusses the Lemma C is proved, the canonical isomorphisms between de Rham cohomology and Eilenberg-MacLane cohomology, Theorem A is reduced to Theorem D. The proofs of Theorems A and C involve the harmonic theory of M with coefficients in a flat vector bundle. The flat vector bundle we shall need is the complexification of the flat vector bundle ad P, denote this flat complex vector bundle by E.

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.**

In this chapter, we study the topology of C ∞-manifolds. We define the de Rham cohomology of a manifold, which is the vector space of closed differential forms modulo exact forms. After sheafifying the construction, we see that the de Rham complex forms a so-called acyclic resolution of the constant sheaf ℝ. We prove a general result that sheaf cohomology can be computed using such resolutions, and deduce a version of de Rham’s theorem that de Rham cohomology is sheaf cohomology with coefficients in ℝ. It follows that de Rham cohomology depends only on the underlying topology. Using a different acyclic resolution that is dual to the de Rham complex, we prove Poincaré duality. This duality makes cohomology, which is normally contravariant, into a covariant theory. We devote a section to explaining these somewhat mysterious covariant maps, called Gysin maps. We end this chapter with the remarkable Lefschetz trace formula, which in principle, calculates the number of fixed points for a map of a manifold to itself.

Combinatorial ideas are developed in this article to study Chern numbers on ample and numerically effective vector bundles. An effective lower bound for Chern numbers of ample vector bundles is established, which makes some progress toward a long-standing question. Along this line we prove that Chern numbers on nef vector bundles obey reverse dominance ordering, which improves upon some classical and recent results. We propose a simultaneous positivity question on (signed) Chern numbers of compact complex or Kähler manifolds whose (co)tangent bundles are semipositive in various senses, and show that it holds true for compact homogeneous complex manifolds.

This note is dedicated to the second author’s teacher, Professor Atsuo Komatsu, in celebration of his seventieth birthday.It is well known [4], [7] that the theory of characteristic classes in general cohomology is essentially based upon one theorem, the Leray-Hirsch Theorem. We further claim [8] that the entire theory could be developed merely from the Künneth Formula if under suitable conditions a truly relative version of the Meyer-Vietoris sequence exists in the general cohomology theory. In his lectures delivered at Aarhus in 1968, Dold [6] set up the necessary machinery including the Leray-Hirsch Theorem to define Chern Classes with values in general cohomology. However, he then stated [6, p. 47] that he “found a difficulty here in choosing adequate orientations (Thorn Classes) for the bundles involved”, and proceeded differently, discarding the “classical” approach used in both ordinary cohomology [9] and K-theory [3]. Later, he [7] published a more categorical work, although the approach to Chern Classes was basically unchanged from his previous work. Consequently, the possibility of the classical direct approach to the theory has remained open.

In this article, we construct Chern classes in rational Deligne cohomology for coherent sheaves on a smooth complex compact manifold. We prove that these classes satisfy the functoriality property under pullbacks, the Whitney formula and the Grothendieck–Riemann–Roch theorem for projective morphisms between smooth complex compact manifolds.

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In Chapter 6, we studied the first de Rham cohomology H 1 (M) of a manifold This measures the difference between exactness and local exactness of 1-forms on M and was shown to have interesting topological applications. Here we generalize these ideas, using the full Grassmann algebra A* (M) to produce a graded algebra H* (M), the de Rham cohomology algebra. The proper generalization of “locally exact 1-form” is “closed p-form”, defined as a p-form that is annihilated by “exterior differentiation”. Exact forms are closed and H P (M) measures the extent to which closed p-forms may fail to be exact. By Stokes’ theorem, the geometric boundary operator and exterior differentiation of forms are mutually adjoint operations in a certain precise sense. This is a generalization of the fundamental theorem of calculus and a powerful tool for computing cohomology. The reader who would like to pursue this theory further could hardly do better than to consult [5].

We define the quantum exterior product ∧ h \wedge _h and quantum exterior differential d h d_h on Poisson manifolds. The quantum de Rham cohomology, which is a deformation quantization of the de Rham cohomology, is defined as the cohomology of d h d_h . We also define the quantum Dolbeault cohomology. A version of quantum integral on symplectic manifolds is considered and the corresponding quantum Stokes theorem is stated. We also derive the quantum hard Lefschetz theorem. By replacing d d by d h d_h and ∧ \wedge by ∧ h \wedge _h in the usual definitions, we define many quantum analogues of important objects in differential geometry, e.g. quantum curvature. The quantum characteristic classes are then studied along the lines of the classical Chern-Weil theory. The quantum equivariant de Rham cohomology is defined in the similar fashion.

New techniques are developed, based on the consideration of the projective bundle associated with a direct sum of two vector bundles, to give a simpler solution of the problem of blowing up Chern classes which was previously solved by Porteous[12] using the Grothendieck Riemann-Roch theorem.

In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the context of complete toric varieties fit nicely with a generalized Hirzebruch‐Riemann‐Roch theorem. As important special cases, we obtain new (or recover well‐known) formulae for the Baum‐Fulton‐MacPherson Todd (or MacPherson's Chern) classes of toric varieties, as well as for the Thom‐Milnor L‐classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of the torus‐orbit decomposition and the motivic properties of the motivic Chern and respectively homology Hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the (dual) Todd classes of closures of orbits. This method even applies to torus‐invariant subspaces of a given toric variety. The obtained formula is then applied to weighted lattice‐point counting in lattice polytopes and their subcomplexes, yielding generalized Pick‐type formulae. Second, in the case of simplicial toric varieties, we compute our characteristic classes by using the Lefschetz‐Riemann‐Roch theorem of Edidin‐Graham in the context of the geometric quotient description of such varieties. In this setting, we define mock Hirzebruch classes of simplicial toric varieties (which specialize to the mock Chern, mock Todd, and mock L‐classes of such varieties) and investigate the difference between the (actual) homology Hirzebruch class and the mock Hirzebruch class. We show that this difference is localized on the singular locus, and we obtain a formula for it in which the contribution of each singular cone is identified explicitly. Finally, the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by Cappell and Shaneson in the early 1990s.© 2015 Wiley Periodicals, Inc.

Let <TEX>$M_c$</TEX> = M(2, 0, c) be the moduli space of O(l)-semistable rank 2 torsion-free sheaves with Chern classes <TEX>$c_1=0\;and\;c_2=c$</TEX> on a K3 surface X, where O(1) is a generic ample line bundle on X. When <TEX>$c=2n\geq4$</TEX> is even, <TEX>$M_c$</TEX> is a singular projective variety equipped with a holomorphic symplectic structure on the smooth locus. In particular, <TEX>$M_c$</TEX> has trivial canonical divisor. In [22], O'Grady asks if there is any symplectic desingularization of <TEX>$M_{2n}$</TEX> for <TEX>$n\geq3$</TEX>. In this paper, we show that there is no crepant resolution of <TEX>$M_{2n}$</TEX> for <TEX>$n\geq3$</TEX>. This obviously implies that there is no symplectic desingularization.

Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent works of Druel and Greb–Guenancia–Kebekus this establishes the Beauville–Bogomolov decomposition for minimal models with trivial canonical class.