Gysin Maps Research Articles

Virtual Riemann–Roch theorems for almost perfect obstruction theories

This is the third in a series of works devoted to constructing virtual structure sheaves and K-theoretic invariants in moduli theory. The central objects of study are almost perfect obstruction theories, introduced by Y.-H. Kiem and the author as the appropriate notion in order to define invariants in K-theory for many moduli stacks of interest, including generalized K-theoretic Donaldson–Thomas invariants. In this paper, we prove virtual Riemann–Roch theorems in the setting of almost perfect obstruction theory in both the non-equivariant and equivariant cases, including cosection localized versions. These generalize and remove technical assumptions from the virtual Riemann–Roch theorems of Fantechi–Göttsche and Ravi–Sreedhar. The main technical ingredients are a treatment of the equivariant K-theory and equivariant Gysin map of sheaf stacks and a formula for the virtual Todd class.

Nonproper intersection products and generalized cycles

We develop intersection theory in terms of the {{mathscr {B}}}-group of a reduced analytic space. This group was introduced in a previous work as an analogue of the Chow group; it is generated by currents that are direct images of Chern forms and it contains all usual cycles. However, contrary to Chow classes, the {{mathscr {B}}}-classes have well-defined multiplicities at each point. We focus on a {{mathscr {B}}}-analogue of the intersection theory based on the Stückrad–Vogel procedure and the join construction in projective space. Our approach provides global {{mathscr {B}}}-classes which satisfy a Bézout theorem and have the expected local intersection numbers. We also introduce {{mathscr {B}}}-analogues of more classical constructions of intersections using the Gysin map of the diagonal. These constructions are connected via a {{mathscr {B}}}-variant of van Gastel’s formulas. Furthermore, we prove that our intersections coincide with the classical ones on cohomology level.

Fundamental classes in motivic homotopy theory

We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated twisted bivariant theory, extending the formalism of Fulton and MacPherson. We import the tools of Fulton’s intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess of intersection, self-intersections, and blow-ups. We also develop a theory of Euler classes of vector bundles in this setting. For the Milnor–Witt spectrum recently constructed by Déglise–Fasel, we get a bivariant theory extending the Chow–Witt groups of Barge–Morel, in the same way the higher Chow groups extend the classical Chow groups. As another application we prove a motivic Gauss–Bonnet formula, computing Euler characteristics in the motivic homotopy category.

LOCALIZATION BY -PERIODIC COMPLEXES AND VIRTUAL STRUCTURE SHEAVES

AbstractIn [12], Kim and the first author proved a result comparing the virtual fundamental classes of the moduli spaces of $\varepsilon $ -stable quasimaps and $\varepsilon $ -stable $LG$ -quasimaps by studying localized Chern characters for $2$ -periodic complexes.In this paper, we study a K-theoretic analogue of the localized Chern character map and show that for a Koszul $2$ -periodic complex it coincides with the cosection-localized Gysin map of Kiem and Li [11]. As an application, we compare the virtual structure sheaves of the moduli space of $\varepsilon $ -stable quasimaps and $\varepsilon $ -stable $LG$ -quasimaps.

K-Theoretic Generalized Donaldson–Thomas Invariants

Abstract We introduce the notion of almost perfect obstruction theory on a Deligne–Mumford stack and show that stacks with almost perfect obstruction theories have virtual structure sheaves, which are deformation invariant. The main components in the construction are an induced embedding of the coarse moduli sheaf of the intrinsic normal cone into the associated obstruction sheaf stack and the construction of a $K$-theoretic Gysin map for sheaf stacks. We show that many stacks of interest admit almost perfect obstruction theories. As a result, we are able to define virtual structure sheaves and $K$-theoretic classical and generalized Donaldson–Thomas invariants of sheaves and complexes on Calabi–Yau three-folds.

Localizing virtual structure sheaves for almost perfect obstruction theories

Abstract Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the K-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a K-theoretic wall-crossing formula for simple $\mathbb{C} ^\ast $ -wall crossings and define K-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.

Quantum singularity theory via cosection localization

Abstract We generalize the cosection localized Gysin map to intersection homology and Borel–Moore homology, which provides us with a purely topological construction of the Fan–Jarvis–Ruan–Witten invariants and some GLSM invariants.

Gysin maps, duality, and Schubert classes

We establish a Gysin formula for Schubert bundles and a strong version of the duality theorem in Schubert calculus on Grassmann bundles. We then combine them to compute the fundamental classes of Schubert bundles in Grassmann bundles, which yields a new proof of the Giambelli formula for vector bundles.

HILBERT STRATIFOLDS AND A QUILLEN TYPE GEOMETRIC DESCRIPTION OF COHOMOLOGY FOR HILBERT MANIFOLDS

In this paper we use tools from differential topology to give a geometric description of cohomology for Hilbert manifolds. Our model is Quillen’s geometric description of cobordism groups for finite-dimensional smooth manifolds [Quillen, ‘Elementary proofs of some results of cobordism theory using steenrod operations’, Adv. Math., 7 (1971)]. Quillen stresses the fact that this construction allows the definition of Gysin maps for ‘oriented’ proper maps. For finite-dimensional manifolds one has a Gysin map in singular cohomology which is based on Poincaré duality, hence it is not clear how to extend it to infinite-dimensional manifolds. But perhaps one can overcome this difficulty by giving a Quillen type description of singular cohomology for Hilbert manifolds. This is what we do in this paper. Besides constructing a general Gysin map, one of our motivations was a geometric construction of equivariant cohomology, which even for a point is the cohomology of the infinite-dimensional space $BG$, which has a Hilbert manifold model. Besides that, we demonstrate the use of such a geometric description of cohomology by several other applications. We give a quick description of characteristic classes of a finite-dimensional vector bundle and apply it to a generalized Steenrod representation problem for Hilbert manifolds and define a notion of a degree of proper oriented Fredholm maps of index $0$.

Some mixed Hodge structures on $$l^{2}$$ -cohomology groups of coverings of Kähler manifolds

We give methods to compute l^2-cohomology groups of a covering manifolds obtained by removing pullback of a (normal crossing) divisor to a covering of a compact K\"ahler manifold. We prove that in suitable quotient categories, these groups admit natural mixed Hodge structure whose graded pieces are given by expected Gysin maps.

Localizing virtual cycles by cosections

We show that a cosection of the obstruction sheaf of a perfect obstruction theory localizes the virtual cycle to the non-surjective locus of the cosection. We give algebraic constructions of localized Gysin maps and localized virtual cycles. Various applications of these constructions are discussed.

An equivariant generalization of the Miller splitting theorem

Let G be a compact Lie group. We build a tower of G-spectra over the suspension spectrum of the space of linear isometries from one G-representation to another. The stable cofibres of the maps running down the tower are certain interesting Thom spaces. We conjecture that this tower provides an equivariant extension of Miller's stable splitting of Stiefel manifolds. We provide a cohomological obstruction to the tower producing a splitting in most cases; however, this obstruction does not rule out a split tower in the case where the Miller splitting is possible. We claim that in this case we have a split tower which would then produce an equivariant version of the Miller splitting, we prove this claim in certain special cases though the general case remains a conjecture. To achieve these results we construct a variation of the functional calculus with useful homotopy-theoretic properties and explore the geometric links between certain equivariant Gysin maps and residue theory.

Multicurves and equivariant cohomology

Let $A$ be a finite abelian group. The author sets up an algebraic framework for studying $A$-equivariant complex-orientable cohomology theories in terms of a suitable kind of equivariant formal group. He computes the equivariant cohomology of many spaces in these terms, including projective bundles (and associated Gysin maps), Thom spaces, and infinite Grassmannians.

Comparing two approaches to the K-theory classification of D-branes

We consider the two main classification methods of D-brane charges via K-theory, in type II superstring theory with vanishing B -field: the Gysin map approach and the one based on the Atiyah–Hirzebruch spectral sequence. Then, we find out an explicit link between these two approaches: the Gysin map provides a representative element of the equivalence class obtained via the spectral sequence. We also briefly discuss the case of rational coefficients, characterized by a complete equivalence between the two classification methods.

Gysin maps in oriented theories

We prove that Gysin maps (transfer maps along closed embeddings of smooth varieties) in an oriented cohomology theory are compatible with compositions of closed embeddings. As opposed to the proof given by Panin, we do not use characteristic classes of vector bundles. Our proof is more straightforward. It is based on the properties of Thom isomorphisms and the deformation to the normal cone isomorphisms. We also provide a simpler proof of the additivity property of Gysin maps based on the use of Gysin maps with support.

On smooth Chas–Sullivan loop product in Quillen's geometric complex cobordism of Hilbert manifolds

In [A.J. Baker, C. Ozel, Complex cobordism of Hilbert manifolds with some applications to flag varieties, Contemp. Math. 258 (2000) 1–19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In [C. Ozel, On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups, in preparation], by using Quinn's Transversality Theorem [F. Quinn, Transversal approximation on Banach manifolds, Proc. Sympos. Pure Math. 15 (1970) 213–222], it has been shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It has been shown that the Thom isomorphism in this theory was satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps has been proved. In [M. Chas, D. Sullivan, String topology, math.GT/9911159, 1999], Chas and Sullivan described an intersection product on the homology of loop space LM. In [R.L. Cohen, J.D.S. Jones, A homotopy theoretic realization of string topology, math.GT/0107187, 2001], R. Cohen and J. Jones described a realization of the Chas–Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. In this paper, we will extend this product on cobordism and bordism theories.

Nonemptiness of symmetric degeneracy loci

Let V be a rank N vector bundle on a d -dimensional complex projective scheme X ; assume that V is equipped with a quadratic form with values in a line bundle L and that S 2 V * ⊗ L is ample. Suppose that the maximum rank of the quadratic form at any point of X is r > 0. The main result of this paper is that if d > N - r , the locus of points where the rank of the quadratic form is at most r - 1 is nonempty. We give some applications to subschemes of matrices, and to degeneracy loci associated to embeddings in projective space. The paper concludes with an appendix on Gysin maps. The main result of the appendix, which may be of independent interest, identifies a Gysin map with the natural map from ordinary to relative cohomology.

Equivariant Gysin maps and pulling back fixed points

We develop a new approach to the pulling back fixed points theorem of W. Browder and use it in order to prove various generalizations of this result.

An Equivariant Analog of the Poincaré―Hopf Theorem

A new method for localization of algebro-topological invariants of smooth manifolds via equivariant tangent vector fields is presented. Main realizations of direct image constructions (the Gysin map and the Becker―Gottlieb transfer map) are calculated for Grassmannizations of complex vector bundles and for a complex-oriented cohomology theory. Bibliography: 12 titles.

A generalization of Coleman's p-adic integration theory

We associate to a scheme X smooth over a p-adic ring a kind of cohomology group H i fp (X,j). For proper X this cohomology has Poincare duality hence Gysin maps and cycle class maps which are reasonably explicit. For zero-cycles we show that the cycle class map is given by Coleman integration. The cohomology theory H fp is therefore interpreted as giving a generalization of Coleman’s theory. We find an embedding H syn 2i (X,i)↪H fp 2i (X,i) where H syn is (rigid) syntomic cohomology. Our main result is an explicit description of the syntomic Abel-Jacobi map in terms of generalized Coleman integration.