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In 1956 Hirzebruch found an explicit formula for the denominators of the Todd polynomials, which was proved later in his joint work with Atiyah. We present a new formula for the Todd polynomials in terms of the "forgotten symmetric functions," which follows from our previous work on complex cobordisms. In particular, this leads to a simpler proof of the Hirzebruch formula and provides new interpretations for the Hirzebruch numbers. [ABSTRACT FROM AUTHOR]

A new framework is proposed for computing general numerical invariants, called Bertini type invariants, associated to a family of polynomial ideals depending on deformation parameters. A deterministic algorithm is introduced for computing parameter dependency of Bertini type invariants. As an application, an algorithm for computing Chern–Schwartz–MacPherson classes of a family of singular projective hypersurfaces is obtained. [ABSTRACT FROM AUTHOR]

A space X is W-trivial if for every real vector bundle α over X the total Stiefel-Whitney class w (α) is 1. It follows from a result of Milnor that if X is an orientable closed smooth manifold of dimension 1, 2, 4 or 8, then X is not W-trivial. In this note we completely characterize W-trivial orientable connected closed smooth manifolds in dimensions 3, 5 and 6. In dimension 7, we describe necessary conditions for an orientable connected closed smooth 7-manifold to be W-trivial. [ABSTRACT FROM AUTHOR]

We prove a general local rigidity theorem for pull-backs of homogeneous forms on reductive symmetric spaces under representations of discrete groups. One application of the theorem is that the volume of a closed manifold locally modelled on a reductive homogeneous space G/H is constant under deformation of the G/H-structure. The proof elaborates on an argument given by Labourie for closed anti-de Sitter 3-manifolds. The core of the work is a reinterpretation of old results of Cartan, Chevalley and Borel, showing that the algebra of G-invariant forms on G/H is generated by "Chern–Weil forms" and "Chern–Simons forms". [ABSTRACT FROM AUTHOR]

Chain total double complexes with reductive differentials for non-abelian simplexes with associated spaces are considered. It is conjectured that corresponding relative cohomology is equivalent to the coset space of vanishing functionals over non-vanishing functionals related to differentials of complexes. The conjecture is supported by the theorem for the case of spaces of correlation functions and generalized connections on vertex operator algebra bundles. [ABSTRACT FROM AUTHOR]

In this paper, we establish an intrinsic Gauss-Bonnet-Chern formula for Finsler manifolds by using the Mathai-Quillen's superconnection formalism, in which no extra vector field is involved. Furthermore, we prove a more general Lichnerowicz formula in this direction through a geometric localization procedure. [ABSTRACT FROM AUTHOR]

We discuss the existence of almost complex structures on closed hyperbolic manifolds of even dimension at least four. We prove that for n = 2 and for all odd n every hyperbolic 2n-manifold has a finite covering admitting an almost complex structure. Conjecturally this should be true for all n. For n = 4 we prove it for arithmetic manifolds. [ABSTRACT FROM AUTHOR]

For any regular Lie algebroid A, the kernel K and the image F of its anchor map ρA together with A itself fit into a short exact sequence, called the Atiyah sequence, of Lie algebroids. We prove that the Atiyah and Todd classes of dg manifolds arising from a regular Lie algebroid respect the Atiyah sequence, i.e., the Atiyah and Todd classes of A restrict to the Atiyah and Todd classes of the bundle K of Lie algebras on the one hand, and project onto the Atiyah and Todd classes of the integrable distribution F ⊆ TM on the other hand. [ABSTRACT FROM AUTHOR]

Let the circle act in a Hamiltonian fashion on a compact symplectic manifold (M , ω) of dimension 2n. Then the S 1 -action has at least n + 1 fixed points. We study the case when the fixed point set consists of precisely n + 1 isolated points. We first show certain equivalence on the first Chern class of M and some particular weight of the S 1 -action at some fixed point. Then we show that the particular weight can completely determine the integral cohomology ring of M, the total Chern class of M, and the sets of weights of the S 1 -action at all the fixed points. We will see that all these data are isomorphic to those of known examples, C P n , or G ~ 2 (R n + 2) with n ≥ 3 odd, equipped with standard circle actions. [ABSTRACT FROM AUTHOR]

By refining a result of Farrell and in terms of the Jiang subgroups of self-maps of smooth manifolds, we obtain a sufficient condition for the vanishing of the signature and the Hirzebruch's χ y -genus. Along this line we show that the χ y -genus of a non-positively curved compact Kähler manifold vanishes when the center of its fundamental group is non-trivial, which realizes an expectation of Farrell's in the Kähler setting. Moreover, in the latter case all the Chern numbers vanish whenever its complex dimension is no more than 4, which also provides some evidence towards a complex version's Hopf Conjecture proposed by the author and Zheng. [ABSTRACT FROM AUTHOR]

The orbifold Euler characteristic of the moduli space ℳ g ; 1 of genus g smooth curves with one marked point (g ≥ 1) was calculated by Harer and Zagier: χ (ℳ g ; 1) = ζ (1 − 2 g) = − B 2 g / (2 g) , where ζ is the Riemann zeta function and B2g is the (2g)th Bernoulli number. We give a shorter proof of this result using only formal power series and classical combinatorics. [ABSTRACT FROM AUTHOR]

The generalized Miller–Morita–Mumford classes of a manifold bundle with fiber M depend only on the underlying τ M -fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for τ M -fibrations, B a u t (τ M) , and its cohomology ring, i.e., the ring of characteristic classes of τ M -fibrations. For a bundle ξ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal ξ -fibration with holonomy in a given connected monoid, together with explicit cocycle representatives for the characteristic classes of the canonical bundle over its total space. This yields tools for computing the rational cohomology ring of B a u t (ξ) as well as the subring generated by the generalized Miller–Morita–Mumford classes. To illustrate, we carry out sample computations for spheres and complex projective spaces. We discuss applications to tautological rings of simply connected manifolds and to the problem of deciding whether a given τ M -fibration comes from a manifold bundle. [ABSTRACT FROM AUTHOR]

We prove that the only relation imposed on the Hodge and Chern numbers of a compact Kähler manifold by the existence of a nowhere zero holomorphic one-form is the vanishing of the Hirzebruch genus. We also treat the analogous problem for nowhere zero closed one-forms on smooth manifolds. [ABSTRACT FROM AUTHOR]

The author gives a definition of orbifold Stiefel-Whitney classes of real orbifold vector bundles over special q-CW complexes (i.e., right-angled Coxeter complexes). Similarly to ordinary Stiefel-Whitney classes, orbifold Stiefel-Whitney classes here also satisfy the associated axiomatic properties. [ABSTRACT FROM AUTHOR]

In this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional Spin -manifold have many non-trivial homotopy groups provided that the manifold admits such a metric. [ABSTRACT FROM AUTHOR]

We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the volume. The proof relies on Chern–Weil theory applied to a connection constructed from Euclidean connections on charts in which the metric tensor is harmonic and has bounded Hölder norm. We generalize this theorem to a Gromov–Hausdorff closed class of rough Riemannian manifolds defined in terms of Hölder regularity. Assuming an additional upper Ricci curvature bound, we show that also the Euler characteristic is bounded proportionally to the volume. Additionally, we remark on a volume comparison theorem for Betti numbers of manifolds with an additional upper bound on sectional curvature. It is a consequence of a result by Bowen. [ABSTRACT FROM AUTHOR]

We focus on the elliptic genera of level N at the cusps of a congruence subgroup for any complete intersection. Writing the first Chern class of a complete intersection as a product of an integral coefficient c1 and a generator of the 2nd integral cohomology group, we mainly discuss the values of the elliptic genera of level N for the complete intersection in the cases of c1 >, =, or < 0. In particular, the values about the Todd genus, Â-genus, and Ak-genus can be derived from the elliptic genera of level N. [ABSTRACT FROM AUTHOR]

We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of Chern forms are strongly nonnegative. This particularly implies that all the Chern numbers of such a holomorphic vector bundle are nonnegative and can be bounded below and above respectively by two special Chern numbers. As applications, we obtain a family of new results on compact connected complex manifolds which are homogeneous or can be holomorphically immersed into complex tori, some of which improve several classical results. [ABSTRACT FROM AUTHOR]

For two differentiable maps between two manifolds of possibly different dimensions, the local and global coincidence homology classes are introduced and studied by Bisi-Bracci-Izawa-Suwa (2016) in the framework of Čech-de Rham cohomology. We take up the problem from the combinatorial viewpoint and give some finer results, in particular for the local classes. As to the global class, we clarify the relation with the cohomology coincidence class as studied by Biasi-Libardi-Monis (2015). In fact they introduced such a class in the context of several maps and we also consider this case. In particular we define the local homology class and give some explicit expressions. These all together lead to a generalization of the classical Lefschetz coincidence point formula. [ABSTRACT FROM AUTHOR]

For a complete symplectic manifold M 2 n , we define the L 2 -hard Lefschetz property on M 2 n . We also prove that the complete symplectic manifold M 2 n satisfies L 2 -hard Lefschetz property if and only if every class of L 2 -harmonic forms contains a L 2 symplectic harmonic form. As an application, we get if M 2 n is a closed symplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler characteristic satisfies the inequality (- 1) n χ (M 2 n) ≥ 0 . [ABSTRACT FROM AUTHOR]

We consider a smooth algebraic variety with an action of a linear algebraic group acting with finitely many orbits. We study equivariant characteristic classes of the orbits, namely the equivariant motivic Chern classes, in the K‐theory of the ambient space. We prove that the motivic Chern class satisfies the axiom system inspired by that of 'K‐theoretic stable envelopes', recently defined by Okounkov and studied in relation with quantum group actions on the K‐theory algebra of moduli spaces. We also give explicit formulas for the equivariant motivic Chern classes of Schubert cells and matrix Schubert cells. Finally, we calculate the equivariant motivic Chern class of the orbits of the A2 quiver representation, which yields formulas for the motivic Chern classes of determinantal varieties and more general degeneracy loci. [ABSTRACT FROM AUTHOR]

We discuss the complex geometry of two complex five-dimensional Kähler manifolds which are homogeneous under the exceptional Lie group G 2 . For one of these manifolds, rigidity of the complex structure among all Kählerian complex structures was proved by Brieskorn; for the other one, we prove it here. We relate the Kähler assumption in Brieskorn's theorem to the question of existence of a complex structure on the six-dimensional sphere, and we compute the Chern numbers of all G 2 -invariant almost complex structures on these manifolds. [ABSTRACT FROM AUTHOR]

We study smooth bundles over surfaces with highly connected almost parallelizable fiber M of even dimension, providing necessary conditions for a manifold to be bordant to the total space of such a bundle and showing that, in most cases, these conditions are also sufficient. Using this, we determine the characteristic numbers realized by total spaces of bundles of this type, deduce divisibility constraints on their signatures and Â‐genera, and compute the second integral cohomology of BDiff+(M) up to torsion in terms of generalized Miller–Morita–Mumford classes. We also prove analogous results for topological bundles over surfaces with fiber M and discuss the resulting obstructions to smoothing them. [ABSTRACT FROM AUTHOR]

Let π: Mn → Pn be an n-dimensional small cover over Pn and λ: ℱ(Pn) → ℤ2n be its characteristic function. The author uses the symbol c(λ) to denote the cardinal number of the image Im(λ). If c(λ) = n + 1 or n + 2, then a necessary and sufficient condition on the existence of spin structure on Mn is given. As a byproduct, under some special conditions, the author uses the second Stiefel-Whitney class to detect when Pn is n-colorable or (n + 1)-colorable. [ABSTRACT FROM AUTHOR]

In this paper we study the mod 2 cohomology ring of the Grasmannian G ~ n , 3 of oriented 3-planes in R n . We determine the degrees of the indecomposable elements in the cohomology ring. We also obtain an almost complete description of the cohomology ring. This description allows us to provide lower and upper bounds on the cup length of G ~ n , 3 . As another application, we show that the upper characteristic rank of G ~ n , 3 equals the characteristic rank of γ ~ n , 3 , the oriented tautological bundle over G ~ n , 3 if n is at least 8. [ABSTRACT FROM AUTHOR]

This article is motivated by the original Casson invariant regarded as an integral lifting of the Rochlin invariant. We aim to defining an integral lifting of the Adams e-invariant of stably framed 3-manifolds, perhaps endowed with some additional structure. We succeed in doing so for manifolds which are links of normal complex Gorenstein smoothable singularities. These manifolds are naturally equipped with a canonical SU (2) -frame. To start we notice that the set of homotopy classes of SU (2) -frames on the stable tangent bundle of every closed oriented 3-manifold is canonically a Z -torsor. Then we define the E ^ -invariant for the manifolds in question, an integer that modulo 24 is the Adams e-invariant. The E ^ -invariant for the canonical frame equals the Milnor number plus 1, so this brings a new viewpoint on the Milnor number of the smoothable Gorenstein surface singularities. [ABSTRACT FROM AUTHOR]

Inspired by the work of Heller and Sasin [1], we construct in this paper Weil homomorphism in a locally free sheaf W of ϕ-fields [2] over a structured space. We introduce the notion of G-consistent, linear connection on this sheaf, what allows us to clearly define Chern, Pontrjagin and Euler characteristic classes. We also show proper equalities between those classes.

We compute the cup-length of flag manifolds F(2, 2, n3) and F(1, 3, 2s+1 − 3) and the height of the third Stiefel–Whitney characteristic class of the canonical vector bundle over the Grassmann manifold F(4, n). [ABSTRACT FROM AUTHOR]

Motivated by recent works on Hamiltonian circle actions satisfying certain minimal conditions, in this paper, we consider Hamiltonian circle actions satisfying an almost minimal condition. More precisely, we consider a compact symplectic manifold (M , ω) admitting a Hamiltonian circle action with fixed point set consisting of two connected components X and Y satisfying dim (X) + dim (Y) = dim (M) . Under certain cohomology conditions, we determine the circle action, the integral cohomology rings of M, X and Y, and the total Chern classes of M, X, Y, and of the normal bundles of X and Y. The results show that these data are unique—they are exactly the same as those in the standard example G ~ 2 (R 2 n + 2) , the Grassmannian of oriented 2-planes in R 2 n + 2 , which is of dimension 4n with (any) n ∈ N , equipped with a standard circle action. Moreover, if M is Kähler and the action is holomorphic, we can use a few different criteria to claim that M is S 1 -equivariantly biholomorphic and S 1 -equivariantly symplectomorphic to G ~ 2 (R 2 n + 2) . [ABSTRACT FROM AUTHOR]

The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincaré complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more generally, the topological L-class, for closed oriented stratified pseudomanifolds that can be equipped with a middle-perverse Verdier self-dual complex of sheaves, determined by Lagrangian sheaves along strata of odd codimension. This class of spaces, called L-pseudomanifolds, contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We apply this result in proving the Brasselet–Schürmann–Yokura conjecture for normal complex projective 3-folds with at most canonical singularities, trivial canonical class and positive irregularity. The conjecture asserts the equality of topological and Hodge L-class for compact complex algebraic rational homology manifolds. [ABSTRACT FROM AUTHOR]

Let M be an even-dimensional, oriented closed manifold. We show that the restriction of a singular Riemannian flow on M to a small tubular neighborhood of each connected component of its singular stratum is foliated diffeomorphic to an isometric flow on the same neighborhood. We then prove a formula that computes characteristic numbers of M as the sum of residues associated with the infinitesimal foliation at the components of the singular stratum of the flow. [ABSTRACT FROM AUTHOR]

We show that many spin 6-manifolds have the homotopy type but not the homeomorphism type of a Kähler manifold. Moreover, for given Betti numbers, there are only finitely many deformation types and hence topological types of smooth complex projective spin threefolds of general type. Finally, on a fixed spin 6-manifold, the Chern numbers take on only finitely many values on all possible Kähler structures. [ABSTRACT FROM AUTHOR]

We prove several basic ring-theoretic results about tautological rings of manifolds W, that is, the rings of generalised Miller-Morita-Mumford classes for fibre bundles with fibre W. Firstly we provide conditions on the rational cohomology of W which ensure that its tautological ring is finitely-generated, and we show that these conditions cannot be completely relaxed by giving an example of a tautological ring which fails to be finitely-generated in quite a strong sense. Secondly, we provide conditions on torus actions on W which ensure that the rank of the torus gives a lower bound for the Krull dimension of the tautological ring of W. Lastly, we give extensive computations in the tautological rings of CP2 and S2×S2. [ABSTRACT FROM AUTHOR]

We study the localization of Futaki–Morita integrals at isolated degenerate zeros by giving a streamlined exposition in the spirit of Bott and implementing the localization procedure for a holomorphic vector field on C P n with a maximally degenerate zero. This yields an essentially unique formula for these Futaki–Morita integral invariants without using a summation over multiple points. [ABSTRACT FROM AUTHOR]

This paper mainly deals with the question of equivalence between equivariant cohomology Chern numbers and equivariant K-theoretic Chern numbers when the transformation group is a torus. By using the equivariant Riemann-Roch relation of Atiyah-Hirzebruch type, it is proved that the vanishing of equivariant cohomology Chern numbers is equivalent to the vanishing of equivariant K-theoretic Chern numbers. [ABSTRACT FROM AUTHOR]

We determine the characteristic rank of the canonical oriented vector bundle over G ˜ 3 , n for all n ≥ 3 , and as a consequence, we obtain the affirmative answer to a conjecture of Korbaš and Rusin. As an application of this result, we calculate the Z 2 -cup-length for a new infinite family of manifolds G ˜ 3 , n . This result confirms the corresponding claim of Fukaya's conjecture. [ABSTRACT FROM AUTHOR]

In this paper we first prove that every differential character can be represented by differential form with singularities. Then we lift the Gauss-Bonnet-Chern theorem for vector bundles to differential characters. [ABSTRACT FROM AUTHOR]

In this paper we show that both of the Green-Schwarz anomaly factorization formula for the gauge group E × E and the Hořava-Witten anomaly factorization formula for the gauge group E8 can be derived through modular forms of weight 14. This answers a question of Schwarz. We also establish generalizations of these factorization formulas and obtain a new Hořava-Witten type factorization formula. [ABSTRACT FROM AUTHOR]

By studying modular invariance properties of some characteristic forms, we get some generalized anomaly cancellation formulas on (4 r − 1)-dimensional manifolds with no assumption that the 3rd de-Rham cohomology of manifolds vanishes. These anomaly cancellation formulas generalize our previous anomaly cancellation formulas on (4 r − 1)-dimensional manifolds. We also generalize our previous anomaly cancellation formulas on (4 r − 1)-dimensional manifolds and the Han-Yu rigidity theorem to the ( a, b) case. [ABSTRACT FROM AUTHOR]

A Gröbner basis for the small quantum cohomology of Grassmannian G is constructed and used to obtain new recurrence relations for Kostka numbers and inverse Kostka numbers. Using these relations it is shown how to determine inverse Kostka numbers which are related to the mod- p Wu formula. [ABSTRACT FROM AUTHOR]

We introduce Dolbeault cohomology valued characteristic classes of Higgs bundles over complex manifolds. Flat vector bundles have characteristic classes lying in odd degree de Rham cohomology and a theorem of Reznikov says that these must vanish in degrees three and higher over compact Kähler manifolds. We provide a simple and independent proof of Reznikov's result and show that our characteristic classes of Higgs bundles and the characteristic classes of flat vector bundles are compatible via the nonabelian Hodge theorem. [ABSTRACT FROM AUTHOR]

The Dolbeault cohomology plays an important role in the study of some ∂ ¯ -invariants of complex and holomorphic vector bundles as ∂ ¯ -Chern classes and Atiyah classes. In this paper we generalize similar invariants and their properties in the tangential Dolbeault cohomology. More exactly, we introduce and we study tangential Atiyah classes for F -holomorphic vector bundles and ∂ ¯ -tangential Chern classes for tangentially smooth vector bundles over a manifold M endowed with a complex foliation F . Also, ∂ ¯ -tangential secondary invariants are studied following similar constructions for Lie algebroids. The notions are introduced by a global formalism that is used in the tangential theory of foliated spaces. [ABSTRACT FROM AUTHOR]

We derive new lower bound for the characteristic rank of the canonical 3-plane bundle γ ˜ n , 3 over the oriented Grassmann manifold G ˜ n , 3 ≅ S O ( n ) / ( S O ( 3 ) × S O ( n − 3 ) ) for all n in range 2 t + 2 t − 1 ≤ n ≤ 2 t + 1 − 4 , t ≥ 3 . Using this knowledge we determine the Z 2 -cup-length of G ˜ 2 t + 2 t − 1 + 1 , 3 and G ˜ 2 t + 2 t − 1 + 2 , 3 , verifying the corresponding claim of T. Fukaya's conjecture. [ABSTRACT FROM AUTHOR]