Bott's Formula Research Articles

On Koszul complex of a supermodule

This paper is devoted to an exposition of the Koszul complex of a supermodule and its Berezinian from an intrinsic and as general as possible point of view. As an application, an analogue to Bott's formula in the supercommutative setting is given, computing the cohomology of twisted differential p-forms on the projective superspace.

Computations of Gromov–Witten invariants of toric varieties

We present the Julia package ToricAtiyahBott.jl, providing an easy way to perform the Atiyah–Bott formula on the moduli space of genus 0 stable maps M‾0,m(X,β) where X is any smooth projective toric variety, and β is any effective 1-cycle. The list of the supported cohomological cycles contains the most common ones, and it is extensible. We provide a detailed explanation of the algorithm together with many examples and applications. The toric variety X, as well as the cohomology class β, must be defined using the package Oscar.jl.

Enumeration of Rational Contact Curves via Torus Actions

We prove that some Gromov–Witten numbers associated to rational contact (Legendrian) curves in any contact complex projective space with arbitrary incidence conditions are enumerative. Also, we use the Bott formula on the Kontsevich space to find the exact value of those numbers. As an example, the numbers of rational contact curves of low degree in P3 and P5 are computed. The results are consistent with existing results.

Holomorphic Atiyah–Bott Formula for Correspondences

Holomorphic Atiyah–Bott Formula for Correspondences

𝜔+-Type Index of GL+(2)-Paths

Abstract In this paper, we establish an ω + {\omega^{+}} -type index theory for paths in the general linear group GL + ⁢ ( 2 ) {\mathrm{GL}^{+}(2)} . This is done by the complete homotopy classification for such paths. We also compare this index theory with the ω index theory for paths in the symplectic group Sp ⁢ ( 2 ) {\mathrm{Sp}(2)} and obtain a generalization of Bott formula for iterated paths in GL + ⁢ ( 2 ) {\mathrm{GL}^{+}(2)} . As applications, the minimal periodic solution problem and the linear stability of general differential systems are studied.

CATEGORICAL PROOF OF HOLOMORPHIC ATIYAH–BOTT FORMULA

Given a $2$-commutative diagramin a symmetric monoidal $(\infty ,2)$-category $\mathscr{E}$ where $X,Y\in \mathscr{E}$ are dualizable objects and $\unicode[STIX]{x1D711}$ admits a right adjoint we construct a natural morphism $\mathsf{Tr}_{\mathscr{E}}(F_{X})\xrightarrow[{}]{~~~~~}\mathsf{Tr}_{\mathscr{E}}(F_{Y})$ between the traces of $F_{X}$ and $F_{Y}$, respectively. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty ,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah–Bott formula (also known as the Holomorphic Lefschetz fixed point formula).

Euler sequence and Koszul complex of a module

We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential $p$-forms of a projective bundle. In particular we generalize Bott's formula for the projective space to a projective bundle over a scheme of characteristic zero.

Affine Partitions and Affine Grassmannians

We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott's formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of Bousquet-Melou-Eriksson, Eriksson-Eriksson and Reiner. In other types the identities appear to be new. For type $A_{n}$, the affine colored partitions form another family of combinatorial objects in bijection with $(n+1)$-core partitions and $n$-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the affine Grassmannians in terms of affine partitions and a generalization of Young's lattice which refines weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted.

Multiple closed geodesics on 3-spheres

This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113–149]) of its linearized Poincaré map contains no 2 × 2 rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with d ⩾ 2 , it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics.

Equivariant cohomology and localization for Lie algebroids

Let M be a manifold carrying the action of a Lie group G, and let A be a Lie algebroid on M equipped with a compatible infinitesimal G-action. Using these data, we construct an equivariant cohomology of A and prove a related localization formula for the case of compact G. By way of application, we prove an analog of the Bott formula.

Affine colored partitions (abstract only)

We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott's formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of Bousquet-Melou-Eriksson, Eriksson-Eriksson and Reiner. In other types the identities appear to be new. Our main application is to determine the rationally smooth Schubert varieties in the affine Grassmannians. Several of the proofs are computer assisted. This is joint work with Steve Mitchell.

Simple Darboux points of polynomial planar vector fields

We are interested in some aspects of the integrability of complex polynomial planar vector fields in finite form. Especially, in the case of simple Darboux points, we deduce the famous Baum–Bott formula from a kind of global residue theorem; our elementary proof essentially relies on Hilbert's Nullstellensatz. As a corollary of our result, we propose formulas relating the various integers involved in the Lagutinskii–Levelt procedure for a Darboux polynomial at the various Darboux points. In particular, from the whole set of our formulas, it is possible to deduce an upper bound on the degree of irreducible Darboux polynomials in classical cases; with respect to such applications, this corollary seems to provide an alternate tool to usual genus formulas. As many people do in these subjects, we illustrate our corollary by giving a new simple proof of the fact that the polynomial Jouanolou derivation y s ∂ x + z s ∂ y + x s ∂ z , with s⩾2, has no Darboux polynomial.

A compactification of the space of twisted cubics

We give an elementary, explicit smooth compactification of a parameter space for the family of twisted cubics. The construction also applies to the family of subschemes defined by determinantal nets of quadrics, e.g., cubic ruled surfaces in $\boldsymbol P^4$, Segre varieties in $\boldsymbol P^5$. It is suitable for applications of Bott's formula to a few enumerative problems.

The Bott Formula for Toric Varieties

The purpose of this paper is to give an explicit formula which allows one to compute the dimension of the cohomology groups of the sheaf $\Omega_{\P}^p(D)$ of p-th differential forms of Zariski twisted by an ample invertible sheaf on a complete simplicial toric variety. The formula involves some combinatorial sums of integer points over all faces of the support polytope for ${\O_X}(D)$. We also introduce a new combinatorial object, the so-called p-th Hilbert-Erhart polynomial, which generalizes the usual notion and behaves similar. Namely, there exists a generalization of the inversion law for a usual Hilbert-Erhart polynomial. Some applications of the Bott formula are discussed.

A Refinement of the Lecture Hall Theorem

Forn⩾1, let Lnbe the set of lecture hall partitions of lengthn, that is, the set ofn-tuples of integersλ=(λ1,…,λn) satisfying 0⩽λ1/1⩽λ2/2⩽…⩽λn/n. Let ⌈λ⌉ be the partition (⌈λ1/1⌉,…,⌈λn/n⌉), and leto(⌈λ⌉) denote the number of its odd parts. We show that the identity∑λ∈Lnq|λ|u|⌈λ⌉|vo(⌈λ⌉)=(1+uvq)(1+uvq2)…(1+uvqn)(1−u2qn+1)(1−u2qn+2)…(1−u2q2n)is equivalent to a refinement of Bott's formula for the affine Coxeter groupCn, obtained by I. G. Macdonald (Math. Ann.199(1972), 161–174) and V. Reiner (Electron. J. Combin.2(1995), R25). The caseu=v=1 of the above identity, called the lecture hall theorem, was proved by us in (Ramanujan J.1(1997), 101–111), and then by Andrews (“Mathematical Essays in Honor of G.-C. Rota,” pp. 1–22, Birkhäuser, Cambridge, MA, 1998). In the present paper, we give two direct proofs of the above identity. The first one is rather short, but requires a bit ofq-calculus; the second one is the first truly bijective proof ever found in the domain of lecture hall partitions. Although we describe our bijection in completely combinatorial terms, it finds its origin in the algebraic context of Coxeter groups. Both proofs are completely independent of all earlier proofs of the Lecture Hall Theorem.

Bott formula of the Maslov-type index theory

In this paper the integer valued !-index theory parameterized by all ! on the unit circle for paths in the symplectic group Sp(2n) is established. Based on this index theory, the Bott formula of the Maslov-type index theory for iterated paths in Sp(2n) is estalished, the mean index for periodic solutions of Hamiltonian systems is dened, and the increasing estimate of the iterated Maslov-type index in terms of the mean index is proved.

Affine Weyl Groups as Infinite Permutations

We present a unified theory for permutation models of all the infinite families of finite and affine Weyl groups, including interpretations of the length function and the weak order. We also give new combinatorial proofs of Bott's formula (in the refined version of Macdonald) for the Poincaré series of these affine Weyl groups.

Lecture Hall Partitions

We prove a finite version of the well-known theorem that says that the number of partitions of an integer N into distinct parts is equal to the number of partitions of N into odd parts. Our version says that the number of “lecture hall partitions of length n ” of N equals the number of partitions of N into small odd parts: 1,3,5, ldots, 2n-1 . We give two proofs: one via Bott's formula for the Poincare series of the affine Coxeter group \(\tilde C_n \), and one direct proof.