K. E. Feldman
In the paper we describe obstructions for the existence of symplectic and Hamiltonian symplectic circle actions on closed compact manifolds in terms of Hirzebruch genera and relations between differential and homotopic invariants of such manifolds.
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Circle actions on oriented 4-manifolds
本文研究环作用于定向四维流形的性质，推导Atiyah-Hirzebruch公式，并通过固定点集的数据构建图，证明在孤立固定点情况下，满足该公式的抽象图对应于四维定向S¹-流形。
We prove that if the circle group acts smooth and unitary on 2n-dimensional stably complex manifold with two isolated fixed points and it is not bound equivariantly, then n=1 or 3. Our proof relies on the rigid Hirzebruch genera.
We establish an S^1-equivariant index theorem for Dirac operators on Z/k-manifolds. As an application, we generalize the Atiyah-Hirzebruch vanishing theorem for S^1-actions on closed spin manifolds to the case of Z/k-manifolds.
Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce many identities concerning the weights on the fixed points. In particular, it gives a criterion to detemine whether or not a symplectic circle action with isolated fixed points is Hamiltonian. As applications, we simplify the proofs of some known results related to symplectic circle actions, due to Godinho, Tolman-Weitsman and Pelayo-Tolman, and generalize so…
Kawakubo and Uchida showed that, if a closed oriented $4k$-dimensional manifold $M$ admits a semi-free circle action such that the dimension of the fixed point set is less than $2k$, then the signature of $M$ vanishes. In this note, by using $G$-signature theorem and the rigidity of the signature operator, we generalize this result to more general circle actions. Combining the same idea with the remarkable Witten-Taubes-Bott rigidity theorem, we explore more vanishing results on spin manifolds admitting such circle actions. Our results are closely related to some earlier results of Conner-Floy…
Let a $k$-dimensional torus $T^k$ act on a $2n$-dimensional compact connected almost complex manifold $M$ with isolated fixed points. As for circle actions, we show that there exists a (directed labeled) multigraph that encodes weights at the fixed points of $M$. This includes the notion of a GKM graph as a special case that weights at each fixed point are pairwise linearly independent. If in addition $k=n$, i.e., $M$ is an almost complex torus manifold, the multigraph is a graph; it has no multiple edges. We show that the Hirzebruch $χ_y$-genus $χ_y(M)=\sum_{i=0}^n a_i(M) \cdot (-y)^i$ of an …
I. Hambleton, A. Korzeniewski and A. Ranicki proved that the signature of a fibre bundle of closed, connected, compatibly oriented PL manifolds is always multiplicative modulo 4. In this paper, we consider the Hirzebruch $χ_y$-genera for odd integers $y$ for a smooth fiber bundle such that the base, fibre, and total space are compact complex algebraic manifolds (in the complex analytic topology, not in the Zariski topology). We show that the Hirzebruch $χ_y$-genera of such a fibre bundle are always multiplicative modulo 4. We also investigate multiplicativity modulo 8 and show that if $y$ is c…
Circle actions on six dimensional oriented manifolds with isolated fixed points
本文研究了六维定向流形上圆作用的固定点数据，通过等变连通和操作将其简化为无固定点作用，揭示了固定点数据的可消去性。
Following Brooks's calculation of the $\hat{A}$-genus of complete intersections, a new and more computable formula about the $\hat{A}$-genus and $α$-invariant will be described as polynomials of multi-degree and dimension. We also give an iterated formula of $\hat{A}$-genus and the necessary and sufficient conditions for the vanishing of $\hat{A}$-genus of complex even dimensional spin complete intersections. Finally, we obtain a general formula about the Hirzebruch genus of complete intersections, and calculate some classical Hirzebruch genera as examples.
The short article [V.M. Buchstaber, E.Yu. Netay, $\mathbb{C}P(2)$-multiplicative Hirzebruch genera and elliptic cohomology, Russian Math. Surveys, 69:4 (2014), 757-759] states results on $\mathbb{C}P(2)$-multiplicative Hirzebruch genera. The aim of the following text is to give a proof of Theorem 3. This proof uses only the technique of functional differential equations.
The rigidity theorem of Witten-Bott-Taubes-Hirzebruch tells us that, if the circle group acts on a closed almost complex (or more generally unitary) manifold whose first Chern class is divisible by a positive integer N greater than 1, then its equivariant elliptic genus of level N is rigid. Applying this to a non-singular compact toric variety, we see that its elliptic genus of level N is rigid if its first Chern class is divisible by N. But, using a vanishing theorem of Hirzebruch, we can show moreover that the genus actually vanishes. In this note we shall extend this result to torus manifol…
It is well known that the two-parametric Todd genus and elliptic functions of level $d$ define $n$-multiplicative Hirzebruch genera, if $d$ divides $n+1$. Both these cases are particular cases of Krichever genera defined by the Baker--Akhiezer functions. In this work the inverse problem is solved. Namely, it is proved that only these families of functions define $n$-multiplicative Hirzebruch genera among all the Krichever genera for all $n$.
On rigidity of complex Hirzebruch genera on $SU$-manifolds
本文证明了在具有环面作用的 $SU$-流形上刚性的复亏格必为椭圆 Krichever 亏格。
We obtain a general lower bound for the number of fixed points of a circle action on a compact almost complex manifold $M$ of dimension $2n$ with nonempty fixed point set, provided the Chern number $c_1c_{n-1}[M]$ vanishes. The proof combines techniques originating in equivariant K-theory with celebrated number theory results on polygonal numbers, introduced by Pierre de Fermat. This lower bound confirms in many cases a conjecture of Kosniowski from 1979, and is better than existing bounds for some symplectic actions. Moreover, if the fixed point set is discrete, we prove divisibility properti…
We consider the geometrical addition law on the elliptic curve in Tate coordinates. It corresponds to the general formal group law over the ring of polynomials with integer coefficients of the parametra of the curve. We study the structure of this law and the differential equation that determines its exponent. We describe a 5-parametric family of Hirzebruch genera with integer values on stably complex manifolds. We introduce the general Krichever genus, which is given by a generalized Baker-Akhiezer function. This function has many of the fundamental properties of the Baker-Akhiezer function, …
We obtain general formulae expressing Hirzebruch genera of a manifold with Z/p-action in terms of invariants of this action (the sets of weights of fixed points). As an illustration, we consider numerous particular cases of well-known genera, in particular, the elliptic genus. We also describe the connection with the so-called Conner-Floyd equations for the weights of fixed points.
The classical multiplicative (Hirzebruch) genera of manifolds have the wonderful property which is called rigidity. Rigidity of a genus h means that if a compact connected Lie group G acts on a manifold X, then the equivariant genus h^G(X) is independent on G, i.e. h^G(X)=h(X). In this paper we are considering the rigidity problem for complex manifolds. In particular, we are proving that a genus is rigid if and only if it is a generalized Todd genus.
Let $E$ be a fiber $F$ bundle over a base $B$ such that $E, F$ and $B$ are smooth compact complex algebraic varieties. In this paper we give explicit formulae for the difference of the Hirzebruch $χ_y$-genus $χ_y(E) - χ_y(F)χ_y(B)$. As a byproduct of the formulae we obtain that the signature of such a fiber bundle is multiplicative mod $4$, i.e. the signature difference $σ(E) -σ(F)σ(B)$ is always divisible by $4$. In the case of $\operatorname{dim}_{\mathbb C}E \leqq 4$ the $χ_y$-genus difference $χ_y(E) - χ_y(F)χ_y(B)$ can be concretely described only in terms of the signature difference $σ(E…
A quasitoric manifold is a smooth 2n-manifold M^{2n} with an action of the compact torus T^n such that the action is locally isomorphic to the standard action of T^n on C^n and the orbit space is diffeomorphic, as manifold with corners, to a simple polytope P^n. The name refers to the fact that topological and combinatorial properties of quasitoric manifolds are similar to that of non-singular algebraic toric varieties (or toric manifolds). Unlike toric varieties, quasitoric manifolds may fail to be complex; however, they always admit a stably (or weakly almost) complex structure, and their co…
Let $M$ be a $Spin$-manifold with $S^1$-action and let $σ\in S^1$ be of finite order. We show that the indices of certain twisted Dirac operators vanish if the action of $σ$ has sufficiently large fixed point codimension. These indices occur in the Fourier expansion of the elliptic genus of $M$ in one of its cusps. As a by-product we obtain a new proof of a theorem of Hirzebruch and Slodowy on involutions.
Graphs for torus actions on oriented manifolds with isolated fixed points and classification in dimension 6
本文研究环面在具有孤立不动点的紧致定向流形上的作用，通过带符号标记的多重图编码不动点数据，并在6维情形下通过连通和与吹胀等操作对多重图及不动点数据进行分类。
We revisit and elucidate the $\widehat{A}$-genus, Hirzebruch's $L$-genus and Witten's $W$-genus, cobordism invariants of special classes of manifolds. After slight modification, involving Hecke's trick, we find that the $\widehat{A}$-genus and $L$-genus arise directly from Jacobi's theta function. For every $k\geq 0,$ we obtain exact formulas for the quasimodular expressions of $\widehat{A}_k$ and $L_k$ as ``traces'' of partition Eisenstein series \[ \widehat{\mathcal{A}}_k(τ)= \operatorname{Tr}_k(ϕ_{\widehat{A}};τ)\ \ \ \ \ \ {\text {and}}\ \ \ \ \ \ \mathcal{L}_k(τ)= \operatorname{Tr}_k(ϕ_L;…
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