Let X be a projective scheme carrying a circle action S with isolated fixed points. We associate a simplicial complex Delta(X,S) of "closure chains" using a refinement of its Morse/Bialynicki-Birula decomposition. If this decomposition is a stratification (e.g. when X is a flag manifold), then Delta(X,S) is just the order complex of the poset of fixed points. For X a toric variety, Delta(X,S) is a triangulation of the moment polytope. We compute some other examples, including a Bott-Samelson manifold and the punctual Hilbert scheme of 4 points in the plane.
Summing over the facets of Delta(X,S), we obtain a positive formula for the Duistermaat-Heckman measure on the moment polytope of X, defined for any torus action extending S. We explain how, through brutal use of partial fractions, this can be extended to an AB/BV-type formula for integrating general classes. Throughout we work with equivariant Chow groups, and do not make any smoothness requirements on X.
Our goal in this work is to develop aspects of Bialynicki-Birula and Morse-Bott theory that can be extended from the classical setting of smooth manifolds to that of complex analytic spaces with a holomorphic $\mathbb{C}^*$ action. We extend prior results on existence of Bialynicki-Birula decompositions for compact, complex Kähler manifolds to non-compact complex manifolds and develop functorial properties of the Bialynicki-Birula decomposition, in particular with respect to blowup along a $\mathbb{C}^*$-invariant, embedded complex submanifold. We deduce the existence of a Bialynicki-Birula de…
In this short note we prove that every maximal torus action on the free algebra is conjugate to a linear action. This statement is the free algebra analogue of a classical theorem of A. Białynicki-Birula.
Recently, V. Chernousov, S. Gille and A. Merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the Bruhat decomposition. Using the method of Bialynicki-Birula, I generalize this decomposition to the case of a (possibly anisotropic) smooth projective variety homogeneous under the action of an isotropic reductive group. This answers a question of N. Karpenko.
Suppose an algebraic torus acts on a complex algebraic variety $X$. Then a great part of information about global invariants of $X$ are encoded in some data localized around the fixed points. The goal of this note is to present a connection between two approaches to localization for $C^*$-action. The homological results are related to $S^1$-action, while from $R^*_{>0}$-action we obtain a geometric decomposition. We study the resulting decompositions of Hirzebruch $χ_y$-genus and their relative versions. We show that via a limit process the second decomposition is obtained from the first one. …
The work of Hausel proves that the Bialynicki-Birula stratification of the moduli space of rank two Higgs bundles coincides with its Shatz stratification. These two stratifications do not coincide in general. Here, we give an approach for the rank three case of the classification of the Shatz stratification in terms of the Bialynicki-Birula stratification.
We prove the existence of Bialynicki-Birula decomposition for Kempf monoids, which is a large class that contains for example monoids with reductive unit group in all characteristics. This extends the existence statements from previous works of Alper-Hall-Rydh and Jelisiejew-Sienkiewicz.
By using Oprea's Bialynicki-Birula decomposition for the stack of genus zero stable maps to flag manifolds. We calculate the Poincaré polynomial of the moduli space in degree one and degree two.
Hyperquot schemes are generalizations of Grothendieck's Quot scheme to partial flags. Using a Bialynicki-Birula decomposition, we obtain combinatorial data for the Betti numbers, and collect this information into the form of rational generating functions for the Poincare polynomials.
For a fixed 2-block Springer fiber, we describe the structure of its irreducible components and their relation to the Bialynicki-Birula paving, following work of Fung. That is, we consider the space of complete flags in C^n preserved by a fixed nilpotent matrix with 2 Jordan blocks, and study the action of diagonal matrices commuting with our fixed nilpotent. In particular, we describe the structure of each component, its set of torus fixed points, and prove a conjecture of Fung describing the intersection of any pair. Then we define a convolution algebra structure on the direct sum of the coh…
We prove that the Hilbert scheme of points on a higher dimensional affine space is non-reduced and has components lying entirely in characteristic p for all primes p. In fact, we show that Vakil's Murphy's Law holds up to retraction for this scheme. Our main tool is a generalized version of the Bialynicki-Birula decomposition.
Using the Bialynicki-Birula method, we determine the additive structure of the integral homology groups of the moduli spaces of semi-stable sheaves on the projective plane having rank and Chern classes (5, 1, 4), (7, 2, 6), respectively, (0, 5, 19). We compute the Hodge numbers of these moduli spaces.
We introduce the notion of Q-filtrable varieties: projective varieties with a torus action and a finite number of fixed points, such that the cells of the associated Bialynicki-Birula decomposition are all rationally smooth. Our main results develop GKM theory in this setting. We also supply a method for building nice combinatorial bases on the equivariant cohomology of any Q-filtrable GKM variety. Applications to the theory of group embeddings are provided.
We generalize the Bialynicki-Birula decomposition to singular schemes and apply it to the Hilbert scheme of points on an affine space. We find an infinite family of small, elementary and generically smooth components of the Hilbert scheme of points of the affine four-space. Our method gives easily verifiable sufficient conditions for proving that a point of the Hilbert scheme is smooth and lies on an elementary component. We also present a necessary condition for smoothability of a finite subscheme given by a homogeneous ideal.
Let H_{ab} be the equivariant Hilbert scheme parametrizing the 0-dimensional subschemes of the affine plane invariant under the natural action of the one-dimensional torus T_{ab}:={(t^{-b},t^a), t\in k^*}. We compute the irreducible components of H_{ab}: they are in one-one correspondence with a set of Hilbert functions. As a by-product of the proof, we give new proofs of results by Ellingsrud and Stromme, namely the main lemma of the computation of the Betti numbers of the Hilbert scheme H^l parametrizing the 0-dimensional subschemes of the affine plane of length l and a description of Bialyn…
For $G$ a finite group acting linearly on $\mathbb{A}^2$, the equivariant Hilbert scheme $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ is a natural resolution of singularities of $\operatorname{Sym}^r(\mathbb{A}^2/G)$. In this paper we study the topology of $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ for abelian $G$ and how it depends on the group $G$. We prove that the topological invariants of $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ are periodic or quasipolynomial in the order of the group $G$ as $G$ varies over certain families of abelian subgroups of $GL_2$. This is done by using the Bialynicki-Biru…
The additive invariants of an algebraic variety is calculated in terms of those of the fixed point set under the action of additive and multiplicative groups, by using Bialynicki-Birula's fixed point formula for a projective algebraicset with a G_m-action or G_a-action. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincare characteristic for the Chow var…
We consider bases for the cohomology space of regular semisimple Hessenberg varieties, consisting of the classes that naturally arise from the Bialynicki-Birula decomposition of the Hessenberg varieties. We give an explicit combinatorial description of the support of each class, which enables us to compute the symmetric group actions on the classes in our bases. We then successfully apply the results to the permutohedral varieties to explicitly write down each class and to construct permutation submodules that constitute summands of a decomposition of cohomology space of each degree. This reso…
Let $C$ be a smooth curve over an algebraically closed field $\mathbf{k}$, and let $E$ be a locally free sheaf of rank $r$. We compute, for every $d>0$, the generating function of the motives $[\mathrm{Quot}_C(E,\boldsymbol{n} )] \in K_0(\mathrm{Var}_{\mathbf{k}})$, varying $\boldsymbol{n} = (0\leq n_1\leq\cdots\leq n_d)$, where $\mathrm{Quot}_C(E,\boldsymbol{n} )$ is the nested Quot scheme of points, parametrising $0$-dimensional subsequent quotients $E \twoheadrightarrow T_d \twoheadrightarrow \cdots \twoheadrightarrow T_1$ of fixed length $n_i = χ(T_i)$. The resulting series, obtained by ex…
We prove the LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any positive quaternion-Kahler manifold of real dimension at most 16 is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most 9 with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contr…
