Luke Gutzwiller, Stephen A. Mitchell
Our main theorem is that the inclusion of a Birkhoff variety in the affine Grassmannian is a homotopy equivalence. We also construct analogues of tubular neighborhoods for Birkhoff and Schubert varieties. We include some observations on torus-equivariant cohomology.
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We compute the integral homology (including torsion), the topological K-theory, and the Hodge structure on cohomology of Calabi-Yau threefold hypersurfaces and complete intersections in Gorenstein toric Fano varieties. The methods are purely topological.
In the theory of algebraic group actions on affine varieties, the concept of a Kempf-Ness set is used to replace the categorical quotient by the quotient with respect to a maximal compact subgroup. By making use of the recent achievements of "toric topology" we show that an appropriate notion of a Kempf-Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate subspace arrangement complements) arising in the Batyrev-Cox "geometric invariant theory" approach to toric varieties. We proceed by studying the cohomology of these "toric" Kempf-Ness sets. In the case …
Chow-Witt rings and topology of flag varieties
本文利用子商丛的 Pontryagin 类计算 A 型部分旗簇的 Witt 层上同调环，证明了实旗流形整上同调挠均为 2-挠，并描述了旗簇的 Chow-Witt 环及其枚举应用。
Let X be the base locus of a linear system W of k quadrics. Let also S be the intersection of W with the discriminant hypersurface in the space of all homogeneous polynomials of degree two. We prove a formula relating the topology of X with the one of S and its (iterated) singular points. As a corollary we prove the sharp bound b(X)\leq O(n)^{k-1}.
We use the non-proper Morse theory of Palais-Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties, and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules, as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As …
We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, V_k and R_k, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of V_k and R_k are analytically isomorphic, if the group is 1-formal; in particular, the tangent cone to V_k at 1 equals R_k. These new obstructions to 1-formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 1 to…
Invariants of real affine varieties based on their complexifications
本文引入了一类基于复化拓扑的实代数集新不变量，计算了球面的这些不变量，从而完全分类了两个球面乘积上的代数向量丛的拓扑同构类，并得到了球面乘积到球面的正则映射的存在性与不存在性新结果，同时证明了新定义的不变量阻碍了实代数集中光滑子流形的弱代数逼近，反驳了Kucharz和Kurdyka的一个猜想。
We consider the interplay of point counts, singular cohomology, étale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree $d$ rational curves in $\mathbb{P}^n$. We deduce as special cases algebro-geometric and arithmetic refinements of topological computations of Segal, Cohen--Cohen--Mann--Milgram, Vassiliev and others.
Let $B$ be a Borel subgroup of $\mathrm{GL}_n(\mathbb{C})$ and $\mathbb{T}$ a maximal torus contained in $B$. Then $\mathbb{T}$ acts on $\mathrm{GL}_{n}(\mathbb{C})/B$ and every Schubert variety is $\mathbb{T}$-invariant. We say that a Schubert variety is of complexity $k$ if a maximal $\mathbb{T}$-orbit in $X_w$ has codimension $k$. In this paper, we discuss topology, geometry, and combinatorics related to Schubert varieties of complexity one.
We give a motivated introduction to the theory of perverse sheaves, culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the Decomposition Theorem, indicate some important applications and examples.
This note is mostly an expository survey, centered on the topology of complements of hyperplane arrangements, their Milnor fibrations, and their boundary structures. An important tool in this study is provided by the degree 1 resonance and characteristic varieties of the complement, and their tight relationship with orbifold fibrations and multinets on the underlying matroid. In favorable situations, this approach leads to a combinatorial formula for the first Betti number of the Milnor fiber and the algebraic monodromy. We also produce a pair of arrangements for which the respective Milnor fi…
In this expository paper we describe a powerful combinatorial formula and its implications in geometry, topology, and algebra. This formula first appeared in the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey discovered it independently five years later, and it played a prominent role in her work to evaluate certain polynomials closely related to Schubert polynomials. Billey's formula relates many pieces of Schubert calculus: the geometry of Schubert varieties, the action of the torus on the flag variety, combinatorial data about permutations, the cohomology of the flag vari…
Cluster-permutohedra and submanifolds of flag varieties with torus actions
本文揭示了cluster-permutohedron与稀疏等谱Hermitian矩阵流形的环面拓扑间的关系，证明面偏序集同构性，并利用同伦性质探讨等变形式性障碍，同时推广该结构以描述Grassmann流形等一大类流形的组合结构。
In this paper we define the adjoint Reidemeister torsion as a differential form on the character variety of a compact oriented 3-manifold with toral boundary, and prove it defines a regular volume form. Then we show that the torsion form can vanish only at singular points of the character variety. In fact, if the singular point corresponds to a reducible character, we show that the torsion does not vanish under a generic hypothesis on the Alexander polynomial, else we relate the vanishing order with the type of singularity. Finally we consider the ideal points added after compactification of t…
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