Robert Friedman, John W. Morgan
Let G be a complex reductive group and let C be a smooth curve of genus at least one. We prove a converse to a theorem of Atiyah-Bott concerning the stratification of the space of holomorphic G-bundles on C. In case the genus of C is one, we establish that one has a stratification in the strong sense. The paper concludes with a characterization of the minimally unstable strata in case G is simple.
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This paper is a companion of the paper "Weil's conjecture for function fields" by J. Lurie and the author. We present a different exposition of essentially the same algebro-geometric proof of the Atiyah-Bott for the cohomology of Bun(G), which subsequently leads to the proof of the Tamagawa number formula for the volume of the automorphic space for function fields.
The $\otimes^\star$-monoidal structure on the category of sheaves on the $\mathrm{Ran}$ space is not pro-nilpotent in the sense of Francis-Gaitsgory. However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the $\mathrm{Ran}$ space and integrating along the $\mathrm{Ran}$ space, i.e. taking factorization homology. Based on ideas sketched by Gaitsgory, we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah…
We present an implementation of the Atiyah-Bott residue formula for $\overline{M}_{0,m}(\mathbb{P}^{n},d)$. We use this implementation to compute a large number of Gromov-Witten invariants of genus $0$, including intersection numbers of rational curves on general complete intersections. We also compute some numbers of rational contact curves satisfying suitable Schubert conditions. Our computations confirm known predictions made by Mirror Symmetry. The code we developed for these problems is publicly available and can also be used for other types of computations.
Given a symmetric monoidal $(\infty,2)$-category $\mathscr E$ we promote the trace construction to a functor. 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).
Atiyah-Bott localization in equivariant Witt cohomology
本文在等变Witt上同调中为SL₂ⁿ和Nⁿ建立了Atiyah-Bott局部化定理及Bott留数公式的类似物，并分析了不同群作用下轨道类型与不动点分解的限制条件。
Interpreting the Ooguri-Vafa symplectic form à la Atiyah-Bott
本文从Atiyah-Bott角度解释Ooguri-Vafa辛形式，并将其与Hitchin模空间的度量进行关联。
We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed by Grothendieck et al.: it relies on the interplay between self dualities of quasi- and ind- coherent sheaves on $X$ and formal deformation theory of Gaitsgory-Rozenblyum. In particular, we give a description of the Todd class in terms of the difference of two formal group structures on the derived loop scheme $\mathcal LX$. The equivariant case is reduced to…
In this paper we prove two formulas for the characters of representations of reductive groups. Both express the character of a representation in terms of the same geometric data attached to it. When specialized to the case of a compact Lie group, one of them reduces to Kirillov's character formula in the compact case, and the other, to an application of the Atiyah-Bott fixed point formula to the Borel-Weil realization of the representation.
We collect Atiyah-Bott Combinatorial Dreams (A.B.C.Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron-F. Sottile, S. Billey-M. Haiman, P. Pragacz, and T. Ikeda-L. Mihalcea-I. Naruse. The relation complements a theorem of A. Kresch-H. Tamvakis in quantum cohomology. Results of A. Buch-V. Ravikumar rule out a similar correspondence in K-theory.
Let $\Fl^a_\la$ be the PBW degeneration of the flag varieties of type $A_{n-1}$. These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$. We prove that $\Fl^a_\la$ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\la$ of $\Fl^a_\la$. The varieties $R_\la$ can be viewed as towers of successive $\bP^1$-fibrations, thus providing an analogue of the classical Bott-Samelson-Demazure-Hansen desingularization. We prove that the varieties $R_\la$ are Frobenius split. This gives us Frobenius splitting for the…
We present a generalization of the classical Schur modules of $GL(N)$ exhibiting the same interplay among algebra, geometry, and combinatorics. A generalized Young diagram $D$ is an arbitrary finite subset of $\NN \times \NN$. For each $D$, we define the Schur module $S_D$ of $GL(N)$. We introduce a projective variety $\FF_D$ and a line bundle $\LL_D$, and describe the Schur module in terms of sections of $\LL_D$. For diagrams with the ``northeast'' property, $$(i_1,j_1),\ (i_2, j_2) \in D \to (\min(i_1,i_2),\max(j_1,j_2)) \in D ,$$ which includes the skew diagrams, we resolve the singularitie…
Let G be a reductive group over an algebraically closed field of positive characteristic. In this article we show an analogue for Morozov theorem for characteristics that are separably good for G (and under additional hypotheses on the group). This theorem characterises, when p= 0, subalgebras of parabolic subgroups of G with respect to their nilradical. If now k is an arbitrary field, let C be a smooth projective and geometrically connected curve. Let G be a reductive C-group scheme, which is the twisted form of a constant C-group. The aforementioned analogue is in particular useful to extend…
It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Lojasiewicz exponent of an analytic function is equal to one half at a critical point, then the function is Morse-Bott and thus its critical set nearby is an analytic submanifold. The main ingredients in our proofs are the Lojasiewicz gradient inequality for an analytic function on a…
Equivariant formality in complex-oriented theories
该研究将有理上同调的等变形式性结果推广至任意复定向上同调理论，借助新的上同调分裂结果完成证明，还建立了相关定理的类似结论并给出其在Morava K-理论中的应用。
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