Algebraic monodromy and obstructions to formality

Formality, Alexander invariants, and a question of Serre

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We show that the I-adic completion of the Alexander invariant of a 1-formal group G is determined solely by the cup-product map in low degrees. It follows that the germs at the origin of the characteristic and resonance varieties of G are analytically isomorphic; in particular, the tangent cone to V_k(G) at 1 equals R_k(G). This provides new obstructions to 1-formality. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerf…

Formality criteria for algebras over operads

We study some formality criteria for differential graded algebras over differential graded operads. This unifies and generalizes other known approaches like the ones by Manetti and Kaledin. In particular, we construct general operadic Kaledin classes and show that they provide obstructions to formality. Moreover, we show that an algebra A is formal if and only if its operadic cohomology spectral sequence degenerates at $E_2$.

球丛的形式性

Formality of Sphere Bundles

本文研究连通紧流形上可定向球丛的形式性，证明了当基流形形式性时，丛的形式性等价于 Bianchi-Massey 张量消失，并给出了基流形非形式性时欧拉类可约球丛的形式性障碍。

Non-formality of planar configuration spaces in characteristic two

We prove that the ordered configuration space of 4 or more points in the plane has a non-formal singular cochain algebra in characteristic two. This is proved by constructing an explicit non trivial obstruction class in the Hochschild cohomology of the cohomology ring of the configuration space, by means of the Barratt-Eccles-Smith simplicial model. We also show that if the number of points does not exceed its dimension, then an euclidean configuration space is intrinsically formal over any ring.

关于传递李叠映射积分性障碍等价性的研究

On the equivalence of the integrability obstructions for transitive Lie algebroids

本文探讨了传递李叠映射积分性障碍的不同视角，揭示了同调方法与同调构造的相互作用，证明了Mackenzie的同调障碍与Crainic和Fernandes的单胞映射以及Meinrenken的粘合构造等价。

Voronov 的 Swiss-Cheese 操作子的非形式性

Non-formality of Voronov's Swiss-Cheese operads

本文证明 Voronov 的一类 Swiss-Cheese 操作子在特征零下非形式，并将形式性障碍定位在元数 $(2,2^n)$，强于此前结果。

Formality and splitting of real non-abelian mixed Hodge structures

We define and construct mixed Hodge structures on real schematic homotopy types of complex projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. We also show that these split on tensoring with the ring R[x] equipped with the Hodge filtration given by powers of (x-i), giving new results even for simply connected varieties. Thus the mixed Hodge structures can be recovered from cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, these all become real variations of mix…

Real non-abelian mixed Hodge structures for quasi-projective varieties: formality and splitting

We define and construct mixed Hodge structures on real schematic homotopy types of complex quasi-projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. We also show that these split on tensoring with the ring R[x] equipped with the Hodge filtration given by powers of (x-i), giving new results even for simply connected varieties. The mixed Hodge structures can thus be recovered from the Gysin spectral sequence of cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, th…

Derived universal Massey products

We define an obstruction to the formality of a differential graded algebra over a graded operad defined over a commutative ground ring. This obstruction lives in the derived operadic cohomology of the algebra. Moreover, it determines all operadic Massey products induced on the homology algebra, hence the name of derived universal Massey product.

正常表面奇点的积分 perverse 障碍：解析确定性和monodromy

Integral Perverse Obstructions for Normal Surface Singularities: Resolution Determinants and Monodromy

研究正常复分析表面奇点的积分pervers障碍，通过解析确定性和monodromy理论揭示其局部不变性。

Topology and geometry of cohomology jump loci

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…

$A$-无穷双模的障碍理论

Obstruction theory for $A$-infinity bimodules

本文构建截断最小 $A$-无穷双模扩张的障碍理论，以新型边缘谱序列和上同调刻画障碍，并推出微分分次双模的内在形式性与近乎形式性结果。

图配置空间、合同腺与形式性

Graphical configuration spaces, Contractads and Formality

本文研究图配置空间的合同腺结构，证明其在特定条件下形式性，并探讨了循环图的共形式性障碍及二维合同腺的细胞结构。

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

A central question in arrangement theory is to determine whether the characteristic polynomial $Δ_q$ of the algebraic monodromy acting on the homology group $H_q(F(\mathcal{A}),\mathbb{C})$ of the Milnor fiber of a complex hyperplane arrangement $\mathcal{A}$ is determined by the intersection lattice $L(\mathcal{A})$. Under simple combinatorial conditions, we show that the multiplicities of the factors of $Δ_1$ corresponding to certain eigenvalues of order a power of a prime $p$ are equal to the Aomoto--Betti numbers $β_p(\mathcal{A})$, which in turn are extracted from $L(\mathcal{A})$. When $…

高亏格下的Goldman-Turaev Lie双代数与Kashiwara-Vergne问题

The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera

本文研究高亏格曲面上的Goldman-Turaev形式化问题，通过定义并求解新的高亏格Kashiwara-Vergne问题，建立了GT形式化映射，并揭示了其与Grothendieck-Teichmüller群的联系。