Chiral Poincaré duality

Poincaré duality for tropical Dolbeault cohomology of non-archimedean Mumford curves

We calculate the tropical Dolbeault cohomology for the analytifications of the projective line and Mumford curves over non-archimedean fields. We show that the cohomology satisfies Poincaré duality and behaves analogously to the cohomology of curves over the complex numbers. Further, we give a complete calculation of the dimension of the cohomology on a basis of the topology.

Poincaré duality for $L^p$ cohomology on subanalytic singular spaces

We investigate the problem of Poincaré duality for $L^p$ differential forms on bounded subanalytic submanifolds of $\mathbb{R}^n$ (not necessarily compact). We show that, when $p$ is sufficiently close to $1$ then the $L^p$ cohomology of such a submanifold is isomorphic to its singular homology. In the case where $p$ is large, we show that $L^p$ cohomology is dual to intersection homology. As a consequence, we can deduce that the $L^p$ cohomology is Poincaré dual to $L^q$ cohomology, if $p$ and $q$ are Hölder conjugate to each other and $p$ is sufficiently large.

Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic $\mathscr{D}$-modules

The aim of this paper is to compute the Frobenius structures of some cohomological operators of arithmetic $\ms{D}$-modules. To do this, we calculate explicitly an isomorphism between canonical sheaves defined abstractly. Using this calculation, we establish the relative Poincaré duality in the style of SGA4. As another application, we compare the push-forward as arithmetic $\ms{D}$-modules and the rigid cohomologies taking Frobenius into account. These theorems will lead us to an analog of "Weil II" and a product formula for $p$-adic epsilon factors.

Tropical Poincaré duality spaces

The tropical fundamental class of a rational balanced polyhedral fan induces cap products between tropical cohomology and tropical Borel-Moore homology. When all these cap products are isomorphisms, the fan is said to be a tropical Poincaré duality space. If all the stars of faces also are such spaces, such as for fans of matroids, the fan is called a local tropical Poincaré duality space. In this article, we first give some necessary conditions for fans to be tropical Poincaré duality spaces and a classification in dimension one. Next, we prove that tropical Poincaré duality for the stars of …

Poincaré/Koszul duality

We prove a duality for factorization homology which generalizes both usual Poincaré duality for manifolds and Koszul duality for $\mathcal{E}_n$-algebras. The duality has application to the Hochschild homology of associative algebras and enveloping algebras of Lie algebras. We interpret our result at the level of topological quantum field theory.

Poincaré duality of wonderful compactifications and tautological rings

Let $g \geq 2$. Let $M_{g,n}^{rt}$ be the moduli space of $n$-pointed genus $g$ curves with rational tails. Let $C_g^n$ be the $n$-fold fibered power of the universal curve over $M_g$. We prove that the tautological ring of $M_{g,n}^{rt}$ has Poincaré duality if and only if the same holds for the tautological ring of $C_g^n$. We also obtain a presentation of the tautological ring of $M_{g,n}^{rt}$ as an algebra over the tautological ring of $C_g^n$. This proves a conjecture of Tavakol. Our results are valid in the more general setting of wonderful compactifications.

椭圆 Schubert 类与庞加莱对偶

Elliptic Schubert Classes and the Poincare Duality

本文通过 Kostant-Kumar 方法，证明了 Schubert 簇的椭圆类满足庞加莱对偶性。

T-spectra and Poincaré Duality

Frank Adams introduced the notion of a complex oriented cohomology theory represented by a commutative ring-spectrum and proved the Poincaré Duality theorem for this general case. In the current paper we consider oriented cohomology theories on algebraic varieties represented by multiplicative symmetric $T$-spectra and prove the Duality theorem, which mimics the result of Adams. This result is held, in particular, for Motivic Cohomology and Algebraic Cobordism of Voevodsky.

p-进解析几何中的模p庞加莱对偶

Mod-p Poincaré Duality in p-adic Analytic Geometry

本文证明了混合特征非阿基米德域上光滑紧合刚性解析空间的模p étale上同调满足庞加莱对偶，通过构造Faltings迹映射联系一般纤维与模p纤维的对偶性，并推广到p进整数及p进有理数系数。

Cycles, cocycles, and duality on tropical manifolds

We prove a Poincaré duality for the Chow rings of smooth fans whose support are tropical linear spaces. As a consequence, we show that cycles and cocycles on tropical manifolds are Poincaré dual to each other. This allows us to define pull-backs of tropical cycles along arbitrary morphisms with smooth target.

Brill-Noether problem on splice quotient singularities and duality of topological Poincaré series

In this manuscript we investigate the analouge of the Brill-Noether problem for smooth curves in the case of normal surface singularities. We determine the maximal possible value of $h^1$ of line bundles without fixed components in the Picard group $\pic^{l'}(\tX)$ in the following cases: for some special Chern classes $l'$ if $\tX$ is a resolution of a splice quotient singularity $(X, 0)$ and for arbitrary Chern classes in the case of weighted homogenous singularities. Motivated by this problem, we define the \emph{virtual cohomology numbers} $h^1_{virt}(l')$ for all Chern classes $l'$ such t…

棱柱上的F-量度与 syntomic 周期类

Syntomic cycle classes and prismatic Poincaré duality

本文引入棱柱上的F-量度，构造syntomic周期类，并证明proper smooth schemes的棱柱庞加莱对偶性。

Combinatorial duality for Poincaré series, polytopes and invariants of plumbed 3-manifolds

Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the `periodic constant' of the topological multivariable Poincaré series (zeta function). This involves a complicated regularization procedure (via quasipolynomials measuring the asymptotic behaviour of the coefficients). We show that the (a Gorenstein type) symmetry of the zeta function combined with Ehrhart-Macdonald-Stanley reciprocity (of Ehrhart theory of polytopes) provide a simple expression for the periodic constant. Using these dualities we a…

De Rham comparison and Poincaré duality for rigid varieties

Over any smooth algebraic variety over a $p$-adic local field $k$, we construct the de Rham comparison isomorphisms for the étale cohomology with partial compact support of de Rham $\mathbb Z_p$-local systems, and show that they are compatible with Poincaré duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over $k$. In particular, we prove finiteness of étale cohomology with partial compact support of any $\mathbb Z_p$-local systems, and e…

非阿基米德几何中的相对Poincaré对偶

Relative Poincaré duality in nonarchimedean geometry

本文证明了Bhatt-Hansen猜想，即沿真态射的导出前推与Verdier对偶交换，并在解析几何中构造了cycle classes与trace maps，从而简化了$p$-adic Poincaré对偶等结论的证明。

Explicit Poincaré duality in the cohomology ring of the SU(2) character variety of a surface

We provide an explicit description of the Poincaré dual of each generator of the rational cohomology ring of the $SU(2)$ character variety for a genus $g$ surface with central extension -- equivalently, that of the moduli space of stable holomorphic bundles of rank 2 and odd degree.

一般低次超曲面和庞加莱对偶的几何马林猜想的逆

A converse to geometric Manin's conjecture for general low degree hypersurfaces and Poincaré duality

该研究通过高亏分数的圆法推广，证明了对于一般低次超曲面，不存在增加Fujita不变量的累积映射，从而在特定范围内建立了这些模空间的庞加莱对偶性。

pro-étale $\mathbf{Q}_p$-局部系的庞加莱对偶

Poincaré duality for pro-étale $\mathbf{Q}_p$-local systems

本文在Banach-Colmez空间框架下证明了真p-adic刚性解析空间上pro-étale Q_p-局部系的有限性、庞加莱对偶及最优上同调消失界，并推广至算术上同调。

T-超曲面的庞加莱对偶性、退化性与实莱夫谢茨性质

Poincaré Duality, Degeneracy, and Real Lefschetz Property for T-Hypersurfaces

研究关于计算T-超曲面同调群的雷诺迪诺 - 肖谱序列，给出其正指标页满足的庞加莱对偶性及消失准则，该准则表明此谱序列在第二页退化的条件是T-超曲面满足实莱夫谢茨超平面截面定理。

Rigid cohomology over Laurent series fields II: Finiteness and Poincaré duality for smooth curves

In this paper we prove that the $\mathcal{E}^\dagger_K$-valued cohomology, introduced in [9] is finite dimensional for smooth curves over Laurent series fields $k((t))$ in positive characteristic, and forms an $\mathcal{E}^\dagger_K$-lattice inside `classical' $\mathcal{E}_K$-valued rigid cohomology. We do so by proving a suitable version of the p-adic local monodromy theory over $\mathcal{E}^\dagger_K$, and then using an étale pushforward for smooth curves to reduce to the case of $\mathbb{A}^1$. We then introduce $\mathcal{E}^\dagger_K$-valued cohomology with compact supports, and again prov…

刚性解析 Hyodo–Kato 理论的 Poincaré 对偶

Poincaré duality for rigid analytic Hyodo--Kato theory

本文利用刚性解析方法构建半稳定格式的紧支集 Hyodo–Kato 理论，引入多种对数刚性上同调，并证明相关结构与 Poincaré 对偶相容。

An Abel map to the compactified Picard scheme realizes Poincaré duality

For a smooth algebraic curve X over a field, applying H_1 to the Abel map X -> Pic (X/\partial X) to the Picard scheme of X modulo its boundary realizes the Poincaré duality isomorphism H_1(X, Z/ n) -> H^1(X/ \partial X, Z/n(1)) = H^1_c(X, Z/n(1)). We show the analogous statement for the Abel map X/\partial X -> Picbar (X/\partial X) to the compactified Picard, or Jacobian, scheme, namely this map realizes the Poincaré duality isomorphism H_1(X/ \partial X, Z/n) -> H^1(X, Z/n(1)). In particular, H_1 of this Abel map is an isomorphism. In proving this result, we prove some results about Picbar …

模 $p$ 皮卡定理对于 $p$-adic 周期域

Mod $p$ Poincaré duality for $p$-adic period domains

本文提出了一类满足模 $p$ 皮卡定理的 $p$-adic 周期域，恢复了几乎良好概形的皮卡定理并推广了德林费尔德对称空间的上同调计算。

Poincaré duality and unimodularity

It is well known that the cup-product pairing on the complementary integral cohomology groups (modulo torsion) of a compact oriented manifold is unimodular. We prove a similar result for the $\ell$-adic cohomology groups of smooth algebraic varieties.

$p$-adic 几何 pro-étale 上同调的對偶性

Duality for $p$-adic geometric pro-étale cohomology

本文证明了 $p$-adic 域上光滑部分真刚性解析簇的 $p$-adic 几何 pro-étale 上同调满足 Poincaré 对偶，通过 Fargues-Fontaine 曲线上的固体层及比较定理归结为经典 Serre 对偶。

庞加莱对偶与超引力

Poincaré Duality and Supergravity

本文通过相对庞加莱对偶理论，建立了超引力中超空间与物理时空的联系，并证明了不同场论表述的等价性。

Quadratic Duality for Chiral Algebras

We introduce a notion of quadratic duality for chiral algebras. This can be viewed as a chiral version of the usual quadratic duality for quadratic associative algebras. We study the relationship between this duality notion and the Maurer-Cartan equations for chiral algebras, which turns out to be parallel to the associative algebra case. We also present some explicit examples.

Chiral Koszul duality

We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld in \cite{bd}, to higher-dimensional varieties. This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen's homotopy theory of differential graded Lie algebras. We prove the equivalence of higher-dimensional chiral and factorization algebras by embedding factorization algebras into a larger category of chiral commutative coalgebras, then realizing this interrelation as a chiral form of Koszul duality. We apply these…