The projective coordinate ring of a projective Poisson scheme $X$ does not usually admit a structure of a Poisson algebra. We show that when $H^1(X,O_X)=H^2(X,O_X)=0$, this can be corrected by embedding $X$ into a canonical one-parameter deformation. The scheme $X$ then becomes the Hamiltonian reduction of the spectrum of the deformed projective coordinate ring with respect to $G_m$. The projection into the base of the deformation is the moment map.
We review nonabelian Poisson structures on affine and projective spaces over $\mathbb{C}$. We also construct a class of examples of nonabelian Poisson structures on $\mathbb{C} P^{n-1}$ for $n>2$. These nonabelian Poisson structures depend on a modular parameter $τ\in\mathbb{C}$ and an additional descrete parameter $k\in\mathbb{Z}$, where $1\leq k<n$ and $k,n$ are coprime. The abelianization of these Poisson structures can be lifted to the quadratic elliptic Poisson algebras $q_{n,k}(τ)$.
Using combinatorial methods, we determine that a projective coordinate ring of the moduli of parabolic principal $\mathrm{SL}_2$-bundles on a marked projective curve is not Gorenstein when the genus and number of marked points are greater than $1$.
This paper is intended both an introduction to the algebraic geometry of holomorphic Poisson brackets, and as a survey of results on the classification of projective Poisson manifolds that have been obtained in the past twenty years. It is based on the lecture series delivered by the author at the Poisson 2016 Summer School in Geneva. The paper begins with a detailed treatment of Poisson surfaces, including adjunction, ruled surfaces and blowups, and leading to a statement of the full birational classification. We then describe several constructions of Poisson threefolds, outlining the classif…
We introduce a weak concept of Morita equivalence, in the birational context, for Poisson modules on complex normal Poisson projective varieties. We show that Poisson modules, on projective varieties with mild singularities, are either rationally Morita equivalent to a flat partial holomorphic sheaf, or a sheaf with a meromorphic flat connection or a co-Higgs sheaf. As an application, we study the geometry of rank two meromorphic rank two $\mathfrak{sl}_2$-Poisson modules which can be interpreted as a Poisson analogous to transversally projective structures for codimension one holomorphic foli…
A general problem is to classify the real forms of a complex variety up to isomorphism. This paper introduces the polar group of a real form $X$ of a complex variety $Y$ as a tool to distinguish such real forms. This group is an invariant of $X$ which encodes information about the residual divisors in the coordinate ring of $Y$ over the coordinate ring of $X$. We calculate polar groups for various curves, including the real line and the algebraic 1-sphere.
Let S be a smooth complex projective surface equipped with a Poisson structure s and also a polarization H. The moduli space M_H(S,P) of stable sheaves on S having a fixed Hilbert polynomial P of degree one has a natural Poisson structure given by s, studied by Tyurin and Bottacin. We prove that the symplectic leaves of M_H(S,P) are the fibers of the natural map from it to the symmetric power of the effective divisor on S given by the singular locus of s.
The total coordinate ring of a normal projective variety
正规簇的全坐标环TC（X）是Cox引入并研究的环在复曲面簇上的推广。考虑一个正规投射簇X，其除子类群为Cl（X），并假设它是一个可生成的自由阿尔贝群.我们定义X的全坐标环为TC（X）= oplus_{D} H ^0（X，O_X（D）），其中上述和是对包含在Cl（X）的一个固定的完备表示系中的X的所有Weil因子取的。本文证明了对任意正规投射簇X，TC（X）是一个UFD，这是本文所证明的一个更一般的定理的推论。（Berchtold和Haussen独立地证明了光滑簇的唯一因子分解。我们还证明了对于X，P^2沿有限个共线点的爆破，TC（X）是Noether的。我们还给出了TC（X）不是Noetherian的例子，但对任意Weil因子D，oplus_nH^0（X，O（nD））是Noetherian的.
In this paper we study the moduli stack of complexes of vector bundles (with chain isomorphisms) over a smooth projective variety $X$ via derived algebraic geometry. We prove that if $X$ is a Calabi-Yau variety of dimension $d$ then this moduli stack has a $(1-d)$-shifted Poisson structure. In the case $d=1$, we construct a natural foliation of the moduli stack by $0$-shifted symplectic substacks. We show that our construction recovers various known Poisson structures associated to complex elliptic curves, including the Poisson structure on Hilbert scheme of points on elliptic quantum projecti…
In this paper, we will show that the Hesselink stratification of a Hilbert scheme of hypersurfaces is independent of the choice of Plücker coordinate and there is a positive relation between the length of Hesselink's worst virtual 1-parameter subgroup and multiplicity of a projective hypersurface.
We consider hypersurfaces in $(\mathbb{P}^1)^n$ that contain a generic sequence of small dynamical height with respect to a split map and project onto $n-1$ coordinates. We show that these hypersurfaces satisfy strong coincidence relations between their points with zero height coordinates. More precisely, it holds that in a Zariski-open dense subset of such a hypersurface $n-1$ coordinates have height zero if and only if all coordinates have height zero. This is a key step in the resolution of the dynamical Bogomolov conjecture for split maps.
We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress in 1975. The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren in 1996. This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform p…
We study the dual ${\rm G}^\ast$ of a standard semisimple Poisson-Lie group ${\rm G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}({\rm G}^\ast)$ can be naturally embedded into a cluster Poisson algebra with a Weyl group action. We prove that $\mathcal{O}({\rm G}^\ast)$ admits a natural basis which has positive integer structure coefficients and satisfies an invariance property with respect to a braid group action. We continue the study of the moduli space $\mathscr{P}_{{\rm G},\mathbb{S}}$ of ${\rm G}$-local systems introduced in \cite{GS3}, and prove th…
