${\rm SL}_3$-laminations as bases for ${\rm PGL}_3$ cluster varieties for surfaces
Hyun Kyu Kim
备注：
ver2: A major change is that the quantization is added. Some terminology changed, hence the title changed / ver3: Relatively minor corrections, and bibliography update / ver4: Some errors corrected, gaps filled, subsection 5.7 added, conjectural section 6 cut down, referee's feedbacks applied. To appear in Memoirs of the AMS
In this paper we partially settle Fock-Goncharov's duality conjecture for cluster varieties associated to their moduli spaces of ${\rm G}$-local systems on a punctured surface $\frak{S}$ with boundary data, when ${\rm G}$ is a group of type $A_2$, namely ${\rm SL}_3$ and ${\rm PGL}_3$. Based on Kuperberg's ${\rm SL}_3$-webs, we introduce the notion of ${\rm SL}_3$-laminations on $\frak{S}$ defined as certain ${\rm SL}_3$-webs with integer weights. We introduce coordinate systems for ${\rm SL}_3$-laminations, and show that ${\rm SL}_3$-laminations satisfying a congruence property are geometric realizations of the tropical integer points of the cluster $\mathscr{A}$-moduli space $\mathscr{A}_{{\rm SL}_3,\frak{S}}$. Per each such ${\rm SL}_3$-lamination, we construct a regular function on the cluster $\mathscr{X}$-moduli space $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$. We show that these functions form a basis of the ring of all regular functions. For a proof, we develop ${\rm SL}_3$ quantum and classical trace maps for any triangulated bordered surface with marked points, and state-sum formulas for them. We construct quantum versions of the basic regular functions on $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$. The bases constructed in this paper are built from non-elliptic webs, hence could be viewed as higher `bangles' bases, and the corresponding `bracelets' versions can also be considered as direct analogs of Fock-Goncharov's and Allegretti-Kim's bases for the ${\rm SL}_2$-${\rm PGL}_2$ case.
We give a Thurston-like definition for laminations on higher Teichmuller spaces associated to a surface $S$ and a semi-simple group $G$ for $G-SL_m$ and $PGL_m$. The case $G=SL_2$ or $PGL_2$ corresponds to the classical theory of laminations. Our construction involves positive configurations of points in the affine building. We show that these laminations are parameterized by the tropical points of the spaces $\X_{G,S}$ and $\A_{G,S}$ of Fock and Goncharov. Finally, we explain how these laminations give a compactification of higher Teichmuller spaces.
We make explicit the equations of any projective $PGL_2(\C)$-variety defined by quadrics. We study their zero-locus and their relationship with the geometry of the Veronese curve.
In this paper, we compute the E-polynomials of the $PGL(2,\mathbb{C})$-character varieties associated to surfaces of genus $g$ with one puncture, for any holonomy around it, and compare it with its Langlands dual case, $SL(2,\mathbb{C})$. The study is based on the stratification of the space of representations and on the analysis of the behaviour of the E-polynomial under fibrations.
Let A be a central division algebra over a field F with ind A = n. In computing canonical p-dimension of projective PGL_1(A)-homogeneous varieties, for p prime, we can reduce to the case of generalized Severi-Brauer varieties X_e(A) with ind A a power of p divisible by e. We prove that canonical 2-dimension (and hence canonical dimension) equals dimension for all X_e(A) with ind A = 2e a power of 2.
Cluster varieties are geometric objects that have recently found applications in several areas of mathematics and mathematical physics. This thesis studies the geometry of a large class of cluster varieties associated to compact oriented surfaces with boundary. The main original contribution of this thesis is to develop the properties of a particular kind of cluster variety called the symplectic double. We show that the symplectic double is birational to a certain moduli space of local systems associated to a doubled surface. We define a version of the notion of measured lamination on such a s…
Let $G$ be a complex reductive group and $\mathcal{X}_{r}G$ denote the $G$-character variety of the free group of rank $r$. Using geometric methods, we prove that $E(\mathcal{X}_{r}SL_{n})=E(\mathcal{X}_{r}PGL_{n})$, for any $n,r\in\mathbb{N}$, where E(X) denotes the Serre (also known as E-) polynomial of the complex quasi-projective variety $X$, settling a conjecture of Lawton-Muñoz in [LM]. The proof involves the stratification by polystable type introduced in [FNZ], and shows moreover that the equality of E-polynomials holds for every stratum and, in particular, for the irreducible stratum …
Borrowing a reduction principle to a recent preprint of G. Faltings (toroidal resolution of some matrix singularities, 1999), we use Lafforgue's compactification of PGL_r^{N+1}/PGL_r to construct a canonical log-smooth toroidal resolution for the bad reduction in a prime p of Shimura varieties of unitary and symplectic type with parahoric level structures at p. Using this result, non-canonical semi-stable resolutions over Z_p[p^{1/ν}] can be derived.
A detailed study of half-BPS line operators of higher rank 4d N=2 theory engineered from six dimensional A_{N-1} (2,0) theory on a bordered Riemann surface with full marked points is performed. Geometrically, each 4d UV line operator is represented by an irreducible bipartite web formed by three junctions on Riemann surface, and such web structure is called higher lamination. Algebraically, the space of UV line operators is identified with the integral tropical a coordinates of the corresponding PGL(N,C) local system, and the space of IR line operator is identified with the cluster X coordinat…
A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and Cohen-Macaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any…
Given an algebraic variety $X\subset\mathbb{P}^N$ with stabilizer $H$, the quotient $PGL_{N+1}/H$ can be interpreted a parameter space for all $PGL_{N+1}$-translates of $X$. We define $X$ to be a $\textit{homogeneous variety}$ if $H$ acts on it transitively, and satisfies a few other properties, such as $H$ being semisimple. Some examples of homogeneous varieties are quadric hypersurfaces, rational normal curves, and Veronese and Segre embeddings. In this case, we construct new compactifications of the parameter spaces $PGL_{N+1}/H$, obtained compactifying $PGL_{N+1}$ to the classically known …
The moduli spaces of flat $\mathrm{SL}_2$- and $\mathrm{PGL}_2$-connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tamás Hausel in Remark 3.30 of "Global topology of the Hitchin system".
For $G = \mathrm{GL}_2, \mathrm{SL}_2, \mathrm{PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the $G$-character variety of a compact Riemann surface $C$ and of the moduli space of $G$-Higgs bundles on $C$ of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.
For G = GL_2, PGL_2 and SL_2 we prove that the perverse filtration associated to the Hitchin map on the cohomology of the moduli space of twisted G-Higgs bundles on a Riemann surface C agrees with the weight filtration on the cohomology of the twisted G character variety of C, when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.
In this paper, we generalise results obtained earlier by John Cremona and the author on the reduction theory of binary forms, which describe positive zero-cycles in P^1, to positive zero-cycles (or point clusters) in projective spaces of arbitrary dimension. This should have applications to more general projective varieties in P^n, by associating a suitable positive zero-cycle to them in an PGL(n+1)-invariant way. We discuss this in the case of (smooth) plane curves.
Let $C$ be an algebraic smooth complex curve of genus $g>1$. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on $C$ and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients $(\PP^{r-1})^{rg}//PGL(r)$, where $r$ is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stabi…
Let $C$ be a hyperelliptic curve of genus $g \geq 3$. We give a new description of the theta map for moduli spaces of rank 2 semistable vector bundles with trivial determinant. In orther to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb{P}^1)^{2g}//\operatorname{PGL(2)}$. Then, we use recent results of Kumar to identify the restriction of the theta map to these GIT quotients with some explicit osculating projection. As a corollary of this construction, we obtain a birational equivalence between the ramification locus of th…
We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and $\operatorname{PGL}_n$. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for $d$ coprime to $n$, that the number of rank $n$ Higgs bundles of deg…
A decorated surface S is a surface with a finite set of special points on the boundary, considered modulo isotopy. Let G be a split reductive group. A pair (G, S) gives rise to a moduli space A(G, S), closely related to the space of G-local systems on S. It has a positive structure. So the set of its integral tropical points is defined. We introduce a rational positive function W on A(G, S), the potential. The condition that its tropicalisation is non-negative determines its subset. For SL(2), we recover the set of integral laminations on S. We prove that when S is a disc with n special points…
