Let X and X' be compact Riemann surfaces of genus at least 3, and let G and G' be nonabelian reductive complex groups. If one component M_G^d(X) of the moduli space for semistable principal G-bundles over X is isomorphic to another component M_{G'}^{d'}(X'), then X is isomorphic to X'.
Let $C$ be an elliptic curve, $w\in C$, and let $S\subset C$ be a finite subset of cardinality at least $3$. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle $\mathcal O_C(w)$ over $(C,S)$ which are semistable with respect to a weight vector $\big(\frac{1}{2}, \dots, \frac{1}{2}\big)$.
The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal crossing singularity and it provides a flat degeneration. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than $1$) and fixed degree such that rank and degree are co-prime.
Let $G$ be a simple simply-connected connected linear algebraic group over $\mathbb{C}$. We proved a $2$-birational Torelli theorem for the moduli space of semistable principal $G$-bundles over a smooth curve of genus $\geq 3$, which says that if two such moduli spaces are $2$-birational then the curves are isomorphic. We also proved a $3$-birational Torelli theorem for the moduli space of stable symplectic parabolic bundles over a smooth curve of genus $\geq 4$.
Let $(X,D)$ and $(X',D')$ be two compact Riemann surfaces of genus $g \geq 4$ with the set of marked points $D \subset X$ and $D' \subset X'$. Fix a parabolic line bundle $L$ with trivial parabolic structure. Let $\mathcal{N}_{\textnormal{Sp}}(2m,α,L)$ and $\mathcal{N}'_{\textnormal{Sp}}(2m,α,L)$ be the moduli spaces of stable symplectic parabolic Higgs bundles over $X$ and $X'$ respectively, with rank $2m$ and fixed parabolic structure $α$, with the symplectic form taking values in $L$. We prove that if $\mathcal{N}_{\textnormal{Sp}}(2m,α,L)$ is isomorphic to $\mathcal{N}'_{\textnormal{Sp}}(2…
Let $g \geq 2$ and let the Torelli map denote the map sending a genus $g$ curve to its principally polarized Jacobian. We show that the restriction of the Torelli map to the hyperelliptic locus is an immersion in characteristic not $2$. In characteristic $2$, we show the Torelli map restricted to the hyperelliptic locus fails to be an immersion because it is generically inseparable; moreover, the induced map on tangent spaces has kernel of dimension $g-2$ at every point.
We construct a moduli space of adequately marked Enriques surfaces that have a supersingular K3 cover over fields of characteristic $p \geq 3$. We show that this moduli space exists as a scheme locally of finite type over $\mathbb{F}_p$. Moreover, there exists a period map from this moduli space to a period scheme and we obtain a Torelli theorem for supersingular Enriques surfaces.
We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is singular: when the cohomology class of a symplectic structure satisfies certain linear equations with integer coefficients, its polar divisor can be partially smoothed, yielding adjacent irreducible components of the moduli space that correspond to possibly non-normal crossings structures. These components are indexed by combinatorial data we call smoothing di…
We develop the global moduli theory of symplectic varieties in the sense of Beauville. We prove a number of analogs of classical results from the smooth case, including a global Torelli theorem. In particular, this yields a new proof of Verbitsky's global Torelli theorem in the smooth case (assuming $b_2\geq 5$) which does not use the existence of a hyperkähler metric or twistor deformations.
We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties. Our main result is a generalization of Ogus' crystalline Torelli theorem for supersingular K3 surfaces. For applications, we answer a slight variant of a question asked by F. Charles on moduli spaces of sheaves on K3 surfaces and give a crystalline Torelli theorem for supersingular cubic fourfolds.
We survey crystalline cohomology, crystals, and formal group laws with an emphasis on geometry. We apply these concepts to K3 surfaces, and especially to supersingular K3 surfaces. In particular, we discuss stratifications of the moduli space of polarized K3 surfaces in positive characteristic, Ogus' crystalline Torelli theorem for supersingular K3 surfaces, the Tate conjecture, and the unirationality of K3 surfaces.
Let $X$ be a compact Riemann surface of genus $g \geq 3$. We consider the moduli space of holomorphic connections over $X$ and the moduli space of logarithmic connections singular over a finite subset of $X$ with fixed residues. We determine the Chow group of these moduli spaces. We compute the global sections of the sheaves of differential operators on ample line bundles and their symmetric powers over these moduli spaces, and show that they are constant under certain condition. We show the Torelli type theorem for the moduli space of logarithmic connections. We also describe the rational con…
Let $C$ be an algebraic curve of genus $g$ and $L$ a line bundle over $C$. Let $\mathcal{MS}_C(n,L)$ and $\mathcal{MO}_C(n,L)$ be the moduli spaces of $L$-valued symplectic and orthogonal bundles respectively, over $C$ of rank $n$. We construct rational curves on these moduli spaces which generalize Hecke curves on the moduli space of vector bundles. As a main result, we show that these Hecke type curves have the minimal degree among the rational curves passing through a general point of the moduli spaces. As its byproducts, we show the non-abelian Torelli theorem and compute the automorphism …
