In this paper we prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces $\mathfrak{M}_{0,n}$ of Riemann spheres with $n$ marked points are multiple zeta values. In order to do this, we introduce a differential algebra of multiple polylogarithms on $\mathfrak{M}_{0,n}$, and prove that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes' formula iteratively, and to exploit the geometry of the moduli spaces to reduce each period integral to multiple zeta values.
We also give a geometric interpretation of the double shuffle relations, by showing that they are two extremal cases of general product formulae for periods which arise by considering natural maps between moduli spaces.
We use the weighted moduli height as defined in \cite{sh-h} to investigate the distribution of fine moduli points in the moduli space of genus two curves. We show that for any genus two curve with equation $y^2=f(x)$, its weighted moduli height $\mathfrak h (\mathfrak{p}) \leq 2^3 \sqrt{3 \cdot 5 \cdot 7} \, \cdot H(f)$, where $H(f)$ is the minimal naive height of the curve as defined in \cite{height}. Based on the weighted moduli height $\mathfrak h$ we create a database of genus two curves defined over $\mathbb Q$ with small $\mathfrak h$ and show that for small such height ($\mathfrak h < 5…
In this paper we study and relate the non-abelian zeta functions introduced by Weng and invariants of the moduli spaces of arbitrary rank stable pairs over curves. We prove a wall-crossing formula for the latter invariants and obtain an explicit formula for these invariants in terms of the motive of a curve. Previously, formulas for these invariants were known only for rank 2 due to Thaddeus and for rank 3 due to Muñoz. Using these results we obtain an explicit formula for the non-abelian zeta functions, we check the uniformity conjecture by Weng for the ranks 2 and 3, and we prove the countin…
The aim of this paper is to investigate the closed subschemes of moduli spaces corresponding to projective varieties which admit an effective action by a given finite group $G$. To achieve this, we introduce the moduli functor $\mathsf{M}^G_h$ of $G$-marked Gorenstein canonical models with Hilbert polynomial $h$, and prove the existence of $\mathfrak{M}_h[G]$, the coarse moduli scheme for $\mathsf{M}^G_h$. Then we show that $\mathfrak{M}_h[G]$ has a proper and finite morphism onto $\mathfrak{M}_h$ so that its image $\mathfrak{M}_h(G)$ is a closed subscheme. In the end we obtain the canonical r…
For any smooth projective moduli space $M$ of Gieseker stable sheaves on a complex projective K3 surface (or an abelian surface) S, we prove that the Chow motive $\mathfrak{h}(M)$ becomes a direct summand of a motive $\bigoplus \mathfrak{h}(S^{k_i})(n_i)$ with $k_i\leq \dim(M)$. The result implies that finite dimensionality of $\mathfrak{h}(M)$ follows from finite dimensionality of $\mathfrak{h}(S)$. The technique also applies to moduli spaces of twisted sheaves and to moduli spaces of stable objects in ${\rm D}^{\rm b}(S,α)$ for a Brauer class $α\in{\rm Br}(S)$. In a similar vein, we investig…
We give a natural construction of unramified over Z framed mixed Tate motives, whose periods are the multiple zeta values. Namely, for each convergent multiple zeta-value we define two boundary divisors A and B in the moduli space M_{0,n+3} of stable curves of genus zero. The corresponding multiple zeta-motive is the n-th cohomology of the pair (M_{0,n+3} -A,B).
For a smooth canonically embedded curve $C$ of genus $9$ together with a pencil $|L|$ of degree $6$, we study the relative canonical resolution of $C\subset X\subset \mathbb{P}^8$, where $X$ is the scroll swept out by the pencil $|L|$. We show that the second syzygy bundle in this resolution of $C\subset X$ is unbalanced. The proof reveals a new geometric connection between the universal Brill--Noether variety $\mathcal{W}^1_{9,6}$ and a moduli space $\mathcal{F}^\mathfrak{h}$ of lattice polarized $K3$ surfaces (for a certain rank $3$ lattice $\mathfrak{h}$). As a by-product we prove the unira…
Let $\mathfrak{Var}_k^G$ denote the category of pairs $(X,σ)$, where $X$ is a variety over $k$ and $σ$ is a group action on $X$. We define the Grothendieck ring for varieties with group actions as the free abelian group of isomorphism classes in the category $\mathfrak{Var}_k^G$ modulo a cutting and pasting relation. The multiplication in this ring is defined by the fiber product of varieties. This allows for motivic zeta-functions for varieties with group actions to be defined. This is a formal power series $\sum_{n=0}^{\infty}[\text{Sym}^n (X,σ)]t^n$ with coefficients in the Grothendieck rin…
The purpose of the present paper is to develop the theory of (pre-)Tango structures and (dormant generic) Miura $\mathfrak{g}$-opers (for a semisimple Lie algebra $\mathfrak{g}$) defined on pointed stable curves in positive characteristic. A (pre-)Tango structure is a certain line bundle of an algebraic curve in positive characteristic, which gives some pathological (relative to zero characteristic) phenomena. In the present paper, we construct the moduli spaces of (pre-)Tango structures and (dormant generic) Miura $\mathfrak{g}$-opers respectively, and prove certain properties of them. One of…
We consider the stable compactification $\bar {\mathfrak H}$ of the moduli space of Horikawa surfaces with $K_X^2 = 2p_g(X) -4$. When $K_X^2 =8\ell$ we show that the closures of the two components $\mathfrak H^{\mathrm I}$ and $\mathfrak H^{\mathrm {II}}$ of the Gieseker moduli space intersect, for $\ell>2$ in a divisor parametrising explicitly described semi-smooth surfaces. With growing $K_X^2$ we find an increasing number of generically non-reduced irreducible components in the same connected component of the moduli space of stable surfaces.
For the TZ metric on the moduli space $\mathscr{M}_{0,n}$ of $n$-pointed rational curves, we construct a Kähler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space $\mathfrak{S}_{g,n}$ as holomorphic fibration $\mathfrak{S}_{g,n}\rightarrow\mathfrak{S}_{g}$ over the Schottky space $\mathfrak{S}_{g}$ of compact Riemann surfaces of genus $g$, where the fibers are configuration spaces of $n$ points. For the tautological line bundles $\mathscr{L}_{i}$ over $\mathfrak{S}_{g,n}$ we define Hermitian metrics $h_{i}$ in terms of Fourier coefficients of a coveri…
Let $C$ be a smooth projective curve of genus $g\geq 2$ over $\mathbb C$. Fix $n\geq 1$, $d\in {\mathbb Z}$. A pair $(E,ϕ)$ over $C$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $C$ and a section $ϕ\in H^0(E)$. There is a concept of stability for pairs which depends on a real parameter $τ$. Let ${\mathfrak M}_τ(n,d)$ be the moduli space of $τ$-polystable pairs of rank $n$ and degree $d$ over $C$. Here we prove that for a generic curve $C$, the moduli space ${\mathfrak M}_τ(n,d)$ satisfies the Hodge Conjecture for $n \leq 4$. For obtaining this, we prove first that…
We study endomorphisms of complete Noetherian local rings in the context of motivic integration. Using the notion of an auto-arc space, we introduce the (reduced) auto-Igusa zeta series at a point, which appears to measure the degree to which a variety is not smooth that point. We conjecture a closed formula in the case of curves with one singular point, and we provide explicit formulas for this series in the case of the cusp and the node. Using the work of Denef and Loeser, one can show that this series will often be rational. These ideas were obtained through extensive calculations in Sage. …
We study the moduli of trigonal curves. We establish the exact upper bound of ${36(g+1)}/(5g+1)$ for the slope of trigonal fibrations. Here, the slope of any fibration $X\to B$ of stable curves with smooth general member is the ratio $δ_B/λ_B$ of the restrictions of the boundary class $δ$ and the Hodge class $λ$ on the moduli space $\bar{\mathfrak{M}}_g$ to the base $B$. We associate to a trigonal family $X$ a canonical rank two vector bundle $V$, and show that for Bogomolov-semistable $V$ the slope satisfies the stronger inequality ${δ_B}/{λ_B}\leq 7+{6}/{g}$. We further describe the rational…
The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let $X$ be a non-singular irreducible complex surface and let $E$ be a vector bundle of rank $n$ on $X$. We use the $m$-elementary transformation of $E$ at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;c_1,c_2)$ to $Hilb_{\, \ma…
In this paper, we study the arithmetic zeta function $$\mathscr{Z}_{\mathcal{X}}(s) = \prod_p \prod_{\substack{x \in \mathcal{X}_p \\ \text{closed}}} \Big( \frac{1}{1-|κ(x)|^{-s}} \Big)^{\mathfrak{m}_{p}(x)}$$ associated to a scheme $\mathcal{X}$ of finite type over $\mathbb{Z}$, where $κ(x)$ denotes the residue field and $\mathfrak{m}_{p}(x)$ the multiplicity of $x$ in $\mathcal{X}_p$. If $\mathcal{X}$ is defined over a finite field, then $\mathscr{Z}_{\mathcal{X}}$ appears naturally in the context of point counting with multiplicities. We prove that $\mathscr{Z}_{\mathcal{X}}$ admits a merom…
We construct a desingularization of the ``main component'' $\bar{\mathfrak M}_{1,k}^0(\Bbb{P}^n,d)$ of the moduli space $\bar{\mathfrak M}_{1,k}(\Bbb{P}^n,d)$ of genus-one stable maps into the complex projective space $\Bbb{P}^n$. As a bonus, we obtain desingularizations of certain natural sheaves over $\bar{\mathfrak M}_{1,k}^0(\Bbb{P}^n,d)$. Such desingularizations are useful for integrating natural cohomology classes on $\bar{\mathfrak M}_{1,k}^0(\Bbb{P}^n,d)$ using localization. In turn, these classes can be used to compute the genus-one Gromov-Witten invariants of complete intersections a…
Using analytic torsion associated to stable bundles, we introduce zeta functions for compact Riemann surfaces. To justify the well-definedness, we analyze the degenerations of analytic torsions at the boundaries of the moduli spaces, the singularities of analytic torsions at Brill-Noether loci, and the asymptotic behaviors of analytic torsions with respect to the degree. These new yet intrinsic zetas, both abelian and non-abelian, are expected to play key roles to understand global analysis and geometry of Riemann surfaces, such as the Tamagawa number conjecture for Riemann surfaces, searched …
We will introduce a regularization for $p$-adic multiple zeta values and show that the generalized double shuffle relations hold. This settles a question raised by Deligne, given as a project in Arizona winter school 2002. Our approach is to use the theory of Coleman functions on the moduli space of genus zero curves with marked points and its compactification. The main ingredients are the analytic continuation of Coleman functions to the normal bundle of divisors at infinity and definition of a special tangential base point on the moduli space.
Rapid developments in genetics and biology have led to phylogenetic methods becoming an important direction in the study of cancer and viral evolution. Although our understanding of gene biology and biochemistry has increased and is increasing at a remarkable rate, the theoretical models of genetic evolution still use the phylogenetic tree model that was introduced by Darwin in 1859 and the generalization to phylogenetic networks introduced by Grant in 1971. Darwin's model uses phylogenetic trees to capture the evolutionary relationships of reproducing individuals [6]; Grant's generalization t…
Let $\mathcal C$ be a smooth irreducible projective curve over the complex numbers and let $G$ be a simple simply-connected complex algebraic group. Let $\mathfrak M=\mathfrak M(G,\mathcal C)$ be the moduli space of semistable principal $G$-bundles on $\mathcal C$. By an earlier result of Kumar-Narasimhan, the Picard group of $\mathfrak M$ is isomorphic with the group of integers. However, in their work the generator of the Picard group was not determined explicitly. The aim of this paper to give the generator `explicitly.' The proof involves an interesting mix of geometry and topology.
We construct a functor that associates to any dg cooperad of dg commutative algebras (satisfying some conditions) an augmented commutative algebra. When applied to the cohomology operad of Francis Brown's moduli spaces it produces an algebra that formally models the algebra of multiple zeta values. We prove that there is an injection from the graded dual of the Grothendieck-Teichmueller Lie algebra into the indecomposables of the algebra associated to the Gerstenhaber cooperad, and that there is a morphism from the algebra associated to Brown's moduli spaces to the algebra associated to the Ge…
$\mathbb{A}^1$-connectedness of moduli of vector bundles on a stacky curve
Indian Institute of Science Education and Research Tirupati（印度科学教育研究所蒂鲁帕提分校）；National Institute of Science Education and Research, HBNI（印度国家科学教育研究中心，HBNI）
For singular metrics, there is no Quillen metric formalism on cohomology determinant. In this paper, we develop an admissible theory, with which the arithmetic Deligne-Riemann-Roch isometry can be established for singular metrics. As an application, we first study Weil-Petersson metrics and Takhtajan-Zograf metrics on moduli spaces of punctured Riemann surfaces, and then give a more geometric interpretation of our determinant metrics in terms of Selberg zeta functions. We end this paper by proposing an arithmetic factorization for Weil-Petersson metrics, cuspidal metrics and Selberg zeta funct…
In this note the geography of minimal surfaces of general type admitting $\mathbb{Z}_2^2$-actions is studied. More precisely, it is shown that Gieseker's moduli space $\mathfrak{M}_{K^2,χ}$ contains surfaces admitting a $\mathbb{Z}_2^2$-action for every admissible pair $(K^2, χ)$ such that $2χ-6\leq K^2\leq 8χ-8$ or $K^2=8χ$. The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBA-compactification $\overline{\mathfrak{M}}_{K^2,χ}$ of Gieseker's moduli space $\mathfrak{M}_{K^2,χ}$ for every admissible pair $(K^2, χ)$ such that $2χ-6\leq K^2\…
In the infinite series of complete families of Calabi-Yau manifolds $\tilde{f}_n: \tilde{\mathcal{X}}_n\rightarrow \mathfrak{M}_{n, n+3}$, where $n$ is an odd number, arising from cyclic covers of $\mathbb{P}^n$ branching along hyperplane arrangements (\cite{SXZ13}), the set of CM points is dense for $n=1, 3$ and finite for $n\geq 5$.
Let $\mathfrak p$ be any point in the moduli space of genus-two curves $\mathcal M_2$ and $K$ its field of moduli. We provide a universal equation of a genus-two curve $\mathcal C_{α, β}$ defined over $K(α, β)$, corresponding to $\mathfrak p$, where $α$ and $β$ satisfy a quadratic $α^2+ b β^2= c$ such that $b$ and $c$ are given in terms of ratios of Siegel modular forms. The curve $\mathcal C_{α, β}$ is defined over the field of moduli $K$ if and only if the quadratic has a $K$-rational point $(α, β)$. We discover some interesting symmetries of the Weierstrass equation of $\mathcal C_{α, β}$. …
The algebra of big zeta values we introduce in this paper is an intermediate object between multiple zeta values and periods of the multiple zeta motive. It consists of number series generalizing multiple zeta values, the simplest examples, which are not multiple zeta series, are Tornheim sums. We show that convergent big zeta values are periods of the moduli space of stable curves of genus zero on one hand and multiple zeta values on the other hand. It gives an alternative way to prove that any such period may be expressed as a rational linear combination of multiple zeta values and a simple …
We prove irreducibility for the space of cyclic covers of fixed numerical type between smooth projective curves, and also for the space of cyclic covers of prime order and of fixed numerical-combinatorial type between moduli-stable projective curves. As an application we describe the irreducible components of the singular locus of the compactified moduli space of curves $\bar{\mahfrak M_g}$, extending the work of Cornalba, who described the irreducible components of the singular locus of the moduli space of curves $\mahfrak M_g}$.
We give a combinatorial description of the rational cohomology of the moduli spaces of pointed genus 1 curves with $n$ marked points and level $N$ structures. More precisely, we explicitly describe the $E_2$ term of the Leray spectral sequence of the forgetful mapping $\EuScript{M}_{1,n}(N)\to\EuScript{M}_{1,1}(N)$ and show that the result is isomorphic to the rational cohomology of $\EuScript{M}_{1,n}(N)$ as a rational mixed Hodge structure equipped with an action of the symmetric group $\mathfrak{S}_n$. The classical moduli space $\EuScript{M}_{1,n}$ is the particular case N=1.
We determine the type of the zeta functions and the range of the dimensions of the moduli spaces of finite flat models of two-dimensional local Galois representations over finite fields. This gives a generalization of Raynaud's theorem on the uniqueness of finite flat models in low ramifications.
Kontsevich's 1997 formula for the deformation quantization of Poisson brackets is a Feynman expansion involving volume integrals over moduli spaces of marked disks. We develop a systematic theory of integration on these moduli spaces via suitable algebras of polylogarithms, and use it to prove that Kontsevich's integrals can be expressed as integer-linear combinations of multiple zeta values. Our proof gives a concrete algorithm for calculating the integrals, which we have used to produce the first software package for the symbolic calculation of Kontsevich's formula.
For $n\ge5$, it is well known that the moduli space $\mathfrak{M_{0,\:n}}$ of unordered $n$ points on the Riemann sphere is a quotient space of the Zariski open set $K_n$ of $\mathbb C^{n-3}$ by an $S_n$ action. The stabilizers of this $S_n$ action at certain points of this Zariski open set $K_n$ correspond to the groups fixing the sets of $n$ points on the Riemann sphere. Let $α$ be a subset of $n$ distinct points on the Riemann sphere. We call the group of all linear fractional transformations leaving $α$ invariant the stabilizer of $α$, which is finite by observation. For each non-trivial f…
The central result of this paper is an identification of the shifted Cartier dual of the moduli stack $\mathcal{M}_{\mathfrak{g}}(C)$ of $\widetilde{G}$-Higgs bundles on $C$ of arbitrary degree (modulo shifts by $Z(\widetilde{G})$) with a quotient of the Langlands dual stack $\mathcal{M}_{^L\mathfrak{g}}(C)$. Via hyperkähler rotation, this may equivalently be viewed as the identification of an SYZ fibration relating Hitchin systems for arbitrary Langlands dual semisimple groups, coupled to nontrivial finite $B$-fields. As a corollary certain self-dual stacks $\frac{\mathcal{M}_{\mathfrak{g}}(C…
We study the symplectic nature of the moduli stack classifying dormant curves over a field $K$ of positive characteristic, i.e., proper hyperbolic curves over $K$ equipped with a dormant indigenous bundle. The central objects of the present paper are the following two Deligne-Mumford stacks. One is the cotangent bundle ${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}}$ of the moduli stack ${^\circledcirc \mathfrak{M}^{^\mathrm{Zzz...}}_{g,K}}$ classifying ordinary dormant curves over $K$ of genus $g$. The other is the moduli stack ${^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$ classify…
In this paper the ubiquity of the Igusa quartic $B \subset \mathbb P^4$ shows up again, this time related to the Prym map $\mathfrak p : \mathcal R_6 \to \mathcal A_5$. We introduce the moduli space $\mathcal X$ of those quartic threefolds $X$ cutting twice a quadratic section of $B$. A general $X$ is $30$-nodal and the intermediate Jacobian $J(X)$ of its natural desingularization is a $5$-dimensional p.p. abelian variety. Let $\frak j: \mathcal X \to \mathcal A_5$ be the period map sending $X$ to $J(X)$, in the paper we study $\frak j$ and its relation to $\frak p$. As is well known the degre…
We construct examples of non-smoothable stable surfaces, that we call Corona surfaces, with invariants in a wide range comprising all possible invariants of smooth minimal surfaces of general type but non-standard second plurigenus. We deduce that the moduli space of stable surfaces $\overline{\mathfrak M}_{k,l}$ has least $k-l+2$ connected components distinguished by the second plurigenus and, in particular, it always has more irreducible components than the Gieseker moduli space with the same invariants.
We build a database of genus 2 curves defined over $\mathbb Q$ which contains all curves with minimal absolute height $h \leq 5$, all curves with moduli height $\mathfrak h \leq 20$, and all curves with extra automorphisms in standard form $y^2=f(x^2)$ defined over $\mathbb Q$ with height $h \leq 101$. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants. The distribution of rational points in the moduli space $\mathcal M_2$ for which the field of moduli is a field of definiti…
We construct level one dominant representations of the affine Kac-Moody algebra $\widehat{\mathfrak{gl}}_k$ on the equivariant cohomology groups of moduli spaces of rank one framed sheaves on the orbifold compactification of the minimal resolution $X_k$ of the $A_{k-1}$ toric singularity $\mathbb{C}^2/\mathbb{Z}_k$. We show that the direct sum of the fundamental classes of these moduli spaces is a Whittaker vector for $\widehat{\mathfrak{gl}}_k$, which proves the AGT correspondence for pure $\mathcal{N}=2$ $U(1)$ gauge theory on $X_k$. We consider Carlsson-Okounkov type Ext-bundles over produc…
We consider two K3 surfaces defined over an arbitrary field, together with a smooth proper moduli space of stable sheaves on each. When the moduli spaces have the same dimension, we prove that if the étale cohomology groups (with Q_ell coefficients) of the two surfaces are isomorphic as Galois representations, then the same is true of the two moduli spaces. In particular, if the field of definition is finite and the K3 surfaces have equal zeta functions, then so do the moduli spaces, even when the moduli spaces are not birational.
Let $\mathcal{A}_g$ denote the moduli stack of principally polarized abelian varieties of dimension $g$. The arithmetic height, or arithmetic volume, of $\overline{\mathcal{A}}_g$, is defined to be the arithmetic degree of the metrized Hodge bundle $\overlineω_g$ on $\overline{\mathcal{A}}_g$. In 1999, Kühn proved a formula for the arithmetic volume of $\overline{\mathcal{A}}_1$ in terms of special values of the Riemann zeta function. In this article, we generalize his result to the case $g=2$.
Using Szenes formula for multiple Bernoulli series we explain how to compute Witten series associated to classical Lie algebras. Particular instances of these series compute volumes of moduli spaces of flat bundles over surfaces, and also certain multiple zeta values.
Consider the Fulton-MacPherson configuration space of $n$ points on $¶^1$, which is isomorphic to a certain moduli space of stable maps to $¶^1$. We compute the cone of effective ${\mathfrak S}_n$-invariant divisors on this space. This yields a geometric interpretation of known asymptotic formulas for the number of integral points of bounded height on compactifications of $\SL_2$ in the space of binary forms of degree $n\ge 3$.
We prove that the degree zero Hochschild-type cohomology of the homology operad of Francis Brown's dihedral moduli spaces is equal to the Grothendieck-Teichmueller Lie algebra plus two classes. This significantly elucidates the (in part still conjectural) relation between the Grothendieck-Teichmueller Lie algebra and (motivic) multiple zeta values.
We compute the Z/\ell and \ell-adic monodromy of every irreducible component of the moduli space M_g^f of curves of genus and and p-rank f. In particular, we prove that the Z/\ell-monodromy of every component of M_g^f is the symplectic group Sp_{2g}(Z/\ell) if g>=3 and \ell is a prime distinct from p. We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank.
The second order casimirs for the affine Krichever--Novikov algebras $\hat{\mathfrak{gl}}_{g,2}$ and $\hat{\mathfrak{sl}}_{g,2}$ are described. More general operators which we call semi-casimirs are introduced. It is proven that the semi-casimirs induce well-defined operators on conformal blocks and, for a certain moduli space of Riemann surfaces with two marked points and fixed jets of local coordinates, there is a natural projection of its tangent space onto the space of these operators. It is (non-formally) explained how semi-casimirs appear in course of operator quantization of the second …
