arXiv subjects

Lingguang Li

Publications and source records attributed to Lingguang Li.

16 recordsLinked to original sources

arXiv2026

Exact sequences of fundamental group schemes for $G$-torsors

Let $k$ be a field, $G$ an affine $k$-group scheme, $X$ a connected scheme proper over $k$, $f:(Y,y)\to(X,x)$ a pointed $G$-torsor, $\mathcal{C}_X$ and $\mathcal{C}_Y $ the Tannakian categories of vector bundles on $X$ and $Y$, such that $f^*\mathcal{C}_X\subseteq \mathcal{C}_Y$, we give the necessary and sufficient conditions for the exactness of the natural sequence of $k$-group schemes $1\rightarrow π(\mathcal{C}_Y,y) \rightarrow π(\mathcal{C}_X,x) \rightarrow G \rightarrow 1.$ This provides a common framework of exact sequences for the $S$-, Nori, $F$-, étale fundamental group schemes and their extended variants. Finally, an isogeny of an elliptic curve shows that the relevant preservation property fails for the local, extended local, and unipotent categories.

math.AG↗

arXiv2026

On the number of Frobenius periodic vector bundles on elliptic curves

This paper counts Frobenius-periodic vector bundles on elliptic curves over an algebraically closed field of characteristic $p>0$. By translating the problem into continuous representations of the étale fundamental group, it derives explicit generating functions and exact-period formulas, with separate treatments of the ordinary and supersingular cases.

math.AG↗

arXiv2026

The Base Change Of Fundamental Group Schemes

Let $k$ be a field, $K/k$ a field extension, $X$ a connected scheme proper over $k$, $x_K\in X_K(K)$ lying over $x\in X(k)$, $\mathcal{C}_X$ and $\mathcal{C}_{X_K}$ the Tannakian categories whose objects consist of vector bundles on $X$ and $X_K$ respectively, $π(\mathcal{C}_X,x)$ and $π(\mathcal{C}_{X_K},x_K)$ the corresponding Tannaka group schemes respectively. We establish a unified criterion determining when the base change homomorphism $π(\mathcal{C}_{X_K},x_K)\rightarrow π(\mathcal{C}_X,x)_K$ is faithfully flat or an isomorphism. As applications, we recover and generalize base change results for the S, Nori, EN, F, EF, ét, Eét, Loc, ELoc, and unipotent-fundamental group schemes under different types of field extensions (e.g., separable, finite Galois, and algebraically closed extensions). Moreover, our approach provides a unified explanation for both positive and negative results, including previously known counterexamples.

math.AG↗

arXiv2026

The Künneth Formula Of Fundamental Group Schemes

Let $k$ be a field, $f:X\rightarrow S$ a proper morphism between connected schemes proper over $k$, $x\in X(k)$ lying over $s\in S(k)$, $X_s$ the fibre of $f$ over $s$, $\mathcal{C}_X$, $\mathcal{C}_{S}$, $\mathcal{C}_{X_s}$ Tannakian categories over $X,S,X_s$ respectively, $π(\mathcal{C}_X,x)$, $π(\mathcal{C}_S,s)$, $π(\mathcal{C}_{X_s},x)$ the Tannaka group schemes respectively. We give a unified criterion for the exactness of the homotopy sequence of Tannakian fundamental group schemes $π(\mathcal{C}_{X_s},x)\rightarrow π(\mathcal{C}_X,x)\rightarrow π(\mathcal{C}_S,s)\rightarrow 1$. In particular, we obtain the equivalent conditions for the Künneth formula of fundamental group schemes for the product $X\times_k Y$ of two connected schemes $X$ and $Y$ proper over $k$. As an application, we obtain the Künneth formula of certain fundamental group schemes over any field, such as S, N, EN, F, EF, ét, Eét, Loc, ELoc and uni-fundamental group schemes.

math.AG↗

arXiv2026

The Lefschetz Type Theorem For Fundamental Group Schemes

Let $k$ be a field, $X$ a connected scheme proper over $k$, $D\subsetneq X$ an ample effective connected divisor, $x\in D(k)$. For Tannakian categories $\mathcal{C}_X$ and $\mathcal{C}_D$ whose objects consist of vector bundles on $X$ and $D$ respectively, we establish general Tannakian criteria for the natural homomorphism $π(\mathcal{C}_D,x)\to π(\mathcal{C}_X,x)$ to be faithfully flat, a closed immersion, or an isomorphism. As applications, under Langer type positivity assumptions, we prove that $π^{\ast}(D,x)\longrightarrow π^{\ast}(X,x)$ is an isomorphism for $\ast\in\{S,N,EN,F, EF,Loc,ELoc,\acute{e}t,E\acute{e}t,uni\}$ over perfect fields.

math.AG↗

arXiv2026

The Birational Invariance Of Fundamental Group Schemes

Let $k$ be a field, $f \colon X \to Y$ a birational morphism of integral connected schemes proper over $k$ with $Y$ normal, $x \in X(k)$ lying over $y \in Y(k)$. For Tannakian categories $\mathcal{C}_X \subset \mathfrak{Vect}(X)$ and $\mathcal{C}_Y \subset \mathfrak{Vect}(Y)$, denote by $π(\mathcal{C}_X,x)$ and $π(\mathcal{C}_Y,y)$ the corresponding Tannaka group schemes. We establish a unified Tannakian criteria for the natural homomorphism $π(\mathcal{C}_X,x)\to π(\mathcal{C}_Y,y)$ to be an isomorphism. As applications, for a birational map $X \dashrightarrow Y$ between smooth projective varieties over a perfect field $k$, we prove that there exists a natural isomorphism $π^{*}(X,x)\cong π^{*}(Y,y)$ for any $* \in \{S,N,EN,F,EF,Loc,ELoc,\acute{e}t, E\acute{e}t,uni\}$. In particular, we prove that the induced homomorphism $π^{str}(X,x)\to π^{str}(Y,y)$ is an isomorphism for any birational morphism $ X \rightarrow Y$.

math.AG↗

arXiv2026

Hecke curves in Frobenius strata of moduli space of rank 2 vector bundles

Let $k$ be an algebraically closed field with characteristic $2$, and let $X$ be a smooth projective algebraic curve of genus $g \geqslant 2$ over $k$. Let $\mathcal{M}^s_X(2,\mathcal{L})$ be the moduli space of rank $2$ stable vector bundles with determinant $\mathcal{L}$ on $X$. The Frobenius stratification measures the instability of bundles in $\mathcal{M}^s_X(r,\mathcal{L})$ under pullback by the Frobenius map. We show that there exists a Frobenius stratum in $\mathcal{M}^s_X(2,\mathcal{L})$ which is covered by Hecke curves.

math.AG↗

arXiv2018

Cone spherical metrics and stable vector bundles

Cone spherical metrics are conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces. A cone spherical metric is called irreducible if each developing map of the metric does not have monodromy lying in ${\rm U(1)}$. We establish on compact Riemann surfaces of positive genera a correspondence between irreducible cone spherical metrics with cone angles being integral multiples of $2π$ and line subbundles of rank two stable vector bundles. Then we are motivated by it to prove a theorem of Lange-type that there always exists a stable extension of $L^*$ by $L$, for $L$ being a line bundle of negative degree on each compact Riemann surface of genus greater than one. At last, as an application of these two results, we obtain a new class of irreducible spherical metrics with cone angles being integral multiples of $2π$ on each compact Riemann surface of genus greater than one

math.AG↗

arXiv2018

Frobenius Stratification of Moduli Spaces of Rank $3$ Vector Bundles in Characteristic $3$, I

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$, $F_X:X\rightarrow X$ the absolute Frobenius morphism. Let $\M^s_X(r,d)$ be the moduli space of stable vector bundles of rank $r$ and degree $d$ on $X$. We study the Frobenius stratification of $\M^s_X(3,0)$ in terms of Harder-Narasimhan polygons of Frobenius pull backs of stable vector bundles and obtain the irreducibility and dimension of each non-empty Frobenius stratum in the case $(p,g)=(3,2)$.

math.AG↗

arXiv2018

Frobenius Stratification of Moduli Spaces of Vector Bundles in Positive characteristic. II

Let $X$ be a smooth projective curve of genus $g(X)\geq 1$ over an algebraically closed field $k$ of characteristic $p>0$, $\M^s_X(r,d)$ the moduli space of stable vector bundles of rank $r$ and degree $d$ on $X$. We study the Frobenius stratification of $\M^s_X(r,d)$ in terms of Harder-Narasimhan polygons of Frobenius pull backs of stable vector bundles and obtain the irreducibility and dimension of each non-empty Frobenius stratum in case $(p,g,r)=(3,2,3)$.

math.AG↗

arXiv2017

Semistability of Rational Principal $GL_n$-Bundles in Positive Characteristic

Let $k$ be an algebraically closed field of characteristic $p>0$, $X$ a smooth projective variety over $k$ with a fixed ample divisor $H$. Let $E$ be a rational $GL_n(k)$-bundle on $X$, and $ρ:GL_n(k)\rightarrow GL_m(k)$ a rational $GL_n(k)$-representation at most degree $d$ such that $ρ$ maps the radical $R(GL_n(k))$ of $GL_n(k)$ into the radical $R(GL_m(k))$ of $GL_m(k)$. We show that if $F_X^{N*}(E)$ is semistable for some integer $N\geq\max\limits_{0<r<m}C^r_m\cdot\log_p(dr)$, then the induced rational $GL_m(k)$-bundle $E(GL_m(k))$ is semistable. As an application, if $\dim X=n$, we get a sufficient condition for the semistability of Frobenius direct image ${F_X}_*(ρ_*(Ω^1_X))$, where $ρ_*(Ω^1_X)$ is the locally free sheaf obtained from $Ω^1_X$ via the rational representation $ρ$.

math.AG↗

arXiv2014

Strong Stability of Cotangent Bundles of Cyclic Covers

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$ of $\dim X\geq 4$ and Picard number $ρ(X)=1$. Suppose that $X$ satisfies $H^i(X,F^{m*}_X(\Omg^j_X)\otimes\Ls^{-1})=0$ for any ample line bundle $\Ls$ on $X$, and any nonnegative integers $m,i,j$ with $0\leq i+j<\dim X$, where $F_X:X\rightarrow X$ is the absolute Frobenius morphism. We prove that by procedures combining taking smooth hypersurfaces of dimension $\geq 3$ and cyclic covers along smooth divisors, if the resulting smooth projective variety $Y$ has ample (resp. nef) canonical bundle $ω_Y$, then $\Omg_Y$ is strongly stable $($resp. strongly semistable$)$ with respect to any polarization.

math.AG↗

arXiv2014

On a Conjecture of Lan-Sheng-Zuo on Semistable Higgs Bundles: Rank 3 Case

Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$ of characteristic $p>2$. We prove that any rank $3$ nilpotent semistable Higgs bundle $(E,θ)$ on $X$ is a strongly semistable Higgs bundle. This gives a partially affirmative answer to a conjecture of Lan-Sheng-Zuo \cite{LanShengZuo12ii}\footnotemark[1]. In addition, we prove a tensor product theorem for strongly semistable Higgs bundles with $p$ satisfying some bounds (Theorem \ref{TensorTheorem}). From this we reprove a tensor theorem for semistable Higgs bundles on the condition that the Lan-Sheng-Zuo conjecture holds (Corollary \ref{TensorStableBundle}).

math.AG↗

arXiv2013

Vanishing Properties of Dual Bass numbers

Let $R$ be a Noetherian ring, $M$ an Artinian $R$-module, $\p\in\Cos_RM$. Then $\cograde_{R_{\p}}\Hom_{R}(R_{\p},M)=\inf\{i | π_{i}(\p,M)>0\}$ and $$π_{i}(\p,M)>0\Rightarrow\cograde_{R_{\p}}\Hom_{R}(R_{\p},M)\leq i\leq\fd_{R_{\p}}\Hom_{R}(R_{\p},M),$$ where $π_{i}(\p,M)$ is the $i$-th dual Bass number of $M$ with respect to $\p$, the integer $\cograde_{R_{\p}}\Hom_{R}(R_{\p},M)$ is the common length of any maximal $\Hom_{R}(R_{\p},M)$-quasi co-regular sequence contained in $\p R_{\p}$, and $\fd_{R_{\p}}\Hom_{R}(R_{\p},M)$ is the flat dimension of $R_{\p}$-module $\Hom_{R}(R_{\p},M)$ (Theorem \ref{Thm:Main}). Besides, we also study the relations among cograde, co-dimension and flat dimension of co-localization module $\Hom_{R}(R_{\p},M)$.

math.AC↗

arXiv2013

Dual Bass Numbers and Co-Cohen Macaulay Modules

In this paper, we give a characterization of co-Cohen Macaulay modules by vanishing properties of the dual Bass numbers of modules. In addition, we show that the co-localization of co-Cohen Macaulay modules preserves co-Cohen Macaulayness under a certain condition.

math.AC↗

arXiv2012

Instability of Truncated Symmetric Powers of sheaves

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Let $F_X:X\rightarrow X$ be the absolute Frobenius morphism, and $\E$ a torsion free sheaf on $X$. We give a upper bound of instability of truncated symmetric powers $\mathrm{T}^l(\E)(0\leq l\leq\rk(\E)(p-1))$ in terms of $L_{\max}(\Omg^1_X)$, $\mathrm{I}(\Omg^1_X)$ and $\mathrm{I}(\E)$ (Theorem \ref{InstabTl}). As an application, We obtain a upper bound of Frobenius direct image ${F_X}_*(\E)$ and some sufficient conditions of slope semi-stability of ${F_X}_*(\E)$. In addition, we study the slope (semi)-stability of sheaves of locally exact (closed) forms $B^i_X$ ($Z^i_X$).

math.AG↗