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Register a new userFiniteness of etale fundamental groups by reduction modulo $p$
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We introduce a spreading out technique to deduce finiteness results for etale fundamental groups of complex varieties by characteristic $p$ methods, and apply this to recover a finiteness result proven recently for local fundamental groups in characteristic $0$ using birational geometry.
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This paper contains three new results. {bf 1}.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of etale fundamental groups introduced in [7,10] and periodic Higgs-de Rham flows introduced in [19]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs-de Rham flows via the category of twisted Fontaine-Faltings module which is also introduced in this paper. {bf 2.}We study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometrically absolutely irreducible crystalline representation. {bf 3.} We investigate the dynamic of self-maps induced by the Higgs-de Rham flow on the moduli spaces of rank-2 stable Higgs bundles of degree 1 on $mathbb{P}^1$ with logarithmic structure on marked points $D:={x_1,,...,x_n}$ for $ngeq 4$ and construct infinitely many geometrically absolutely irreducible $mathrm{PGL_2}(mathbb Z_p^{mathrm{ur}})$-crystalline representations of $pi_1^text{et}(mathbb{P}^1_{{mathbb{Q}}_p^text{ur}}setminus D)$. We find an explicit formula of the self-map for the case ${0,,1,,infty,,lambda}$ and conjecture that a Higgs bundle is periodic if and only if the zero of the Higgs field is the image of a torsion point in the associated elliptic curve $mathcal{C}_lambda$ defined by $ y^2=x(x-1)(x-lambda)$ with the order coprime to $p$.
In this paper we focus on pairs consisting of the affine $N$-space and multiideals with a positive exponent. We introduce a method lifting to characteristic 0 which is a kind of the inversion of modulo p reduction. By making use of it, we prove that Mustata-Nakamuras conjecture and some uniform bound of divisors computing log canonical thresholds descend from characteristic 0 to certain classes of pairs in positive characteristic. We also pose a problem whose affirmative answer gives the descent of the statements to the whole set of pairs in positive characteristic.
We consider the KZ differential equations over $mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $mathbb F_p$. We study the space of polynomial solutions of these differential equations over $mathbb F_p$, constructed in a previous work by V. Schechtman and the second author. Using Hasse-Witt matrices we identify the space of these polynomial solutions over $mathbb F_p$ with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these polynomial solutions over $mathbb F_p$ and the hypergeometric solutions over $mathbb C$.
Let X be a smooth complex projective variety with basepoint x. We prove that every rigid integral irreducible representation $pi_1(X,x)to SL (3,{mathbb C})$ is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by K. Corlette and the second author in the rank 2 case and answers one of their questions.
Let $k$ be an uncountable algebraically closed field of characteristic $0$, and let $X$ be a smooth projective connected variety of dimension $2p$, appropriately embedded into $mathbb P^m$ over $k$. Let $Y$ be a hyperplane section of $X$, and let $A^p(Y)$ and $A^{p+1}(X)$ be the groups of algebraically trivial algebraic cycles of codimension $p$ and $p+1$ modulo rational equivalence on $Y$ and $X$ respectively. Assume that, whenever $Y$ is smooth, the group $A^p(Y)$ is regularly parametrized by an abelian variety $A$ and coincides with the subgroup of degree $0$ classes in the Chow group $CH^p(Y)$. In the paper we prove that the kernel of the push-forward homomorphism from $A^p(Y)$ to $A^{p+1}(X)$ is the union of a countable collection of shifts of a certain abelian subvariety $A_0$ inside $A$. For a very general section $Y$ either $A_0=0$ or $A_0$ coincides with an abelian subvariety $A_1$ in $A$ whose tangent space is the group of vanishing cycles $H^{2p-1}(Y)_{rm van}$. Then we apply these general results to sections of a smooth cubic fourfold in $mathbb P^5$.
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