
Doc. Math., 2013, Volume 18, Pages 547–619
(Mi docma2)




Projective varieties with bad semistable reduction at 3 only
V. Abrashkin^{ab} ^{a} Steklov Mathematical Institute, Gubkina str. 8, 119991 Moscow,
Russia
^{b} Department of Mathematical Sciences, Durham University, Science Laboratories, South Rd, Durham DH1 3LE, United Kingdom
Abstract:
Suppose $F = W(k)[1/p]$ where $W(k)$ is the ring of Witt vectors with coefficients in algebraically closed field $k$ of characteristic $p\ne2$. We construct integral theory of $p$adic semistable representations of the absolute Galois group of $F$ with Hodge–Tate weights from $[0, p)$. This modification of Breuil’s theory results in the following application in the spirit of the Shafarevich Conjecture. If $Y$ is a projective algebraic variety over $\mathbb{Q}$ with good reduction modulo all primes $l\ne3$ and semistable reduction modulo $3$ then for the Hodge numbers of $Y_{\mathbb{C}}=Y\otimes_{\mathbb{Q}}\mathbb{C}$, one has $h^2(Y_{\mathbb{C}})=h^{1,1}(Y_{\mathbb{C}})$.
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MSC: 11S20, 11G35, 14K15 Received: 30.04.2013
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