A Clemens–Schmid type exact sequence over a local basis
DOI:
https://doi.org/10.33044/revuma.2163Abstract
Let $k$ be a finite field of characteristic $p$ and let $X \rightarrow \operatorname{Spec} k[[t]]$ be a semistable family of varieties over $k$. We prove that there exists a Clemens–Schmid type exact sequence for this family. We do this by constructing a larger family defined over a smooth curve and using a Clemens–Schmid exact sequence in characteristic $p$ for this new family.
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P. Berthelot, Dualité de Poincaré et formule de Künneth en cohomologie rigide, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 5, 493–498. MR 1692313.
P. Berthelot and A. Ogus, Notes on Crystalline Cohomology, Princeton University Press, Princeton, NJ, 1978. MR 0491705.
B. Chiarellotto and C. Lazda, Combinatorial degenerations of surfaces and Calabi–Yau threefolds, Algebra Number Theory 10 (2016), no. 10, 2235–2266. MR 3582018.
B. Chiarellotto and N. Tsuzuki, Cohomological descent of rigid cohomology for étale coverings, Rend. Sem. Mat. Univ. Padova 109 (2003), 63–215. MR 1997987.
B. Chiarellotto and N. Tsuzuki, Clemens–Schmid exact sequence in characteristic $p$, Math. Ann. 358 (2014), no. 3-4, 971–1004. MR 3175147.
R. Elkik, Solutions d'équations à coefficients dans un anneau hensélien, Ann. Sci. École Norm. Sup. (4) 6 (1973), 553–603. MR 0345966.
G. Hernandez-Mada, A monodromy criterion for the good reduction of $K3$ surfaces, Rend. Semin. Mat. Univ. Padova 145 (2021), 73–92. MR 4261646.
O. Hyodo and K. Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, Astérisque No. 223 (1994), 221–268. MR 1293974.
T. Ito, Weight-monodromy conjecture over equal characteristic local fields, Amer. J. Math. 127 (2005), no. 3, 647–658. MR 2141647.
K. Kato, Logarithmic structures of Fontaine–Illusie, in Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988), 191–224, Johns Hopkins Univ. Press, Baltimore, MD, 1989. MR 1463703.
D. R. Morrison, The Clemens–Schmid exact sequence and applications, in Topics in Transcendental Algebraic Geometry (Princeton, N.J., 1981/1982), 101–119, Ann. of Math. Stud., 106, Princeton Univ. Press, Princeton, NJ, 1984. MR 0756848.
Y. Nakkajima, Liftings of simple normal crossing log $K3$ and log Enriques surfaces in mixed characteristics, J. Algebraic Geom. 9 (2000), no. 2, 355–393. MR 1735777.
J. R. Pérez-Buendía, A Kulikov-type classification theorem for a one parameter family of K3-surfaces over a $p$-adic field and a good reduction criterion, Ann. Math. Qué. 43 (2019), no. 2, 411–434. MR 3996075.
A. Shiho, Relative Log Convergent Cohomology and Relative Rigid Cohomology I. https://arxiv.org/abs/0707.1742 [math.NT], 2008.
R. G. Swan, Néron–Popescu desingularization, in Algebra and Geometry (Taipei, 1995), 135–192, Lect. Algebra Geom., 2, Int. Press, Cambridge, MA, 1998. MR 1697953.
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