Kazuya Kato, Takeshi Saito
For an $\ell$-adic sheaf on a variety of arbitrary dimension over a perfect field, we define the Swan class measuring the wild ramification as a 0-cycle class supported on the ramification locus. We prove a Lefschetz trace formula for open varieties and a generalization of the Grothendieck-Ogg-Shararevich formula using the Swan class.
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For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. The aim of this paper is to completely clarify this problem in the case of the generic curve C of given genus. Using degeneration techniques and a few facts about the birational geometry of moduli spaces of stable pointed curves we determine precisely under which conditio…
Let X be a nonsingular projective algebraic variety, and let S be a line bundle on X. Let A = (a_1,..., a_n) be a vector of integers. Consider a map f from a pointed curve (C,x_1,...,x_n) to X satisfying the following condition: the line bundle f*(S) has a meromorphic section with zeroes and poles exactly at the marked points x_i with orders prescribed by the integers a_i. A compactification of the space of maps based upon the above condition is given by the moduli space of stable maps to rubber over X. The main result of the paper is an explicit formula (in tautological classes) for the push-…
We study the ramification divisors of projections of a smooth projective variety onto a linear subspace of the same dimension. We prove that the ramification divisors vary in a maximal dimensional family for a large class of varieties. Going further, we study the map that associates to a linear projection its ramification divisor. We show that this map is dominant for most (but not all!) varieties of minimal degree, using (linked) limit linear series of higher rank. We find the degree of this map in some cases, extending the classical appearance of Catalan numbers in the geometry of rational n…
Brill-Noether loci of pencils with prescribed ramification on moduli of curves and on Severi varieties on $K3$ surfaces
研究了具有指定分支的圆柱面在曲线模空间和$K3$表面塞维里品种上的Brill-Noether定位，并探讨了其在$K3$表面对应关系和循环中的应用。
We define generalizations of classical invariants of wild ramification for coverings on a variety of arbitrary dimension over a local field. For an l-adic sheaf, we define its Swan class as a 0-cycle class supported on the wild ramification locus. We prove a formula of Riemann-Roch type for the Swan conductor of cohomology together with its relative version, assuming that the local field is of mixed characteristic. We also prove the integrality of the Swan class for curves over a local field as a generalization of the Hasse-Arf theorem. We derive a proof of a conjecture of Serre on the Artin c…
Let K and F be complete discrete valuation fields of residue characteristic p>0. Let m be a positive integer no more than their absolute ramification indices. Let s and t be their uniformizers. Let L/K and E/F be finite extensions such that the modulo s^m of the extension O_L/O_K and modulo t^m of O_E/O_F are isomorphic. Let j=<m be a positive rational number. In this paper, we prove that the ramification of L/K is bounded by j if and only if the ramification of E/F is bounded by j. As an application, we prove that the categories of finite separable extensions of K and F whose ramifications ar…
Let X be a projective variety, possibly singular. A generalized Albanese variety of X was constructed by Esnault, Srinivas and Viehweg over algebraically closed base field as a universal regular quotient of the relative Chow group of 0-cycles by Levine-Weibel. In this paper, we obtain a functorial description of the Albanese of Esnault-Srinivas-Viehweg over a perfect base field, using duality theory of 1-motives with unipotent part.
Classically the ramification filtration of the Galois group of a complete discrete valuation field is defined in the case where the residue field is perfect. In this paper, we define without any assumption on the residue field, two ramification filtrations and study some of their properties.
We prove that, defined with respect to versal flags, the product of two relative Bott-Samelson varieties over the flag bundle is a resolution of singularities of a relative Richardson variety. This result generalizes Brion's resolution of singularities of Richardson varieties to the relative setting. It reflects the phenomenon that the local geometry of a relative Richardson variety is completely governed by the two intersecting relative Schubert varieties, studied by Chan-Pflueger. We also prove an analogous theorem in the case of relative Grassmannian Richardson varieties, thereby furnishing…
We define tautological relations for the moduli space of stable maps to a target variety. Using the double ramification cycle formula for target varieties of Janda-Pandharipande-Pixton-Zvonkine, we construct nontrivial tautological relations parallel to Pixton's double ramification cycle relations for the moduli of curves. Examples and applications are discussed.
We prove an A'Campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field $K$. As a first application, we relate the orders of the tame monodromy eigenvalues on the $\ell$-adic cohomology of a $K$-curve to the geometry of a relatively minimal $sncd$-model, and we show that the semi-stable reduction theorem and Saito's criterion for cohomological tameness are immediate consequences of this result. As a second application, we compute the error term in the trace formula for smooth and proper $K$-varieties. We see that the validity of the t…
We compute the characteristic cycle of a rank one sheaf on a smooth surface over a perfect field of positive characteristic. We construct a canonical lifting on the cotangent bundle of Kato's logarithmic characteristic cycle using ramification theory and prove the equality of the characteristic cycle and the canonical lifting. As corollaries, we obtain a computation of the singular support in terms of ramification theory.
Perfect quadratic forms give a toroidal compactification of the moduli space of principally polarized abelian g-folds that is Q-factorial and whose ample classes are characterized, over any base. In characteristic zero it has canonical singularities if g is at least 5, and is the canonical model (in the sense of Mori and Reid) if g is at least 12.
We show that compatible systems of $\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on $\ell$-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.
Let $L/K$ be an extension of complete discrete valuation fields, and assume that the residue field of $K$ is perfect and of positive characteristic. The residue field of $L$ is not assumed to be perfect. In this paper, we prove a formula for the Swan conductor of the image of a character $χ\in H^1(K, \mathbb{Q}/\mathbb{Z})$ in $H^1(L, \mathbb{Q}/\mathbb{Z})$ for $χ$ sufficiently ramified. Further, we define generalizations $ψ_{L/K}^{\mathrm{ab}}$ and $ψ_{L/K}^{\mathrm{AS}}$ of the classical Hasse-Herbrand $ψ$-function and prove a formula for $ψ_{L/K}^{\mathrm{ab}}(t)$ for sufficiently large $t…
Let $k$ be an $F$-finite field containing an infinite perfect field of positive characteristic. Let $(X, Δ)$ be a projective log canonical pair over $k$. In this note we show that, for a semi-ample divisor $D$ on $X$, there exists an effective $\mathbb{Q}$-divisor $Δ' \sim_{\mathbb Q} Δ+D$ such that $(X, Δ')$ is log canonical if there exists a log resolution of $(X, Δ)$.
We prove the equality of two non-logarithmic ramification filtrations defined by Matsuda and Abbes-Saito for the abelianized absolute Galois group of a complete discrete valuation field in positive characteristic. We also compute the refined Swan conductor and the characteristic form of a character of the fundamental group of a smooth separated scheme over a perfect field of positive characteristic by using sheaves of Witt vectors.
We define a motivic conductor for any presheaf with transfers $F$ using the categorical framework developed for the theory of motives with modulus by Kahn-Miyazaki-Saito-Yamazaki. If $F$ is a reciprocity sheaf this conductor yields an increasing and exhaustive filtration on $F(L)$, where $L$ is any henselian discrete valuation field of geometric type over the perfect ground field. We show if $F$ is a smooth group scheme, then the motivic conductor extends the Rosenlicht-Serre conductor; if $F$ assigns to $X$ the group of finite characters on the abelianized étale fundamental group of $X$, then…
Forms of almost homogeneous varieties over perfect fields
本文研究完美域上几乎齐次变种的形，扩展Luna-Vust理论至完美域，并应用于复数上SL(2)-三重形的实形确定。
Algebraization of rigid analytic varieties and formal schemes via perfect complexes
本文将Toën和Vaquié的定理扩展到非阿基米德和形式环境，证明了光滑且良好的刚性解析簇代数化当且仅当其完美复形的范畴也是光滑且良好的。
We prove that abelian varieties of small dimension over discrete valuated, stricty henselian ground fields with perfect residue class field obtain semistable reduction after a tamely ramified extension of the ground field. Using this result we obtain perfectness results for Grothendieck's pairing.
Brauer group of varieties over local fields of finite characteristic
本文在正特征局部域上研究了代数簇的Brauer群，证明了Kato挠 filtration 与评估 filtration 的等价性，并扩展了相关结果到正特征情况。
Bounding ramification with coherent sheaves
本文引入具有由E界限制的层的复形范畴，研究其与有限推进的兼容性。
We study the Chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties (ppav). We describe the subring generated by divisor classes, and compute the class of the partial compactification of the universal zero section, which turns out to lie in this subring. Our formula extends the results for the zero section of the universal uncompactified family. The partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of the moduli space of ppav. Our form…
In this paper, we give a recipe for producing infinitely many non-divisible codimension $2$ cycles on a product of two or more very general Abelian varieties. In the process, we introduce the notion of "field of definition" for cycles in the Chow group modulo (a power of) a prime. We show that for a quite general class of codimension $2$ cycles we call "primitive cycles," the field of definition is a ramified extension of the function field of a modular variety. This ramification allows us to use Nori's isogeny method \cite{N} (modified by Totaro \cite{T}) to produce infinitely many non-divisi…
We prove Bloch's formula for the Chow group of 0-cycles with modulus on smooth projective varieties over finite fields. The proof relies on two new results in global ramification theory.
Let $K$ be a complete discrete valuation field whose residue field is perfect and of positive characteristic, let $X$ be a connected, proper scheme over $\mathcal{O}_K$, and let $U$ be the complement in $X$ of a divisor with simple normal crossings. Assume that the pair $(X,U)$ is strictly semi-stable over $\mathcal{O}_K$ of relative dimension one and $K$ is of equal characteristic. We prove that, for any smooth $\ell$-adic sheaf $\mathscr{G}$ on $U$ of rank one, at most tamely ramified on the generic fiber, if the ramification of $\mathscr{G}$ is bounded by $t+$ for the logarithmic upper rami…
The refined Swan conductor is defined by K.\ Kato \cite{KK2}, and generalized by T.\ Saito \cite{wild}. In this part, we consider some smooth $l$-adic étale sheaves of rank $p$ such that we can be define the $rsw$ following T.\ Saito, on some smooth dense open subscheme $U$ of a smooth separated scheme X of finite type over a perfect fields $κ$ of characteristic $p>0$. We give an explicit expression of $rsw(\mathcal{F})$ in some situation. As a consequence, we show that it is integral.
Let p be a rational prime, k be a perfect field of characteristic p and K be a finite totally ramified extension of the fraction field of the Witt ring of k. Let G be a finite flat commutative group scheme over O_K killed by some p-power. In this paper, we prove a description of ramification subgroups of G via the Breuil-Kisin classification, generalizing the author's previous result on the case where G is killed by p>2. As an application, we also prove that the higher canonical subgroup of a level n truncated Barsotti-Tate group G over O_K coincides with lower ramification subgroups of G if t…
Perfectoidness via Sen Theory and Applications to Shimura Varieties
本文通过发展一般赋值环上的p进Hodge理论，建立Sen判据的几何类比，证明一般Shimura簇在p处无穷层的完备化茎具有完美oid性，并由此验证Calegari–Emerton关于整值完备化上同调高次消失的猜想。
Kato's Ramification filtration via de Rham-Witt complex and applications
通过de Rham-Witt复形的分层，研究Kato分层的描述及其在对偶定理和Lefschetz定理中的应用。
We prove non-commutative reciprocity laws on an algebraic surface defined over a perfect field. These reciprocity laws claim the splittings of some central extensions of globally constructed groups over some subgroups constructed by points or projective curves on a surface. For a two-dimensional local field with a finite last residue field the constructed local central extension is isomorphic to a central extension which comes from the case of tame ramification of the Abelian two-dimensional local Langlands correspondence suggested by M. Kapranov.
We prove that for any proper smooth formal scheme $\frak X$ over $\mathcal O_K$, where $\mathcal O_K$ is the ring of integers in a complete discretely valued nonarchimedean extension $K$ of $\mathbb Q_p$ with perfect residue field $k$ and ramification degree $e$, the $i$-th Breuil-Kisin cohomology group and its Hodge-Tate specialization admit nice decompositions when $ie<p-1$. Thanks to the comparison theorems in the recent works of Bhatt, Morrow and Scholze, we can then get an integral comparison theorem for formal schemes when the cohomological degree $i$ satisfies $ie<p-1$, which generalize…
Perfectoid covers of abelian varieties
本文证明了在剩余特征为 $p$ 的代数闭非阿基米德域上的阿贝尔簇 $A$ 存在作为 $\varprojlim_{[p]}A$ 波浪号极限的 perfectoid 空间，且该证明适用于更广泛的 abeloid 簇类。
The abelian fundamental group with modulus in mixed characteristic
本文在混合特征下定义了带模的交换基本群，并证明了严格半稳定概形上限制到足够大次数超曲面诱导交换基本群同构的Lefschetz型定理。
We classify group schemes in terms of their Cartier modules. We also prove the equivalence of different definitions of the tangent space and the dimension for these group schemes; in particular, the minimal dimension of a formal group law that contains $S$ as a closed subgroup is equal to the minimal number of generators for the affine algebra of $S$. As an application the following reduction criteria for Abelian varieties are proved. Let $K$ be a mixed characteristic local field, let its residue field have characteristic $p$, $L$ be a finite extension of $K$, let $\mathfrak{O}_K\subset\mathfr…
This article is devoted to studying the ramification of Galois torsors and of $\ell$-adic sheaves in characteristic $p>0$ (with $\ell\not=p$). Let $k$ be a perfect field of characteristic $p>0$, $X$ be a smooth, separated and quasi-compact $k$-scheme, $D$ be a simple normal crossing divisor on $X$, $U=X-D$, $Λ$ be a finite local ${\mathbb Z}_\ell$-algebra, $F$ be a locally constant constructible sheaf of $Λ$-modules on $U$. We introduce a boundedness condition on the ramification of $F$ along $D$, and study its main properties, in particular, some specialization properties that lead to the fun…
Given a triple cover p: X --> Y of varieties, we produce a new variety Z and a birational morphism f: Z --> X which is an isomorphism away from the fat-point ramification locus of p. The variety Z has a natural interpretation in terms of the data describing the triple cover, and the morphism f has an elegant geometric description.
The notion of modulus is a striking feature of Rosenlicht-Serre's theory of generalized Jacobian varieties of curves. It was carried over to algebraic cycles on general varieties by Bloch-Esnault, Park, Rülling, Krishna-Levine. Recently, Kerz-Saito introduced a notion of Chow group of $0$-cycles with modulus in connection with geometric class field theory with wild ramification for varieties over finite fields. We study the non-homotopy invariant part of the Chow group of $0$-cycles with modulus and show their torsion and divisibility properties. Modulus is being brought to sheaf theory by Kah…
Let $k$ be a field of characteristic $p>0$ not necessarily perfect. Using Berthelot's theory of arithmetic $\mathcal{D}$-modules, we construct a $p$-adic formalism of Grothendieck's six operations for realizable $k$-schemes of finite type.
We classify the $\mathbb{G}_{a}$-actions on normal affine varieties defined over any field that are horizontal with respect to a torus action of complexity one. This generalizes previous results that were available for perfect ground fields (cf. [Flenner-Zaidenberg2005, Liendo2010, L-Liendo2016]).
Perfectoidness of Proper Shimura Varieties With Limpid Integral Models
本文证明任何具有清澈整数模型的适当 Shimura 簇在无限层级上成为完美体空间，提出绕过通用阿贝尔概形的新几何方法，并推广至所有适当 Shimura 簇。
A perfect PAC field containing an algebraically closed field is known to be $C_1$, i.e., every degeneration of a Fano complete intersection has a point. We prove that also every degeneration of a separably rationally connected variety has a point.
Characterizing perfectoid covers of abelian varieties
本文借助 Hodge–Tate 滤过刻画 Abel 簇的 perfectoid 有限进 étale 覆盖，计算几何 Sen 态射并证明相关 $p$-进 Lie 挠子猜想。
We classify almost homogeneous normal varieties of Albanese codimension $1$, defined over an arbitrary field. We prove that such a variety has a unique normal equivariant completion. Over a perfect field, the group scheme of automorphisms of this completion is smooth, except in one case in characteristic $2$, and we determine its (reduced) neutral component.
Let X be a smooth proper variety over a perfect field k of arbitrary characteristic. Let D be an effective divisor on X with multiplicity. We introduce an Albanese variety Alb(X, D) of X of modulus D as a higher dimensional analogon of the generalized Jacobian of Rosenlicht-Serre with modulus for smooth proper curves. Basing on duality of 1-motives with unipotent part (which are introduced here), we obtain explicit and functorial descriptions of these generalized Albanese varieties and their dual functors. We define a relative Chow group of zero cycles w.r.t. the modulus D and show that Alb(X,…
Let $\V$ be a mixed characteristic complete discrete valuation ring with perfect residue field $k$. We solve Berthelot's conjectures on the stability of the holonomicity over smooth projective formal $\V$-schemes. Then we build a category of complexes of arithmetic $\D$-modules over quasi-projective $k$-varieties with bounded, $F$-holonomic cohomology. We get its stability under Grothendieck's six operations.
We prove that smooth, projective, $K$-trivial, weakly ordinary varieties over a perfect field of characteristic $p>0$ are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our work, together with Langer's results, implies that varieties of the above type have strongly semistable tangent bundles with respect to any polarization.
Factorization centers in dimension two and the Grothendieck ring of varieties
本文研究双有理映射的分解中心，证明完美域上光滑射影曲面的双有理自同构的弱分解中胀开与胀缩中心同构，由此推出无法用曲面双有理自同构构造0维簇的非平凡L-等价，且有理曲面的有理性中心是良定义的。
We present a notion of $Δ$-stability and stability filtration in arbitrary categories which is equivalent to the existence of Harder-Narasimhan (HN) sequences on objects. Indeed it is equivalent to the existence of a zero morphism, a partial order on objects, and a collection of some universal sequences. In additive categories embedded in an ambient triangulated category, we could obtain a numerical polynomial or central charge of objects by calculating the Euler characteristic of slope sequences and objects, inducing a partial order and HN sequences. In the case of normal projective surfaces …
The cohomological dimension of a field is the largest degree with non-vanishing Galois cohomology. Serre's "Conjecture II" predicts that for every perfect field of cohomological dimension $2$, every torsor over the field for a semisimple, simply connected algebraic group is trivial. A field is perfect and "pseudo algebraically closed" (PAC) if every geometrically irreducible curve over the field has a rational point. These have cohomological dimension $1$. Every transcendence degree $1$ extension of such a field has cohomological degree $2$. We prove Serre's "Conjecture II" for such fields of …
Embeddings of Certain Exceptional Shimura Varieties into Siegel Modular Varieties
本文定义了一类包含某些例外群局部Shimura变种的局部Shimura变种，并为该类构建了从(G, μ)-显示到p-可除群的函子。作为应用，证明了对于该类，局部Shimura变种在无穷级可代表且完美oid。考虑该类的全局对应物，将某些例外Shimura变种嵌入到Siegel模变种中。特别是，证明它们在无穷级是完美oid。
Let $G$ be a semi-simple algebraic group over a perfect field $k$. A lot of progress has been made recently in computing the Chow motives of projective $G$-homogenous varieties. When $k$ has positive characteristic, a broader class of $G$-homogeneous varieties appear. These are varieties over which $G$ acts transitively with possibly non-reduced isotropy subgroup. In this paper we study these varieties which we call ${\it \mbox{projective pseudo-homogeneous varieties}}$ for $G$ inner type over $k$ and prove that their motives satisfy Rost nilpotence. We also find their motivic decompositions a…
We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projective variety, obtained by pinching a smooth variety along a finite subscheme. In contrast, no Witt group of dimension at least 3 over a perfect field of prime characteristic is isogenous to the Picard variety obtained by this construction.
Existence of global Néron models beyond semi-abelian varieties
本文证明了完美剩余域情形下非半阿贝尔代数群的Bosch-Lütkebohmert-Raynaud猜想，给出了非完美剩余域的反例，并分类了完美剩余域上的单有理有界幂幺群。
Perfectoid unitary Shimura varieties and $p$-adic Eichler-Shimura map I
本文通过完美体构造和收敛Eichler-Shimura映射，研究单位ary群上的$p$-adic自动机形与特征种几何及$p$-adic $L$-函数的关系。
We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface A of Picard number one we find the following: The Kummer variety K^n(A) is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if n is a perfect square. And this is the case if and only if K^n(A) carries a divisor with vanishing Beauville-Bogomolov square.
Rigidity of rationally connected smooth projective varieties from dynamical viewpoints
本文从动力系统视角研究了有理连通光滑射影簇的刚性，证明了其成为环面簇或$(\mathbb{P}^1)^{\times n}$的充要条件分别取决于其自同态的分支除子不变性与特征值性质。
Over a field of positive characteristic, a semisimple algebraic group $G$ may have some nonreduced parabolic subgroup $P$. In this paper, we study the Schubert and Bott-Samelson-Demazure-Hansen (BSDH) varieties of $G/P$, with $P$ nonreduced, when the base field is perfect. It is shown that in general the Schubert and BSDH varieties of such a $G/P$ are not normal, and the projection of the BSDH variety onto the Schubert variety has nonreduced fibers at closed points. When the base field is finite, the generalized convolution morphisms between BSDH varieties are also studied. It is shown that th…
Let $A_K$ be an abelian variety with semistable reduction over a strictly henselian field of positive characteristic with perfect residue class field. We show that there is a close connection between the pairings of Grothendieck, Bester/Bertapelle and Shafarevic. In particular, we show that the pairing of Bester/Bertapelle can be used to describe the $p$-part of Grothendieck's pairing in the semistable reduction case, thus proving a conjecture of Bertapelle.
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