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Two varieties $Z$ and $\widetilde Z$ are said to be related by extremal transition if there exists a degeneration from $Z$ to a singular variety $\overline Z$ and a crepant resolution $\widetilde Z \to \overline Z$. In this paper we compare the genus-zero Gromov--Witten theory of toric hypersurfaces related by extremal transitions arising from toric blow-up. We show that the quantum $D$-module of $\widetilde Z$, after analytic continuation and restriction of a parameter, recovers the quantum $D$-module of $Z$. The proof provides a geometric explanation for both the analytic continuation and restriction parameter appearing in the theorem., 53 pages, comments welcome

We construct extremal metrics on the total space of certain destabilising test configurations for strictly semistable K\"ahler manifolds. This produces infinitely many new examples of manifolds admitting extremal K\"ahler metrics. It also shows for such metrics a new phenomenon of jumping of the complex structure along fibres., Comment: 29 pages, comments very welcome!

We study the dimension of loci of special line bundles on stable curves and for a fixed semistable multidegree. In case of total degree $d = g - 1$, we characterize when the effective locus gives a Theta divisor. In case of degree $g - 2$ and $g$, we show that the locus is either empty or has the expected dimension. This leads to a new characterization of semistability in these degrees. In the remaining cases, we show that the special locus has codimension at least $2$. If the multidegree in addition is non-negative on each irreducible component of the curve, we show that the special locus contains an irrreducible component of expected dimension., 23 pages, 4 Figures. v2: The paper has been split in two, and this is a rewritten version of the first part. v3: some clarifications added and minor changes in presentation. Final version, to appear in Math. Ann. 20 pages, 2 Figures

For an abeloid variety A over a complete algebraically closed field extension K of $$\mathbb {Q}_p$$ Q p , we construct a p-adic Corlette–Simpson correspondence, namely an equivalence between finite-dimensional continuous K-linear representations of the Tate module and a certain subcategory of the Higgs bundles on A. To do so, our central object of study is the category of vector bundles for the v-topology on the diamond associated to A. We prove that any pro-finite-étale v-vector bundle can be built from pro-finite-étale v-line bundles and unipotent v-bundles. To describe the latter, we extend the theory of universal vector extensions to the v-topology and use this to generalise a result of Brion by relating unipotent v-bundles on abeloids to representations of vector groups.

We study hyperelliptic curves y^2=f(x) over local fields of odd residue characteristic. We introduce the notion of a "cluster picture" associated to the curve, that describes the p-adic distances between the roots of f(x), and show that this elementary combinatorial object encodes the curve's Galois representation, conductor, whether the curve is semistable, and if so, the special fibre of its minimal regular model, the discriminant of its minimal Weierstrass equation and other invariants., Comment: 93 pages

Funder: Royal Society, Funder: Centre National de la Recherche Scientifique; doi: http://dx.doi.org/10.13039/501100004794, Fujita and Li have given a characterisation of K-stability of a Fano variety in terms of quantities associated to valuations, which has been essential to all recent progress in the area. We introduce a notion of valuative stability for arbitrary polarised varieties, and show that it is equivalent to K-stability with respect to test configurations with integral central fibre. The numerical invariant governing valuative stability is modelled on Fujita’s $$\beta $$ β -invariant, but includes a term involving the derivative of the volume. We give several examples of valuatively stable and unstable varieties, including the toric case. We also discuss the role that the $$\delta $$ δ -invariant plays in the study of valuative stability and K-stability of polarised varieties.

We prove Kawaguchi-Silverman conjecture for all surjective endomorphisms on every smooth rationally connected variety admitting an int-amplified endomorphism., Comment: 25 pages

We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular., Comment: 57 pages, Example 3.28 added, some proofs in Section 4 expanded

In the 1980's Colliot-Th\'{e}l\`{e}ne, Sansuc, Kato and S. Saito proposed conjectures related to local-to-global principles for $0$-cycles on arbitrary smooth projective varieties over a number field. We give some evidence for these conjectures for a product $X=E_1\times E_2$ of two elliptic curves. In the special case when $X=E\times E$ is the self-product of an elliptic curve $E$ over $\mathbb{Q}$ with potential complex multiplication, we show that the places of good ordinary reduction are often involved in a Brauer-Manin obstruction for $0$-cycles over a finite base change. We give many examples when these $0$-cycles can be lifted to global ones., Comment: 26 pages, Appendix by author two

Based on an explicit formula of the generating series for the $n$-point psi-class intersection numbers (cf. Bertola et. al. [4]), we give a novel proof of a conjecture of Delecroix et. al. [9] regarding the large genus uniform leading asymptotics of the psi-class intersection numbers. We also investigate polynomiality phenomenon in the large genera., Comment: Corrected a mistake in Lemma 8; added the proof of the identity (54); added a few more details. 35 pages

We establish the relationship between the cohomology of a certain sheaf on the intersection lattice of a hyperplane arrangement introduced by Yuzvinsky and the cohomology of the coherent sheaf on punctured affine space, respectively projective space associated to the module of logarithmic vector fields along the arrangement. Our main result gives a K\"unneth formula connecting the cohomology theories, answering a question by Yoshinaga. This, in turn, provides a characterization of the projective dimension of the module of logarithmic vector fields and yields a new proof of Yuzvinsky's freeness criterion. Furthermore, our approach affords a new formulation of Terao's freeness conjecture and a more general problem., Comment: 20 pages, v3 several small changes. Some parts of the introduction are rewritten with a new numbering of the main theorems. Also added a final section with concluding remarks

Let $M$ be a super Riemann surface with holomorphic distribution $\mathcal{D}$ and $N$ a symplectic manifold with compatible almost complex structure $J$. We call a map $\Phi\colon M\to N$ a super $J$-holomorphic curve if its differential maps the almost complex structure on $\mathcal{D}$ to $J$. Such a super $J$-holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super $J$-holomorphic curves as a smooth subsupermanifold of the space of maps $M\to N$., Comment: v2: added a section on surjectivity of the Dirac operator and examples; fixed dimension formula and typos

Given a fixed product of non-isogenous abelian varieties at least one of which is general, we show how to construct complete families of indecomposable abelian varieties whose very general fiber is isogenous to the given product and whose connected monodromy group is a product of symplectic groups or is a unitary group. As a consequence, we show how to realize any product of symplectic groups of total rank $g$ as the connected monodromy group of a complete family of $g'$-dimensional abelian varieties for any $g'\ge g$. These methods also yield a construction of a new Kodaira fibration with fiber genus $4$., 22 pages

The moduli space of Higgs bundles can be constructed as a quotient of an infinite-dimensional space and hence admits an orbit type decomposition. In this paper, we show that the orbit type decomposition is a complex Whitney stratification such that each stratum is a complex symplectic submanifold and hence admits a complex Poisson bracket. Moreover, these Poisson brackets glue to a Poisson bracket on the structure sheaf of the moduli space so that the moduli space is a stratified complex symplectic space, 22 pages

We study the evaluation maps given by elements of the Brauer group of varieties over local fields. We show constancy of the aforementioned maps in several interesting cases., corrected minor mistakes, added details to some proofs

We show that a weak version of the canonical bundle formula holds for fibrations of relative dimension one. We provide various applications thereof, for instance, using the recent result of Xu and Zhang, we prove the log non-vanishing conjecture for three-dimensional klt pairs over any algebraically closed field $k$ of characteristic $p>5$. We also show the log abundance conjecture for threefolds over $k$ when the nef dimension is not maximal, and the base point free theorem for threefolds over the algebraic closure of any finite field of characteristic $p>2$., 33 pages; the article has been substantially changed and the results are stronger: we now show log non-vanishing for all three-dimensional klt pairs of characteristic $p>5$, and log abundance when the nef dimension is not maximal. Further, we corrected a mistake in Lemma 4.3(1) (now Lemma 5.1(1)): the effectivity of adjoint divisors on surfaces holds only up to numerical equivalence

We study a pair of Calabi-Yau threefolds X and M, fibered in non-principally polarized Abelian surfaces and their duals, and an equivalence D^b(X) = D^b(M), building on work of Gross, Popescu, Bak, and Schnell. Over the complex numbers, X is simply connected while pi_1(M) = (Z/3)^2. In characteristic 3, we find that X and M have different Hodge numbers, which would be impossible in characteristic 0. In an appendix, we give a streamlined proof of Abuaf's result that the ring H^*(O) is a derived invariant of complex threefolds and fourfolds. A second appendix by Alexander Petrov gives a family of higher-dimensional examples to show that h^{0,3} is not a derived invariant in any positive characteristic., Comment: 32 pages, Macaulay2 and Magma code as ancillary files, with an appendix by Alexander Petrov

Fix a positive integer n and a finite field $${\mathbb {F}}_q$$ F q . We study the joint distribution of the rank $${{\,\mathrm{rk}\,}}(E)$$ rk ( E ) , the n-Selmer group $$\text {Sel}_n(E)$$ Sel n ( E ) , and the n-torsion in the Tate–Shafarevich group "Equation missing" as E varies over elliptic curves of fixed height $$d \ge 2$$ d ≥ 2 over $${\mathbb {F}}_q(t)$$ F q ( t ) . We compute this joint distribution in the large q limit. We also show that the “large q, then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains.

We refine Cossec and Dolgachev's classification of extra-special Enriques surfaces, providing a complete and concise proof., 9 pages

We show that any strongly rigid-noetherian Huber ringAis sheafy. In particular, we positively answer Problem 31 in the Nonarchimedean Scottish Book.

We refine the geometric Satake equivalence due to Ginzburg, Beilinson-Drinfeld, and Mirkovi\'c-Vilonen to an equivalence between mixed Tate motives on the double quotient $L^+ G \backslash LG / L^+ G$ and representations of Deligne's modification of the Langlands dual group $\hat G$., Comment: final published version (v2 contains a new section on sheaves vs. functions, v3 has an overhauled appendix and many more minor edits)

We give necessary and sufficient conditions for a cdh sheaf to satisfy Milnor excision, following ideas of Bhatt and Mathew. Along the way, we show that the cdh infinity-topos of a quasi-compact quasi-separated scheme of finite valuative dimension is hypercomplete, extending a theorem of Voevodsky to nonnoetherian schemes. As an application, we show that if E is a motivic spectrum over a field k which is n-torsion for some n invertible in k, then the cohomology theory on k-schemes defined by E satisfies Milnor excision., Comment: 28 pages. v3: final version, to appear in Mathematische Annalen; v2: new title and minor corrections

We study Severi type inequalities for big line bundles on irregular varieties via cohomological rank functions. We show that these Severi type inequalities on an irregular variety $X$ are related to some natural defined birational invariants of a general fiber $F$ of the Albanese morphism of $X$. As applications, we provide a new lower bound of volumes of irregular threefolds, a sharp lower bound of varieties of maximal Albanese dimension and of general type, and show that the canonical model of such a variety with the minimal volume should be a flat double covers of a principally polarized abelian variety $(A, \Theta)$ branched over $D\in |2\Theta|$., Comment: correct some typos and add some references

Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. For concise forms of minimal border rank, we show that the condition of vanishing Hessian is equivalent to being wild. This is proven by making a detour through structure tensors of smoothable and Gorenstein algebras. The equivalence fails in the non-minimal border rank regime. We exhibit an infinite series of minimal border rank wild forms of every degree $d\geq 3$ as well as an infinite series of wild cubics. Inspired by recent work on border apolarity of Buczy\'nska and Buczy\'nski, we study the border varieties of sums of powers $\underline{\mathrm{VSP}}$ of these forms in the corresponding multigraded Hilbert schemes., Comment: 25 pp., main result upgraded to an equivalence

Using the connection discovered by Hassett between the Noether-Lefschetz moduli space of special cubic fourfolds of discriminant 42 and the moduli space F_{22} of polarized K3 surfaces of genus 22, we show that the universal K3 surface over F_{22} is unirational., 17 pages. Final version, to appear in Mathematische Annalen

In this paper, we prove that the zero-locus of any global holomorphic log-one-form on a projective log-smooth pair $\left(X,D\right)$ of log-general type must be non-empty. Applying this result, we give an answer to the algebraic hyperbolicity part of Shafarevich's conjecture, with the generic fiber being Kawamata-log-terminal (klt) and of log-general type., Comment: 22 pages, Comments welcome

We exhibit the first examples of closed 7-dimensional Riemannian manifolds with holonomy G_2 that are homeomorphic but not diffeomorphic. These are also the first examples of closed Ricci-flat manifolds that are homeomorphic but not diffeomorphic. The examples are generated by applying the twisted connected sum construction to Fano 3-folds of Picard rank 1 and 2. The smooth structures are distinguished by the generalised Eells-Kuiper invariant introduced by the authors in arXiv:1406.2226., Comment: v4: minor corrections; 28 pages; to appear in Math. Ann

A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be rationally connected themselves. We prove that smooth, complex, weighted complete intersections of low enough degree are rationally simply connected. This result has strong arithmetic implications for weighted complete intersections defined over the function field of a smooth, complex curve. Namely, it implies that these varieties satisfy weak approximation at all places, that R-equivalence of rational points is trivial, and that the Chow group of zero cycles of degree zero is zero.

\textit{Harmonic amoebas} are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced in \cite{Kri}, the consideration of such objects suggests to enlarge the scope of tropical geometry. In the present paper, we introduce the notion of harmonic morphisms from tropical curves to affine spaces and show how these morphisms can be systematically described as limits of families of harmonic amoeba maps on Riemann surfaces. It extends previous results about approximation of tropical curves in affine spaces and provides a different point of view on Mikhalkin's approximation Theorem for regular phase-tropical morphisms, as stated e.g. in \cite{Mikh06}. The results presented here follow from the study of imaginary normalised differentials on families of punctured Riemann surfaces and suggest interesting connections with compactifications of moduli spaces., 42 pages, 4 figures, final version

First the Griffiths line bundle of a $\mathbf Q$-VHS $\mathscr V$ is generalized to a Griffiths character ${\rm grif}(\mathbf G, \mu,r)$ associated to any triple $(\mathbf G, \mu, r)$, where $\mathbf G$ is a connected reductive group over an arbitrary field $F$, $\mu \in X_*(\mathbf G)$ is a cocharacter (over $\overline{F}$) and $r:\mathbf G \to GL(V)$ is an $F$-representation; the classical bundle studied by Griffiths is recovered by taking $F=\mathbf Q$, $\mathbf G$ the Mumford-Tate group of $\mathscr V$, $r:\mathbf G \to GL(V)$ the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to ${\rm grif}(\mathbf G, \mu, r)$. The more general setting also gives rise to the Griffiths bundle in the analogous situation in characteristic $p$ given by a scheme mapping to a stack of $\mathbf G$-Zips. When $\mathbf G$ is $F$-simple, we show that, up to positive multiples, the Griffiths character ${\rm grif}(\mathbf G,\mu,r)$ (and thus also the Griffiths line bundle) is essentially independent of $r$ with central kernel, and up to some identifications is given explicitly by $-\mu$. As an application, we show that the Griffiths line bundle of a projective $\mathbf G{\rm -Zip}^{\mu}$-scheme is nef., Comment: Math. Ann., to appear

We study the rational Picard group of the projectivized moduli space of holomorphic n-differentials on complex genus g stable curves. We define (n - 1) natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the universal curve., 26 pages, 1 figure, comments are welcome

We study the complexity of approximating complex zero sets of certain $n$-variate exponential sums. We show that the real part, $R$, of such a zero set can be approximated by the $(n-1)$-dimensional skeleton, $T$, of a polyhedral subdivision of $\mathbb{R}^n$. In particular, we give an explicit upper bound on the Hausdorff distance: $\Delta(R,T) =O\left(t^{3.5}/\delta\right)$, where $t$ and $\delta$ are respectively the number of terms and the minimal spacing of the frequencies of $g$. On the side of computational complexity, we show that even the $n=2$ case of the membership problem for $R$ is undecidable in the Blum-Shub-Smale model over $\mathbb{R}$, whereas membership and distance queries for our polyhedral approximation $T$ can be decided in polynomial-time for any fixed $n$., Comment: 18 pages, 3 figures. This version corrects an erroneous proof of Theorem 1.1, and a small typo in Assertion (3) of Theorem 1.5, in the published version

An algebraic variety is called $$\mathbb {A}^{1}$$ -cylindrical if it contains an $$\mathbb {A}^{1}$$ -cylinder, i.e. a Zariski open subset of the form $$Z\times \mathbb {A}^{1}$$ for some algebraic variety Z. We show that the generic fiber of a family $$f:X\rightarrow S$$ of normal $$\mathbb {A}^{1}$$ -cylindrical varieties becomes $$\mathbb {A}^{1}$$ -cylindrical after a finite extension of the base. This generalizes the main result of Dubouloz and Kishimoto (Nagoya Math J 223:1–20, 2016) which established this property for families of smooth $$\mathbb {A}^{1}$$ -cylindrical affine surfaces. Our second result is a criterion for existence of an $$\mathbb {A}^{1}$$ -cylinder in X which we derive from a careful inspection of a relative Minimal Model Program run from a suitable smooth relative projective model of X over S.

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., Proof of Proposition 2.2 clarified, one reference added. Comments still welcome!

We consider the stack $\mathcal{L}og_X$ parametrizing log schemes over a log scheme $X$, and weak and strong properties of log morphisms via $\mathcal{L}og_X$, as defined by Olsson. We give a concrete combinatorial presentation of $\mathcal{L}og_X$, and prove a simple criterion of when weak and strong properties of log morphisms coincide. We then apply this result to the study of logarithmic regularity, derive its main properties, and give a chart criterion analogous to Kato's chart criterion of logarithmic smoothness., Comment: 19 pages, final version

We show that every polynomially integrable planar outer convex billiard is elliptic., Comment: To appear in Mathematische Annalen. 26 pages. Minor improvement of presentation

A Nikulin configuration is the data of $16$ disjoint smooth rational curves on a K3 surface. According to a well known result of Nikulin, if a K3 surface contains a Nikulin configuration $\mathcal{C}$, then $X$ is a Kummer surface $X=Km(B)$ where $B$ is an Abelian surface determined by $\mathcal{C}$. Let $B$ be a generic Abelian surface having a polarization $M$ with $M^{2}=k(k+1)$ (for $k>0$ an integer) and let $X=Km(B)$ be the associated Kummer surface. To the natural Nikulin configuration $\mathcal{C}$ on $X=Km(B)$, we associate another Nikulin configuration $\mathcal{C}'$; we denote by $B'$ the Abelian surface associated to $\mathcal{C}'$, so that we have also $X=Km(B')$. For $k\geq2$ we prove that $B$ and $B'$ are not isomorphic. We then construct an infinite order automorphism of the Kummer surface $X$ that occurs naturally from our situation. Associated to the two Nikulin configurations $\mathcal{C},$ $\mathcal{C}'$, there exists a natural bi-double cover $S\to X$, which is a surface of general type. We study this surface which is a Lagrangian surface in the sense of Bogomolov-Tschinkel, and for $k=2$ is a Schoen surface., 22 pages, refereed version

We use the theory of twisted stable maps to Deligne-Mumford stacks to construct compactifications of the moduli space of pairs $(X \to C, S + F)$ where $X \to C$ is a fibered surface, $S$ is a sum of sections, $F$ is a sum of marked fibers, and $(X, S + F)$ is a stable pair in the sense of the minimal model program. This generalizes the work of Abramovich-Vistoli, who compactified the moduli space of fibered surfaces without marked fibers. Furthermore, we compare our compactification to Alexeev's space of stable maps and the KSBA compactification of stable pairs. As an application, we describe the boundary of the compactification of the moduli space of elliptic surfaces., Comment: 22 pages. Improved exposition and expanded the introduction

We provide a proof that every Schubert variety of a semi-infinite flag variety is projectively normal. This gives us an interpretation of a Demazure module of a global Weyl module of a current Lie algebra as the (dual) space of the space of global sections of a semi-infinite Schubert variety. Moreover, we give geometric realizations of Feigin-Makedonskyi's generalized Weyl modules, and the $t = \infty$ specialization of non-symmetric Macdonald polynomials., 30pages, v3: corrected Corollary 5.2 and vanishing theorems in section 6, v4: title changed, many small changes, to appear in Math. Ann

In this article we show that the motive of an affine quadric {q = 1} determines the respective quadratic form., Mathematische Annalen

We show that many spin 6-manifolds have the homotopy type but not the homeomorphism type of a Kaehler manifold. Moreover, for given Betti numbers, there are only finitely many deformation types and hence topological types of smooth complex projectve spin threefolds of general type. Finally, on a fixed spin 6-manifold, the Chern numbers take on only finitely many values on all possible Kaehler structures., 24 pages; final version, to appear in Mathematische Annalen

We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy group is generated by Picard-Lefschetz transformations associated to vanishing cycles of singular spectral curves. We construct such vanishing cycles explicitly and use this to show that the $SL(n,\mathbb{C})$ monodromy group is a {\em skew-symmetric vanishing lattice} in the sense of Janssen. Using the classification of vanishing lattices over $\mathbb{Z}$, we completely determine the structure of the monodromy groups of the $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$ Hitchin fibrations. As an application we determine the image of the restriction map from the cohomology of the moduli space of Higgs bundles to the cohomology of a non-singular fibre of the Hitchin fibration., 32 pages

In an algebro-geometric way, we completely determine whether smooth del Pezzo surfaces are K-(semi)stable or not., Comment: 30 pages. This article is an extension of the authors' previous article, arXiv:1607.00518. The full content of arXiv:1607.00518 is included in this article

Sheng and Zuo's characteristic forms are invariants of a variation of Hodge structure. We show that they characterize Gross's canonical variations of Hodge structure of Calabi-Yau type over (Hermitian symmetric) tube domains.

Let W be a subset of the set of real points of a real algebraic variety X. We investigate which functions $$f: W \rightarrow \mathbb {R}$$ are the restrictions of rational functions on X. We introduce two new notions: curve-rational functions (i.e., continuous rational on algebraic curves) and arc-rational functions (i.e., continuous rational on arcs of algebraic curves). We prove that under mild assumptions the following classes of functions coincide: continuous hereditarily rational (introduced recently by the first named author), curve-rational and arc-rational. In particular, if W is semialgebraic and f is arc-rational, then f is continuous and semialgebraic. We also show that an arc-rational function defined on an open set is arc-analytic (i.e., analytic on analytic arcs). Furthermore, we study rational functions on products of varieties. As an application we obtain a characterization of regular functions. Finally, we get analogous results in the framework of complex algebraic varieties.

Let $f(x,y)$ be a complex irreducible formal power series without constant term. One may solve the equation $f(x,y)=0$ by choosing either $x$ or $y$ as independent variable, getting two finite sets of Newton-Puiseux series. In 1967 and 1968, Abhyankar and Zariski published proofs of an \emph{inversion theorem}, expressing the \emph{characteristic exponents} of one set of series in terms of those of the other ones. In fact, a more general theorem, stated by Halphen in 1876 and proved by Stolz in 1879, relates also the \emph{coefficients} of the characteristic terms of both sets of series. This theorem seems to have been completely forgotten. We give two new proofs of it and we generalize it to a theorem concerning equations with an arbitrary number of variables., 27 pages. This is the final published version. The introduction and several proofs were modified according to the recommendations of the referee. The bibliography was augmented, Mathematische Annalen. Online first on 03.12.2016

Let k be a perfect field of characteristic p > 0, and let G be a finite group. We consider the pointed G-curves over k associated by Harbater, Katz, and Gabber to faithful actions of G on k[[t]] over k. We use such "HKG G-curves" to classify the automorphisms of k[[t]] of p-power order that can be expressed by particularly explicit formulas, namely those mapping t to a power series lying in a Z/pZ Artin-Schreier extension of k(t). In addition, we give necessary and sufficient criteria to decide when an HKG G-curve with an action of a larger finite group J is also an HKG J-curve., 25 pages. The introduction was rewritten. The final publication is available at Springer via http://dx.doi.org/10.1007/s00208-016-1490-2

We establish the canonical class inequality for families of higher dimensional projective manifolds. As an application, we get a new inequality between the Chern numbers of 3-folds with smooth families of minimal surfaces of general type over a curve, $c_1^3, 9 pages

Let F(X_d) be a smooth Fano variety of lines of a hypersurface X_d of degree d. In this paper, we prove the Griffiths group Griff_1(F(X_d)) is trivial if the hypersurface X_d is of 2-Fano type. As a result, we give a positive answer to a question of Professor Voisin about the first Griffiths groups of Fano varieties in some cases. Base on this result, we prove that CH_2(X_d)=\mathbb{Z} for a complex smooth $3$-Fano hypersurface X_d whose Fano variety of lines is smooth., Comment: Appear in Mathematische Annalen September 2016, 28 pages

Let $X$ be a connected affine homogenous space of a linear algebraic group $G$ over $\C$. (1) If $X$ is different from a line or a torus we show that the space of all algebraic vector fields on $X$ coincides with the Lie algebra generated by complete algebraic vector fields on $X$. (2) Suppose that $X$ has a $G$-invariant volume form $\omega$. We prove that the space of all divergence-free (with respect to $\omega$) algebraic vector fields on $X$ coincides with the Lie algebra generated by divergence-free complete algebraic vector fields on $X$ (including the cases when $X$ is a line or a torus). The proof of these results requires new criteria for algebraic (volume) density property based on so called module generating pairs., Comment: 21 pages