We prove that the relative log de Rham cohomology groups of a projective semistable log smooth degeneration admit a natural \textit{limiting} mixed Hodge structure. More precisely, we construct a family of increasing filtrations and a family of nilpotent endomorphisms on the relative log de Rham cohomology groups and show that they satisfy a part of good properties of a nilpotnet orbit in several variables.
We generalize Illusie's result to prove the decomposition of the de Rham complex with smooth horizontal coefficients for a semistable $S$-morphism $f:X\ra Y$ which is liftable over $\Z/p^2\Z$. As an application, we prove the Kollár vanishing theorem in positive characteristic for a semistable $S$-morphism $f:X\ra Y$ which is liftable over $\Z/p^2\Z$, where all concerned horizontal divisors are smooth over $Y$.
We explicitly construct a $V$-normal crossing Gorenstein canonical model of the relative symmetric products of a local semistable degeneration of surfaces without a triple point by means of toric geometry. Using this model, we calculate the stringy $E$-polynomial of the relative symmetric product. We also construct a minimal model of degeneration of Hilbert schemes explicitly.
We determine the asymptotic behavior of the Arakelov metric, the Arakelov-Green's function, and the Faltings delta-invariant for arbitrary one-parameter families of complex curves with semistable degeneration. The leading terms in the asymptotics are given a combinatorial interpretation in terms of S. Zhang's theory of admissible Green's functions on polarized metrized graphs.
Let $X\rightarrow C$ be a totally real semistable degeneration over a smooth real curve $C$ with degenerate fiber $X_0$. Assuming that the irreducible components of $X_0$ are simple from a cohomological point of view, we give a bound for the individual Betti numbers of a real smooth fiber near $0$ in terms of the complex geometry of the degeneration. This generalizes previous work of Renaudineau-Shaw, obtained via combinatorial techniques, for tropical degenerations of hypersurfaces in smooth toric varieties. The main new ingredient is the use of real logarithmic geometry, which allows to work…
Consider a family of Fano varieties $π: X \longrightarrow B\ni o$ over a curve germ with a smooth total space $X$. Assume that the generic fiber is smooth and the special fiber $F=π^{-1}(o)$ has simple normal crossings. Then $F$ is called a semistable degeneration of Fano varieties. We show that the dual complex of $F$ is a simplex of dimension $\leq \mathrm{dim}\ F$. Simplices of any admissible dimension can be realized for any dimension of the fiber. Using this result and the Minimal Model Program in dimension $3$ we reproduce the classification of the semistable degenerations of del Pezzo s…
We study the $E_1$-degeneration of the logarithmic Hodge to de Rham spectral sequence of the special fiber of a semistable family over a discrete valuation ring. On the one hand, we prove that the $E_1$-degeneration property is invariant under admissible blow-ups. Assuming functorial resolution of singularities over $\mathbb{Z}$, this implies that the $E_1$-degeneration property depends only on the generic fiber. On the other hand, we show by explicit examples that the decomposability of the logarithmic de Rham complex is not invariant under admissible blow-ups, which answer negatively an open…
In this paper we answer two long-standing questions in the classification of $G$-torsors on curves for an almost simple, simply connected algebraic group $G$ over the field of complex numbers. The first question is to give an intrinsic definition of (semi)stability for a $G$-torsor on an {\em irreducible nodal curve} and the second one is the construction of a flat degeneration of the moduli space of semistable $G$-torsors when the smooth curve degenerates to an irreducible nodal curve. A generalization of the classical Bruhat-Tits group schemes to two-dimensional regular local rings and an ap…
We define an overconvergent version of the Hyodo-Kato complex for semistable varieties $Y$ over perfect fields of positive characteristic, and prove that its hypercohomology tensored with $\mathbb{Q}$ recovers the log-rigid cohomology when $Y$ is quasi-projective. We then describe the monodromy operator using the overconvergent Hyodo-Kato complex. Finally, we show that overconvergent Hyodo-Kato cohomology agrees with log-crystalline cohomology in the projective semistable case.
In this article we prove that the log Hodge de Rham spectral sequences of certain proper log smooth schemes of Cartier type in characteristic $p>0$ degenerate at $E_1$. We also prove that the log Kodaira vanishings for them hold when they are projective. We formulate the log weak Lefschetz conjecture for log crystalline cohomologies and prove that it is true in certain cases.
We constructed a projective moduli space of semistable torsion free sheaves with `fixed determinant' on a reducible curve. When a family of smooth curves degenerates to the reducible curve, our moduli space is a degeneration of the moduli spaces of semistable vector bundles with a fixed determinant on the smooth curves. Moreover, the moduli space is reduced and seminormal.
We study the vanishing cycles of a one-parameter smoothing of a complex analytic space and show that the weight filtration on its perverse cohomology sheaf of the highest degree is quite close to the monodromy filtration so that its graded pieces have a modified Lefschetz decomposition. We describe its primitive part using the weight filtration on the perverse cohomology sheaves of the constant sheaves. As a corollary we show in the local complete intersection case that 1 is not an eigenvalue of the monodromy on the reduced Milnor cohomology at any points if and only if the total space and the…
We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a p-adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable etale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morph…
We study the problem of how the dual complex of the special fiber of an snc degeneration $\cX_R$ changes under products. We view the dual complex as a skeleton inside the Berkovich space associated to $X_K$. Using the Kato fan, we define a skeleton $\Sk(\cX_R)$ when the model $\cX_R$ is log-regular. We show that if $\cX_R$ and $\cY_R$ are log-regular, and at least one is semistable, then $\Sk(\cX_R\times_R \cY_R) \simeq \Sk(\cX_R)\times \Sk(\cY_R)$. The essential skeleton $\Sk(X_K)$, defined by Mustaţă and Nicaise, is a birational invariant of $X_K$ and is independent of the choice of $R$-mode…
One may construct a large class of Calabi-Yau varieties by taking anticanonical hypersurfaces in toric varieties obtained from reflexive polytopes. If the intersection of a reflexive polytope with a hyperplane through the origin yields a lower-dimensional reflexive polytope, then the corresponding Calabi-Yau varieties are fibered by lower-dimensional Calabi-Yau varieties. A top generalizes the idea of splitting a reflexive polytope into two pieces. In contrast to the classification of reflexive polytopes, there are infinite families of equivalence classes of tops. Tops may be used to describe …
We study duality theorems for the relative logarithmic de Rham-Witt sheaves on semi-stable schemes $X$ over a local ring $\mathbb{F}_q[[t]]$, where $\mathbb{F}_q$ is a finite field. As an application, we obtain a new filtration on the maximal abelian quotient $π^{\text{ab}}_1(U)$ of the étale fundamental groups $π_1(U)$ of an open subscheme $U \subseteq X$, which gives a measure of ramification along a divisor $D$ with normal crossing and $\text{Supp}(D) \subseteq X-U$. This filtration coincides with the Brylinski-Kato-Matsuda filtration in the relative dimension zero case.
We revisit the non-commutative Hodge-to-de Rham Degeneration Theorem of the first author, and present its proof in a somewhat streamlined and improved form that explicitly uses spectral algebraic geometry. We also try to explain why topology is essential to the proof.
We introduce a version of the Cartier isomorphism for de Rham cohomology valid for associative, not necessarily commutative algebras over a field of positive characteristic. Using this, we imitate the well-known argument of P. Deligne and L. Iluusie and prove, in some cases, a conjecture of M. Kontsevich which claims that the Hodge-to-de Rham, a.k.a. Hochschild-to-cyclic spectral sequence degenerates.
For a semistable degeneration of surfaces without a triple point, we show that two models of degeneration of Hilbert scheme of points of the family, Gulbrandsen-Halle-Hulek degeneration given in [GHH] and the one given by the author in [N], are actually isomorphic.
Let $k $ be the algebraic closure of a finite field of odd characteristic $p$ and $X$ a smooth projective scheme over the Witt ring $W(k)$ which is geometrically connected in characteristic zero. We introduce the notion of Higgs-de Rham flow and prove that the category of periodic Higgs-de Rham flows over $X/W(k)$ is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the étale fundamental group $π_1(X_K)$ of the generic fiber of $X$, after Fontaine-Laffaille and Faltings. Moreover, we prove that every semistable Higgs bund…
In a Type III degeneration of K3-surfaces the dual graph of the central fibre is a triangulation of the 2-sphere. We realise the tetrahedral, octahedral and especially the icosahedral triangulation in families of K3-surfaces, preferably with the associated symmetry groups acting.
We overview our recent work defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.
We establish a positive characteristic analogue of intersection cohomology theory for variations of Hodge structure. It includes: a) the de Rham-Higgs comparison theorem for the intersection de Rham complex; b) the $E_1$-degeneration theorem for the intersection de Rham complex of a periodic de Rham bundle; c) the Kodaira-Saito vanishing theorem for the intersection cohomology groups of a periodic Higgs bundle. These results generalize the decomposition theorem of Deligne-Illusie and the de Rham-Higgs theorem of Ogus-Vologodsky, the $E_1$-degneration theorem of Deligne-Illusie, Illusie, Faltin…
We classify irreducible d-semistable degenerations of primary Kodaira surfaces. As an application we construct a canonical completion for the moduli space of primary Kodaira surfaces.
We provide a construction of examples of semistable degeneration via toric geometry. The applications include a higher dimensional generalization of classical degeneration of K3 surface into 4 rational components, an algebraic geometric version of decomposing K3 as the fiber sum of two E(1)'s as well as it's higher dimensional generalizations, and many other new examples.
We construct relative moduli spaces of semistable pairs on a family of projective Deligne-Mumford stacks. We define moduli stacks of stable orbifold Pandharipande-Thomas pairs on stacks of expanded degenerations and pairs, and then show they are separated and proper Deligne-Mumford stacks of finite type. As an application, we present the degeneration formula for the absolute and relative orbifold Pandharipande-Thomas invariants.
For a simple normal crossing variety $X$, we introduce the concepts of prelog Chow ring, saturated prelog Chow group, as well as their counterparts for numerical equivalence. Thinking of $X$ as the central fibre in a (strictly) semistable degeneration, these objects can intuitively be thought of as consisting of cycle classes on $X$ for which some initial obstruction to arise as specializations of cycle classes on the generic fibre is absent. Cycle classes in the generic fibre specialize to their prelog counterparts in the central fibre, thus extending to Chow rings the method of studying smoo…
We provide a new formalism of de Rham--Witt complexes in the logarithmic setting. This construction generalizes a result of Bhatt--Lurie--Mathew, and agrees with those of Hyodo--Kato and Matsuue for log-smooth schemes of log-Cartier type. We then apply our formalism to obtain a more direct proof of the log crystalline comparison of A_inf-cohomology in the case of semistable reduction, which is established by Cesnavicius--Koshiwara.
We study an obstruction to prescribing the dual complex of a strict semistable degeneration of an algebraic surface. In particular, we show that if $Δ$ is a complex homeomorphic to a 2-dimensional manifold with negative Euler characteristic, then $Δ$ is not the dual complex of any semistable degeneration. In fact, our theorem is somewhat more general and applies to some complexes homotopy equivalent to such a manifold. Our obstruction is provided by the theory of tropical complexes.
In this paper, we consider the weight $i$ de Rham--Gauss--Manin bundles on a smooth variety arising from a smooth projective morphism $f:X\_U\lrar U$ for $i\geq 0$. We associate to each weight $i$ de Rham bundle, a certain parabolic bundle on $S$ and consider their parabolic Chern characters in the rational Chow groups, for a good compactification $S$ of $U$. We show the triviality of the alternating sum of these parabolic bundles in the (positive degree) rational Chow groups. This removes the hypothesis of semistable reduction in the original result of this kind due to Esnault and Viehweg.
In this paper we study some properties of degenerations of surfaces whose general fibre is a smooth projective surface and whose central fibre is a reduced, connected surface $X \subset IP^r$, $r \geq 3$, which is assumed to be a union of smooth projective surfaces, in particular of planes. Our original motivation has been a series of papers of G. Zappa which appeared in the 1940-50's regarding degenerations of scrolls to unions of planes. Here, we present a first set of results on the subject; other aspects are still work in progress and will appear later. We first study the geometry and the …
We use a version of the method of Deligne-Illusie to prove that the Hodge-to-de Rham, a.k.a. Hochschild-to-cyclic spectral sequence degenerates for a large class of associative, not necessariyl commutative DG algebras. This proves, under some assumption, a conjecture by Kontsevich and Soibelman made in math.RA/0606241. The approach is similar to my earlier paper math.AG/0511665, but the proof is more straightforward, and the underlying algebraic topology notions are explicitly described. The paper is independent of math.AG/0511665 and in a sense, supercedes it.
Results due to Druel and Beauville show that the blowup of the intermediate Jacobian of a smooth cubic threefold X in the Fano surface of lines can be identified with a moduli space of semistable sheaves of Chern classes c_1=0, c_2=2, c_3=0 on X. Here we further identify this space with a space of matrix factorizations. This has the advantage that this description naturally generalizes to singular and even reducible cubic threefolds. In this way, given a degeneration of X to a reducible cubic threefold X_0, we obtain an associated degeneration of the above moduli spaces of semistable sheaves.
In a beautiful paper Deligne and Illusie proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. In a recent paper Arinkin, Căldăraru and the author of this paper gave a geometric interpretation of the problem of Deligne-Illusie showing that the triviality of a certain line bundle on a derived scheme implies the the Deligne-Illusie result. In the present paper we generalize these ideas to logarithmic schemes and using the theory of twisted derived intersection of logarithmic schemes we obtain the Hodge theorem for the logarithmic de Rham complex…
We introduce a notion of a Hodge-proper stack and extend the method of Deligne-Illusie to prove the Hodge-to-de Rham degeneration in this setting. In order to reduce the statement in characteristic $0$ to characteristic $p$, we need to find a good integral model of a stack (a so-called spreading), which, unlike in the case of schemes, need not to exist in general. To address this problem we investigate the property of spreadability in more detail by generalizing standard spreading out results for schemes to higher Artin stacks and showing that all proper and some global quotient stacks are Hod…
We consider a semistable degeneration of K3 surfaces, equipped with an effective divisor that defines a polarisation of degree two on a general fibre. We show that the map to the relative log canonical model of the degeneration maps every fibre to either a sextic hypersurface in P(1,1,1,3) or a complete intersection of degree (2,6) in P(1,1,1,2,3). Furthermore, we find an explicit description of the hypersurfaces and complete intersections that can arise, thereby giving a full classification of the possible singular fibres.
We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field $K$ of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction of Kulikov models is closely related to an earlier construction of Künnemann, which produces semistable models of Abelian varieties. It is well-known that the special fibre of a strict Kulikov mo…
We show that a particular subfunctor of the relative logarithmic Picard functor for families of aligned, log semistable curves over a regular base scheme and smooth over an open dense subscheme of the base is representable by a smooth algebraic space which is slightly more separated than the classical relative Picard functor. We show the existence of its maximal separated quotient and give a new functorial interpretation in terms of logarithmic geometry for the Néron model of the relative Picard functor of the smooth locus when we restrict to transversal pull backs from spectra of discrete val…
This paper studies the derived de Rham cohomology of F_p and p-adic schemes, and is inspired by Beilinson's recent work. Generalising work of Illusie, we construct a natural isomorphism between derived de Rham cohomology and crystalline cohomology for lci maps of such schemes, as well logarithmic variants. These comparisons give derived de Rham descriptions of the usual period rings and related maps in p-adic Hodge theory. Placing these ideas in the skeleton of Beilinson's construction leads to a new proof of Fontaine's crystalline conjecture and Fontaine-Jannsen's semistable conjecture.
We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fiber consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fiber is a syzygy bundle and which become trivial when restricted to each projective line in the special fiber. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamen…
We answer negatively an open problem of Illusie on the DR-decomposability of the log de Rham complex of the special fiber of a semi-stable reduction over the Witt ring. We also show that $E_1$ degeneration of the Hodge to log de Rham spectral sequence does not imply DR-decomposability of semi-stable varieties.
In this paper, we first prove a Donaldson-Uhlenbeck-Yau theorem over projective normal varieties smooth in codimension two. As a consequence we deduce the polystability of (dual) tensor products of stable reflexive sheaves, and we give a new proof of the Bogomolov-Gieseker inequality, along with a precise characterization of the case of equality. This also improves several previously known algebro-geometric results on normalized tautological classes. We study the limiting behavior of semistable bundles over a degenerating family of projective normal varieties. In the case of a family of stable…
We introduce a "limiting Frobenius structure" attached to any degeneration of projective varieties over a finite field of characteristic p which satisfies a p-adic lifting assumption. Our limiting Frobenius structure is shown to be effectively computable in an appropriate sense for a degeneration of projective hypersurfaces. We conjecture that the limiting Frobenius structure relates to the rigid cohomology of a semistable limit of the degeneration through an analogue of the Clemens-Schmidt exact sequence. Our construction is illustrated, and conjecture supported, by a selection of explicit ex…
