Bert van Geemen
To a Hodge structure V of weight k with CM by a field K we associate Hodge structures V_{-n/2} of weight k+n for n positive and, under certain circumstances, also for n negative. We show that these `half twists' come up naturally in the Kuga-Satake varieties of weight two Hodge structures with CM by an imaginary quadratic field.
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We establish the twisted functoriality in nonabelian Hodge theory in positive characteristic. As an application, we obtain a purely algebraic proof of the fact that the pullback of a semistable Higgs bundle with vanishing Chern classes is again semistable.
We show that certain abelian varieties A have the property that for every Hodge structure V in the cohomology of A, every effective Tate twist of V occurs in the cohomology of some abelian variety. We deduce the general Hodge conjecture for certain non-simple abelian varieties of type IV.
Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index one or two in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.
A Hodge structure V of weight k on which a CM field acts defines, under certain conditions, a Hodge structure of weight k-1, its half twist. In this paper we consider hypersurfaces in projective space with a cyclic automorphism which defines an action of a cyclotomic field on a Hodge substructure in the cohomology. We determine when the half twist exists and relate it to the geometry and moduli of the hypersurfaces. We use our results to prove the existence of a Kuga-Satake correspondance for certain cubic 4-folds.
Let $k$ be a perfect field of odd characteristic and $X$ a smooth algebraic variety over $k$ which is $W_2$-liftable. We show that the exponent twisiting of the classical Cartier descent gives an equivalence of categories between the category of nilpotent Higgs sheaves of exponent $\leq p-1$ over $X/k$ and the category of nilpotent flat sheaves of exponent $\leq p-1$ over $X/k$, and it is equivalent up to sign to the inverse Cartier and Cartier transforms for these nilpotent objects constructed in the nonabelian Hodge theory in positive characteristic by Ogus-Vologodsky. In view of the crucial…
We show that certain one-dimensional hypergeometric differential systems underlie objects of the category of irregular mixed Hodge modules, which was recently introduced by Sabbah, and compute the irregular Hodge filtration for them. We also provide a comparison theorem between two different types of Fourier-Laplace transformation for algebraic integrable twistor $\mathcal{D}$-modules.
Let A be an abelian variety over C such that the semisimple part of the Hodge group of A is a product of copies of SU(p,1) for some p>1. We show that any effective Tate twist of a Hodge structure occurring in the cohomology of A is isomorphic to a Hodge structure in the cohomology of some abelian variety.
We show that any effective Hodge structure of CM-type occurs (without having to take a Tate twist) in the cohomology of some CM abelian variety over C. As a consequence we get a simple proof of the theorem (due to Hazama) that the usual Hodge conjecture for the class of all CM abelian varieties implies the general Hodge conjecture for the same class.
Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds and a relative ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*Ω^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. Using the explicit formula for the curvature tensor of these direct images, we prove that the determinant of the twisted Hodge filtration $F^p_L=\oplus_{i\geq p}R^{n-i}Ω^i_{\mathcal{X}/S}(L)$ is (semi-) positive on the base $S$ if $L$ itself is (semi-) positive on $\mathcal{X}$.
We define and construct the real analytic moduli stack of pluriharmonic bundles on a compact Kaehler manifold X, and show how this is equipped with Hodge and quaternionic structures. This stack maps to the de Rham moduli stack, giving rise to preferred sections of the Deligne-Hitchin twistor stack. We then show how the non-abelian mixed Hodge structures on Malcev homotopy types X can be extended to objects over the pluriharmonic moduli stack.
The purpose of this paper is to introduce the notion of mixed twistor structure, a generalization of the notion of mixed Hodge structure. The utility of this notion is to make possible a theory of weights for various things surrounding arbitrary representations of the fundamental group of a smooth projective variety. We give some examples of generalizations of classical results for variations of mixed Hodge structure, to the twistor setting. This supports a ``meta-theorem'' (which we state but don't prove) that one can everywhere replace the word ``Hodge'' by the word ``twistor''. We show that…
$p$-adic non-abelian Hodge theory for curves via moduli stacks
本文通过模堆的典范扭转，将曲线上的 $p$-adic 非阿贝尔 Hodge 理论几何化，并恢复 Faltings 对应为同胚，同时将表示堆映射到半稳定 Higgs 丛的开子堆。
Let f : X --> Y be a holomorphic map of complex manifolds, which is proper, Kahler, and surjective with connected fibers, and which is smooth over Y-Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X, and consider direct image sheaves F = R^qf_*(K_{X/Y} \otimes E) for q \geq 0. In our previous paper, we discussed the Nakano semi-positivity of F with respect to the so-called Hodge metric, when the map f is smooth. In this paper, we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(F) ``over…
The Hodge numerical invariants of a variation of Hodge structure over a smooth quas--projective variety are a measure of complexity for the global twisting of the limit mixed Hodge structure when it degenerates. These invariants appear in inequalities which they may have correction terms, called Arakelov inequalities. One may investigate the correction term to make them into equalities, also called Arakelov equalities. We investigate numerical Arakelov type (in)equilities for a family of surfaces fibered by curves. Our method uses Arakelov identities in a weight 1 and also in a weight 2 variat…
Let G be a reductive group over an algebraically closed field of positive characteristic. Let C be a smooth projective curve over k. We give a description of the moduli space of flat G-bundles in terms of the moduli space of G-Higgs bundles over the Frobenius twist C' of C. This description can be regarded as the non-abelian Hodge theory for curves in positive characteristic.
We study a generalization of Hodge structures which first appeared in the work of Cecotti and Vafa. It consists of twistors, that is, holomorphic vector bundles on P^1, with additional structure, a flat connection on C^*, a real subbundle and a pairing. We call these objects TERP-structures. We generalize to TERP-structures a correspondence of Cattani, Kaplan and Schmid between nilpotent orbits of Hodge structures and polarized mixed Hodge structures. The proofs use work of Simpson and Mochizuki on variations of twistor structures and a control of the Stokes structures of the poles at zero and…
Tame Parahoric Nonabelian Hodge Correspondence in Positive Characteristic over Algebraic Curves
本文证明了正特征代数曲线上驯顺 $G$-局部系统与对数 $G$-Higgs 丛间的非阿贝尔 Hodge 对应，并引入 parahoric 群概型语言完整刻画了非紧情形。
We show that the Fourier-Laplace transform of a regular holonomic module over the Weyl algebra of one variable, which generically underlies a variation of polarized Hodge structure, underlies itself an integrable variation of polarized twistor structure.
Topology of projective Tate-Shafarevich twists
研究通过扭转操作改变纤维丛的拓扑性质，证明在特定条件下，扭转后的空间与原空间的理查德·贝乌维-博戈莫洛夫-福基斯晶格是Hodge相似的。
For a reductive group $G$ we equip the category of $G_\mathcal{O}$-equivariant polarizable pure Hodge modules on the affine Grassmannian $\mathrm{Gr}_G$ with a structure of neutral Tannakian category. We show that it is equivalent to a twisted tensor product of the category of representations of the Langlands dual group and the category of pure polarizable Hodge structures.
In this paper, we use non-abelian Hodge Theory to study Kodaira type vanishings and its generalizations. In particular, we generalize Saito vanishing using Mixed Twistor D-modules. We also generalize it to a Kawamata-Viehweg type vanishing using Q-divisors, and we also prove a relative version for a projective morphism.
On the invariance of irregular Hodge numbers under crepant birational equivalences
研究朗道 - 金兹堡模型\((U,f)\)中扭曲德拉姆上同调\(\mathrm{H}^k_{\mathrm{dR}}(U,f)\)的非正则霍奇数，证明其在crepant双有理等价下不变，类似巴蒂列夫 - 孔采维奇定理对卡拉比 - 丘流形霍奇数的结论。
We prove that, for every totally real number field E_0, there exists a weight three variation of Hodge structure of Calabi-Yau type defined over the rational numbers with associated endomorphism algebra E_0 such that the unique irreducible factor of Calabi-Yau type of the corresponding real variation of Hodge structure is the canonical real VHS of CY type over the Hermitian symmetric domain II_6, associated to the real group SO^*(12). The main point is a rationality result for the half spin representations of a form of the group SO^*(4m) defined over a number field.
We show that the usual Hodge conjecture implies the general Hodge conjecture for certain abelian varieties of type III, and use this to deduce the general Hodge conjecture for all powers of certain 4-dimensional abelian varieties of type III. We also show the existence of a Hodge structure M such that M occurs in the cohomology of an abelian variety, but the Tate twist M(1) does not occur in the cohomology of any abelian variety, even though it is effective.
We show that if the Stokes matrix of a connection with a pole of order two and no ramification gives rise, when added to its adjoint, to a positive semi-definite Hermitian form, then the associated integrable twistor structure (or TERP structure, or non-commutative Hodge structure) is pure and polarized.
The algebraic Hodge theorem was proved in a beautiful 1987 paper by Deligne and Illusie, using positive characteristic methods. We argue that the central algebraic object of their proof can be understood geometrically as a line bundle on a derived scheme. In this interpretation, the Deligne-Illusie result can be seen as a proof that this line bundle is trivial under certain assumptions. We give a criterion for the triviality of this line bundle in a more general context. The proof uses techniques from derived algebraic geometry, specifically arguments which show the formality of certain derive…
For $p$ a prime number and $q$ a non trivial $p$th root of 1, we present the main steps of the construction of a local $q$-deformation of the "Simpson correspondence in characteristic $p$" found by Ogus and Vologodsky in 2005. The construction is based on the Morita-equivalence between a ring of $q$-twisted differential operators and its center. We also explain the expected relations between this construction and those recently done by Bhatt and Scholze. For the sake of readability, we limit ourselves to the case of dimension 1.
Semistable Non Abelian Hodge theorem in positive characteristic
本文证明对于任意约化群G，曲线上半稳定G-Higgs丛的模空间是半稳定平坦G-联络模空间的扭曲形式，并由此推出Hitchin态射分解定理在两种情形下形式相同。
Let SU_X(n,L) be the moduli space of rank n semistable vector bundles with fixed determinant L on a smooth projective genus g>1 curve X. Let SU_X^s(n,L) denote the open subset parameterizing stable bundles. We show that for small i, the mixed Hodge structure on H^i(SU_X^s(n, L), Q) is independent of the degree of L, and hence pure of weight i. Moreover any simple factors is, up to Tate twisting, isomorphic to a summand of a tensor power of H^1(X,Q). A more precise statement for i = 3, yields a Torelli theorem complementing earlier work of several authors. This is a replacement of our preprint …
We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of Möller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in the torsion of a factor of the Jacobians. This statement can be viewed as a splitting of certain mixed Hodge structures.
We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of non-commutative Hodge structures, investigate existence and variations, and propose explicit construction and classification techniques. We study the main examples of non-commutative Hodge structures coming from a symplectic or a complex geometry possibly twisted by a potential. We study the interactions of the new Hodge theoretic invariants with mirror symmetry and derive non-commutative analogues of some…
Calculating Hodge and weight filtrations on localizations
本文介绍Macaulay2包MixedHodgeModules，用于计算多项式环局部化上的Hodge与权重滤过及相关不变量，并支持计算多种复形。
We show that the limit, by rescaling, of the `new supersymmetric index' attached to the Fourier-Laplace transform of a polarized variation of Hodge structure on a punctured affine line is equal to the spectral polynomial attached to the same object. We also extend the definition by Deligne of a Hodge filtration on the de Rham cohomology of a exponentially twisted polarized variation of complex Hodge structure and prove a E_1 degeneration property for it.
The subject of the present work is the de Rham part of non-commutative Hodge structures on the periodic cyclic homology of differential graded algebras and categories. We discuss explicit formulas for the corresponding connection on the periodic cyclic homology viewed as a bundle over the punctured formal disk. Our main result says that for the category of matrix factorizations of a polynomial the formulas reproduce, up to a certain shift, a well-known connection on the associated twisted de Rham cohomology which plays a central role in the geometric approach to the Hodge theory of isolated si…
Inscription, twistors, and $p$-adic periods
本文提出刻痕$v$-sheaves理论，用于研究$p$-adic周期，构建了Hodge和Hodge-Tate周期映射的刻痕版本，并引入了新的张量 functor 和几何Sen morphism 结构。
Lagrangian correspondences of nonabelian Hodge type and shifted twistor structures
本文建立平坦与Higgs完美复形的导出栈之间的Lagrange对应，并证明光滑射影簇上的解析Deligne-Hitchin-Simpson模栈具有规范平移预扭曲结构。
An isolated hypersurface singularity comes equipped with many different pairings on different spaces, the intersection form and the Seifert form on the Milnor lattice, a polarizing form for a mixed Hodge structure on a dual space, and a flat pairing on the cohomology bundle. This paper describes them and their relations systematically in an abstract setting. We expect applications also in other areas than singularity theory. A good part of the paper is elementary, but not well known: the classification of irreducible Seifert form pairs, the polarizing form on the generalized eigenspace with ei…
On Hodge--Witt cohomology of Drinfeld's upper half space over a finite field
本文研究有限域上 Drinfeld 上半空间的 Hodge–Witt 上同调，通过建立 Witt 向量上的微分算子理论，赋予相关局部上同调群以微分算子代数模结构，并证明其有限生成性。
In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is able to check conjugacy of a given braid to one of E. Artin's generators in any power, and compute its root. Moreover, the braid element which conjugates a given half-twist to one of E. Artin's generators in any power can be restored. The result is applicable to calculations of braid monodromy of branc…
On any Calabi-Yau manifold X one can define a certain sheaf of chiral N=2 superconformal field theories, known as the chiral de Rham complex of X. It depends only on the complex structure of X, and its local structure is described by a simple free field theory. We show that the cohomology of this sheaf can be identified with the infinite-volume limit of the half-twisted sigma-model defined by E. Witten more than a decade ago. We also show that the correlators of the half-twisted model are independent of the Kahler moduli to all orders in worldsheet perturbation theory, and that the relation to…
For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and automorphic lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz numbers of Frobenius-twisted Hecke correspondences acting on the compactly supported étale cohomology. Our proof is an adaptation of the arguments of Langlands and Rapoport \cite{LR87} of deriving the Kottwitz's formula from their conjectural description of the set of mod-$p$ points of Shimura variety (Langlands-Rapoport conjecture), which replaces their Galois gerb theoretic arguments by more geometric …
In this note, we investigate pluri-half-anticanonical systems on the so called LeBrun twistor spaces. We determine its dimension, the base locus, structure of the associated rational map, and also structure of general members, in precise form. In particular, we show that if n>2 and m>1, the base locus of the system |mK^{-1/2}| on nCP^2 consists of two non-singular rational curves, along which any member has singularity, and that if we blow-up these curves, then the strict transform of a general members of |mK^{-1/2}| becomes an irreducible non-singular surface. We also show that if n>3 and m>n…
We calculate the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,F_q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,C)-character variety. The calculation also leads to several conjectures about the cohomology of M_n: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theo…
Since the braid group was discovered by E. Artin, the question of its conjugacy problem has been solved by Garside and Birman, Ko and Lee. However, the solutions given thus far are difficult to compute with a computer, since the number of operations needed is extremely large. Meanwhile, random algorithms used to solve difficult problems such as primality of a number were developed, and the random practical methods have become an important tool. We give a random algorithm, along with a conjecture of how to improve its convergence speed, in order to identify elements in the braid group, which ar…
The group described in this paper appeared while studying fundamental groups of complements of branch curves. It turned out that a certain quotient of the braid group acts on those fundamental groups and studying this action is essential for understanding the structure of the fundamental groups. We describe here the quotient of the Artin braid group by commutators of transversal half-twists and we investigates its group actions. We denote the quotient by B_n~ and refer to the groups which admit an action of B_n~ , as B_n~-groups. We distinguish special elements in B_n~-groups which we call pri…
For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and a lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz number of an arbitrary Frobenius-twisted Hecke correspondence acting on the compactly supported étale cohomology and verify another conjecture of Kottwitz \cite{Kottwitz90} on the stabilization of that formula. The main ingredients of our proof of the formula are a recent work of Kisin \cite{Kisin17} on Langlands-Rapoport conjecture and the theory of Galois gerbs developed by Langlands and Rapoport \cite{LR87}. Es…
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