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In 2010, Brasselet, Schürmann, and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky–MacPherson |$L$| -class |$L_{*}(X)$| and the Hirzebruch homology class |$T_{1,*}(X)$| for a compact complex algebraic variety |$X$| that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions that we find of independent interest. [ABSTRACT FROM AUTHOR]

Let |${\mathcal{M}}_{\operatorname{Dol}}(X,G)$| denote the hyperkähler moduli space of |$G$| -Higgs bundles over a smooth projective curve |$X$|⁠. In the context of four dimensional supersymmetric Yang–Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of |${\mathcal{M}}_{\operatorname{Dol}}(X,G)$|⁠. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne–Hitchin twistor space |$\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$|⁠. Following Gaiotto's suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack , a derived analytic stack with corresponding moduli space |$\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$|⁠ , defined as a gluing between two analytic Hodge stacks along the Riemann–Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence. [ABSTRACT FROM AUTHOR]

We construct a Leray model for a discrete polymatroid with arbitrary building set and we prove a generalized Goresky–MacPherson formula. The first row of the model is the Chow ring of the polymatroid; we prove Poincaré duality, Hard Lefschetz, and Hodge–Riemann theorems for the Chow ring. Furthermore, we provide a relative Lefschetz decomposition with respect to the deletion of an element. [ABSTRACT FROM AUTHOR]

Let |$E$| be a holomorphic vector bundle over a compact Kähler manifold |$(X,\omega)$| with negative sectional curvature |$sec\leq -K<0$| and |$D_{E}$| be the Chern connection on |$E$|⁠. In this article, we show that if |$C:=|[\Lambda ,i\Theta (E)]|\leq c_{n}K$|⁠ , then |$(X,E)$| satisfy a family of Chern number inequalities. The main idea in our proof is to study the |$L^{2}$| |$\bar {\partial }_{\tilde {E}}$| -harmonic forms on lifting bundle |$\tilde {E}$| over the universal covering space |$\tilde {X}$|⁠. We also observe that there is a close relationship between the eigenvalue of the Laplace–Beltrami operator |$\Delta _{\bar {\partial }_{\tilde {E}}}$| and the Euler characteristic of |$X$|⁠. Precisely, if there is a line bundle |$L$| on |$X$| such that |$\chi ^{p}(X,L^{\otimes m})$| is not constant for some integers |$p\in [0,n]$|⁠ , then the Euler characteristic of |$X$| satisfies |$(-1)^{n}\chi (X)\geq (n+1)+\lfloor \frac {c_{n}K}{2nC} \rfloor $|⁠. [ABSTRACT FROM AUTHOR]

We provide a new Hodge theoretical characterization of the set of complex numbers that arises from the complete list, due to A. Beauville, of semistable families of elliptic curves over |${\mathbb {P}}^1$| with four singular fibers. The characterization is approached via a detailed analysis of the periodicity of the uniformizing Higgs bundle attached to |${\mathbb {P}}^1$| minus four points over the field of complex numbers. [ABSTRACT FROM AUTHOR]

We study the nonnegativity of stringy Hodge numbers of a projective variety with Gorenstein canonical singularities, which was conjectured by Batyrev. We prove that the |$(p,1)$| -stringy Hodge numbers are nonnegative, and for three-folds we obtain new results about the stringy Hodge diamond, which hold even when the stringy |$E$| -function is not a polynomial. We also use the Decomposition Theorem and mixed Hodge theory to prove Batyrev's conjecture for a class of four-folds. [ABSTRACT FROM AUTHOR]

We prove results concerning the behavior of Hodge ideals under restriction to hypersurfaces or fibers of morphisms, and addition. The main tool is the description of restriction functors for mixed Hodge modules by means of the V-filtration. [ABSTRACT FROM AUTHOR]

In this article, the author calculates the weights of the tautological classes of character varieties using the functorial mixed Hodge structure. Topics discussed include cohomology of linear algebraic group; property of Hodge structures induced by algebraic maps; and existence of internal hom in the category of simplicial schemes.

In this paper, we prove that, over an algebraically closed field of odd characteristic, a weakly tame automorphism of a K3 surface of finite height can be lifted over the ring of Witt vectors of the base field. Also we prove that a non-symplectic tame automorphism of a supersingular K3 surface or a symplectic tame automorphism of a supersingular K3 surface of Artin-invariant at least 2 can be lifted to the ring of Witt vectors. Using these results, we prove, for a weakly tame K3 surface of finite height, there is a lifting over the ring of Witt vectors to which the entire automorphism group extends. Also we prove a K3 surface equipped with a purely non-symplectic automorphism of a certain high order is unique up to isomorphism. The following is the outline of the paper. In Section 1, we review some background of the problem and state the main results. In Section 2, we recall the deformation theory of K3 surfaces defined over a field of positive characteristic and prove that formal liftings of a K3 surface over the Witt vectors are classified by the Hodge structure on the second crystalline cohomology. In Section 3, we prove the main results based on the classification of liftings and some properties of the crystalline cohomology of K3 surfaces. In Section 4, we prove, using the main result, the uniqueness of a tame purely non-symplectic automorphism of some high order of a K3 surface over a field of positive characteristic. [ABSTRACT FROM AUTHOR]

A nondegenerate CM Hodge structure is irreducible. For Weight 1, Ribet's Nondegeneracy Theorem informs us that the converse is true for Hodge numbers (p, p), p being an odd prime. We extend this to all odd weight CM Hodge structures. [ABSTRACT FROM AUTHOR]

We establish that the extended Robba rings associated to a perfect nonarchimedean field of characteristic p, which arise in p-adic Hodge theory as certain completed localizations of the ring of Witt vectors, are strongly noetherian Banach rings; that is, the completed polynomial ring in any number of variables over such a Banach ring is noetherian. This enables Huber's theory of adic spaces to be applied to such rings. We also establish that rational localizations of these rings are principal ideal domains and that étale covers of these rings (in the sense of Huber) are Dedekind domains. [ABSTRACT FROM AUTHOR]

In this paper, we study birational immersions from a very general smooth plane curve to a nonrational surface with pg =q = 0 to treat dominant rational maps from a very general surface X of degree ≥5 in ℙ3 to smooth projective surfaces Y. Based on the classification theory of algebraic surfaces, Hodge theory, and deformation theory, we prove that there is no dominant rational map from X to Y unless Y is rational or Y is birational to X. [ABSTRACT FROM AUTHOR]

In this paper, we study birational immersions from a very general smooth plane curve to a nonrational surface with pg =q = 0 to treat dominant rational maps from a very general surface X of degree ≥5 in ℙ3 to smooth projective surfaces Y. Based on the classification theory of algebraic surfaces, Hodge theory, and deformation theory, we prove that there is no dominant rational map from X to Y unless Y is rational or Y is birational to X. [ABSTRACT FROM AUTHOR]

We study rank 1 flat bundles over solvmanifolds whose cohomologies are nontrivial. By using Hodge theoretical properties for all topologically trivial rank 1 flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson–Gordon's result for nilmanifolds. [ABSTRACT FROM AUTHOR]

Fix K/Qp a finite extension and let L/K be an infinite, strictly arithmetically profinite extension in the sense of Fontaine-Wintenberger [24]. Let XK(L) denote its associated norm field. The goal of this paper is to associate to L/K, in a canonical and functorial way, a p-adically complete subring A+L/K?Ã+ whose reduction modulo p is contained in the valuation ring of XK(L). When the extension L/K is of a special form, which we call a φ-iterate extension, we prove that XK(L) is (at worst) a finite purely inseparable extension of Frac(A+L/K/pA+L/K). The class of φ-iterate extensions includes all Lubin-Tate extensions, as well as many other extensions such as the non-Galois "Kummer" extension occurring in the work of Faltings, Breuil, and Kisin. In particular, our work provides a canonical and functorial construction of every characteristic zero lift of the norm fields that have thus far played a foundational role in (integral) p-adic Hodge theory, as well as many other cases which have yet to be studied. [ABSTRACT FROM AUTHOR]

This paper determines all the possible endomorphism algebras for polarizable Q-Hodge structures with Hodge numbers (n, 0, . . . , 0,n). This generalizes the classification of the possible endomorphism algebras of abelian varieties by Albert and Shimura. As with abelian varieties, the most interesting feature of the classification is that in certain cases, every Hodge structure on which a given algebra acts must have extra endomorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we complete the determination of the index of factoriality of moduli spaces of semistable sheaves on an abelian or projective K3 surface S. If v is a Mukai vector and H a generic polarization, let Mv(S,H) be the moduli space of H-semistable sheaves on S with Mukai vector v. First, we describe in terms of v the pure weight-2 Hodge structure and the Beauville form on the second integral cohomology of the symplectic resolutions of Mv(S,H) (when S is K3) and of the fiber Kv(S,H) of the Albanese map of Mv(S,H) (when S is abelian). Then, if S is K3, we show that Mv(S,H) is either locally factorial or 2-factorial, and we give an example of both cases. If S is abelian, we show that Mv(S,H) and Kv(S,H) are 2-factorial. [ABSTRACT FROM PUBLISHER]

Based on the work of Lazarsfeld and Popa, we use the derivative complex associated to the bundle of holomorphic p-forms to provide inequalities for all the Hodge numbers of a special class of irregular compact Kähler manifolds. For three-folds and four-folds, we give an asymptotic bound for all the Hodge numbers in terms of the irregularity. As a byproduct, via the Bernstein–Gel’fand–Gel’fand correspondence, we also bound the regularity of the exterior cohomology modules of bundles of holomorphic p-forms. [ABSTRACT FROM AUTHOR]

We use Hodge theory to define a filtration on the Chow groups of a smooth, projective algebraic variety. Assuming the generalized Hodge conjecture and a conjecture of Bloch-Beilinson, we show that this filtration terminates at the codimension of the algebraic cycle class, thus providing a complete set of period-type invariants for a rational equivalence class of algebraic cycles. [ABSTRACT FROM AUTHOR]

We show that rational Hodge classes on a compact Kähler variety X are not in general generated by the Chern classes of analytic coherent sheaves on X. We also show in an appendix that analytic coherent sheaves on a compact Kähler variety X do not in general admit a finite resolution by locally free sheaves. [ABSTRACT FROM AUTHOR]

In general, if M is a moduli space of stable sheaves on X, there is a unique element α in the Brauer group of M such that a πM∗ α−1-twisted universal sheaf exists on X × M. In this paper, we study the situation when X and M are K3 surfaces, and we identify α in terms of Mukai's map between the cohomology of X and of M (defined by means of a quasi-universal sheaf). We prove that the derived category of sheaves on X and the derived category of α-twisted sheaves on M are equivalent. This suggests a conjecture which describes, in terms of Hodge isometries of lattices, when derived categories of twisted sheaves on two K3 surfaces are equivalent. If proven true, this conjecture would generalize a theorem of Orlov and a recent result of Donagi-Pantev. [ABSTRACT FROM AUTHOR]

We consider integrable KdV type hierarchy associated naturally with arbitrary semi-simple Frobenius manifolds. Equations defining hierarchy have Lax form and admit bihamiltonian description. This hierarchy is responsible for extension of corresponding semi-infinite variation of Hodge structures by generating variations along higher times {tα,r} satisfying (∂ / ∂ tα,r)L(t)⊆ħ−rL(t). [ABSTRACT FROM AUTHOR]

We use Hodge theory and a construction of Merkulov to construct A∞-structures on de Rham cohomology and Dolbeault cohomology. [ABSTRACT FROM AUTHOR]

The article presents a mathematical research which describes the presence of a strong homotopy algebra in the classical Hodge theory of Kähler manifolds. It provides a description of the said algebra and considers certain algebraic data that could trigger its emergence. It highlights theorems and equations to show the strong homotopy algebras of a Kähler manifold.

Batyrev has defined the stringy E-function for complex varieties with at most log terminal singularities. It is a rational function in two variables if the singularities are Gorenstein. Furthermore, if the variety is projective and its stringy E-function is a polynomial, Batyrev defined its stringy Hodge numbers essentially as the coefficients of this E-function, generalizing the usual notion of Hodge numbers of a nonsingular projective variety. He conjectured that they are nonnegative. We prove this for a class of ‘mild’ isolated singularities (the allowed singularities depend on the dimension). As a corollary we obtain a proof of Batyrev's conjecture for threefolds in full generality. In these cases, we also give an explicit description of the stringy Hodge numbers, and we suggest a possible generalized definition of stringy Hodge numbers if the E-function is not a polynomial. [ABSTRACT FROM PUBLISHER]