相关论文: $L^2$-Extension of Adjoint bundles and Koll\'ar's …

We generalize Koll\'ar's conjecture (including torsion freeness, injectivity theorem, vanishing theorem and decomposition theorem) to Saito's $S$-sheaves twisted by a $\mathbb{Q}$-divisor. This gives a uniform treatment for various kinds of…

In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We…

We give a formalism of arithmetic mixed sheaves including the case of arithmetic mixed Hodge structures, and show the nonvanishing of certain higher extension groups, and also the nontriviality of the second Abel-Jacobi map for zero cycles…

These are the lecture notes based on earlier papers with some additional new results. New and simple proofs are given for local freeness theorem and the semipositivity theorem. A decomposition theorem for higher direct images of dualizing…

We generalize Koll\'ar's package (including torsion freeness, injectivity theorem, vanishing theorem and decomposition theorem) to polystable locally abelian parabolic Higgs bundles twisted by a multiplier ideal sheaf associated with an…

We prove an extension of the nonabelian Hodge theorem in which the underlying objects are twisted torsors over a smooth complex projective variety. In the prototypical case of $GL_n$-torsors, one side of this correspondence consists of…

Let X be a proper smooth variety over the complex numbers. We consider the generalized Albanese variety Alb(X,Y) of X of modulus Y, which is a higher dimensional analogue of the generalized Jacobian variety with modulus of Rosenlicht-Serre.…

In this paper, we introduce a coherent subsheaf of Saito's $S$-sheaf, which is a combination of the $S$-sheaf and the multiplier ideal sheaf. We construct its $L^2$-Dolbeault resolution, which generalizes MacPherson's conjecture on the…

The goal is to verify the Hodge conjecture (and some related conjectures) for certain moduli spaces. It is shown that the (generalized) Hodge conjecture holds for the projective moduli spaces of vector bundles over an abelian or K3 surface…

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 M\"oller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in…

We formulate and establish a generalization of Koll\'ar's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Koll\'ar's torsion-freeness, Koll\'ar's vanishing theorem, and a…

We express the kernel of Griffiths' Abel-Jacobi map by using the inductive limit of Deligne cohomology in the generalized sense (i.e. the absolute Hodge cohomology of A. Beilinson). This generalizes a result of L. Barbieri-Viale and V.…

In this paper, we define a certain Hodge-theoretic structure for an arbitrary variety X over the complex number field by using the theory of mixed Hodge module due to Morihiko Saito. We call it an arithmetic Hodge structure of X. It is…

Let $f:X\to Y$ be a morphism of complex manifolds. Suppose that $X$ is a K\"ahler manifold. Let $(\mathcal{T},\mathcal{S})$ be a regular polarized pure twistor $\mathcal{D}$-module of weight $w$ on $X$ whose support is proper over $Y$. We…

In this paper we explain how non-abelian Hodge theory allows one to compute the $L^2$ cohomology or middle perversity higher direct images of harmonic bundles and twistor D-modules in a purely algebraic manner. Our main result is a new…

We give a new proof of the Hodge conjecture for abelian fourfolds of Weil type with discriminant 1 and all of their powers. The Hodge conjecture for these abelian fourfolds was proven by Markman using hyperholomorphic sheaves on…

We study the push-forward of Hopf--Galois extensions as the algebraic counterpart of the pullback of principal bundles. We apply the theory of twisted tensor product algebras to endow covariant extensions of modules along a map $\mathsf{F}$…

We show that a purely algebraic structure, a two-dimensional scattering diagram, describes a large part of the wall-crossing behavior of moduli spaces of Bridgeland semistable objects in the derived category of coherent sheaves on…

Let X be a locally symmetric space associated to a reductive algebraic group G defined over Q. L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of X; they were introduced in…

We survey recent advances in non-abelian Hodge theory in the "mixed" setting of non-proper algebraic varieties. We then describe how these tools are used to construct algebraic Shafarevich morphisms and prove a version of the linear…