Vanishing of the higher direct images of the structure sheaf

We prove that the higher direct images of the structure sheaf under a birational and projective morphism between excellent and regular schemes vanish.

ON THE IRREGULAR HODGE FILTRATION OF EXPONENTIALLY TWISTED MIXED HODGE MODULES

Given a mixed Hodge module$\mathcal{N}$and a meromorphic function$f$on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module$\mathcal{N}\otimes \mathcal{E}^{f}$, which extends the construction of Esnaultet al.($E_{1}$-degeneration of the irregular Hodge filtration (with an appendix by Saito),J. reine angew. Math.(2015), doi:10.1515/crelle-2014-0118). We show the strictness of the push-forward filtered${\mathcal{D}}$-module through any projective morphism${\it\pi}:X\rightarrow Y$, by using the theory of mixed twistor${\mathcal{D}}$-modules of Mochizuki. We consider the example of the rescaling of a regular function$f$, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss–Manin systems of$f$in terms of the Harder–Narasimhan filtration of the Kontsevich bundles associated with$f$.

Quotients of an affine variety by an action of a torus

Let X be an affine T-variety. We study two different quotients for the action of T on X: the toric Chow quotient X/C T and the toric Hilbert scheme H. We introduce a notion of the main component H 0 of H, which parameterizes general T-orbit closures in X and their flat limits. The main component U 0 of the universal family U over H is a preimage of H 0. We define an analogue of a universal family WX over the main component of X/C T. We show that the toric Chow morphism restricted on the main components lifts to a birational projective morphism from U 0 to W X. The variety W X also provides a geometric realization of the Altmann-Hausen family. In particular, the notion of W X allows us to provide an explicit description of the fan of the Altmann-Hausen family in the toric case.

Reconstructing schemes from the derived category

We generalize Bondal and Orlov's Reconstruction Theorem for a Gorenstein schemeXand a projective morphismX→Twhose (relative) dualizing sheaf is eitherT-ample orT-antiample.

Dynamical construction of Kähler-Einstein metrics

In this article, we give a new construction of a Kähler-Einstein metric on a smooth projective variety with ample canonical bundle. As a consequence, for a dominant projective morphismf:X→Swith connected fibers such that a general fiber has an ample canonical bundle, and for a positive integerm, we construct a canonical singular Hermitian metrichE,monwith semipositive curvature in the sense of Nakano.

Dynamical construction of Kähler-Einstein metrics

In this article, we give a new construction of a Kähler-Einstein metric on a smooth projective variety with ample canonical bundle. As a consequence, for a dominant projective morphism f: X → S with connected fibers such that a general fiber has an ample canonical bundle, and for a positive integer m, we construct a canonical singular Hermitian metric hE,m on with semipositive curvature in the sense of Nakano.

On relative base point freeness of adjoint bundle

We give an effective result on the relative base point freeness of an adjoint bundle for a pair of a projective morphism and a relatively ample line bundle.