Related papers: Symmetry and Variation of Hodge Structures

Let $X$ be a projective toric variety of dimension $n$ and let $L$ be a ample line bundle on $X$. For $k \geq 0$, it is in general difficult to determine whether $L^{\otimes k}$ is very ample and whether it additionally gives a projectively…

We use techniques of Alper-Hall-Rydh to prove a local structure theorem for smooth morphisms between smooth stacks around points with linearly reductive stabilizers. This implies that the good moduli space of a smooth stack over a base has…

We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional…

Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. Our formulas have many…

Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed…

Let $\Cal E$ be a very ample vector bundle of rank two on a smooth complex projective threefold $X$. An inequality about the third Segre class of $\Cal E$ is provided when $K_X+\det \Cal E$ is nef but not big, and when a suitable positive…

As a formulation of 'codimension-two arguments' in invariant theory, we define a (rational) almost principal bundle. It is a principal bundle off closed subsets of codimension two or more. We discuss the behavior of the category of…

In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each $n \ge 3$ we construct $4$-manifolds $E$ admitting at least $n$ distinct fiberings $p_i: E \to \Sigma_{g_i}$ as a…

We give a proof of generalizations of the classical Arakelov inequality valid for the degree $d$ of the relative canoincal bundle of a family of curves of genus $g$ over a complete curve of genus $p$ under the assumption that the monodromy…

In this paper we study the $\mathbb{C}^*$-fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to…

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with…

Let $G$ be a complex connected reductive algebraic group that acts on a smooth complex algebraic variety $X$, and let $E$ be a $G$-equivariant algebraic vector bundle over $X$. A section of $E$ is regular if it is transversal to the zero…

A good canonical projection of a surface $S$ of general type is a morphism to the 3-dimensional projective space P^3 given by 4 sections of the canonical line bundle. To such a projection one associates the direct image sheaf F of the…

We bound from below the complexity of the top Chern class of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section…

Let C be a smooth projective curve over the field of the complex numbers. We consider Brill-Noether loci over the moduli of maps from C to the Grassmannian G(m,n) and the corresponding Quot schemes of quotients of a trivial vector bundle on…

Fujino gave a proof in [Fuj03] for the semi-ampleness of the moduli part in the canonical bundle formula in the case when the general fibers are K3 surfaces or Abelian varieties. We show a similar statement when the general fibers are…

Let $C$ be a curve of genus two. We denote by $SU_C(3)$ the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over $C$, and by $J^d$ the variety of line bundles of degree $d$ on $C$. In particular, $J^1$ has a…

We describe the singular cohomology ring, the K-ring of complex vector bundles, the Chow ring, and the Grothendieck ring of coherent sheaves of the total space of the fibre bundle with base space an irreducible nonsingular complete…

Let L --> X be a complex line bundle over a compact connected Riemann surface. We consider the abelian vortex equations on L when the metric on the surface has finitely many point degeneracies or conical singularities and the line bundle…

The moduli space $M_g^{trop}$ of tropical curves of genus $g$ is a generalized cone complex that parametrizes metric vertex-weighted graphs of genus $g$. For each such graph $\Gamma$, the associated canonical linear system $\vert…