Related papers: Deligne pairing and determinant bundle

We generalize the construction of a moduli space of semistable pairs parametrizing isomorphism classes of morphisms from a fixed coherent sheaf to any sheaf with fixed Hilbert polynomial under a notion of stability to the case of projective…

We consider a smooth Lagrangian subvariety Y in a smooth algebraic variety X with an algebraic symplectic from. For a vector bundle E on Y and a choice Oh of deformation quantization of the structure sheaf of X, we establish when E admits a…

Scheme-theoretic methods are used to classify ternary quadratic forms with values in line bundles over arbitrary schemes and to canonically determine the isomorphisms between them. The association of a quadratic bundle to its even Clifford…

A symplectic bundle over an algebraic curve has a natural invariant $\sLag$ determined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give…

Let $f: X \to Z$ be a fibration from a normal projective variety $X$ of dimension $n$ onto a normal curve $Z$ over a perfect field of characteristic $p>2$. Let $(X, B)$ be a dlt pair such that the induced pair on a general fibre is log…

Let $X$ be a smooth projective variety defined over an algebraically closed field, and let $L$ be an ample line bundle over $X$. We prove that for any smooth hypersurface $D$ on $X$ in the complete linear system $| L^{\otimes d}|$, the…

On the basis of Brylinski's work, we introduce a notion of equivariant smooth Deligne cohomology group, which is a generalization of both the ordinary smooth Deligne cohomology and the ordinary equivariant cohomology. Using the cohomology…

Let $X={\bf P}^2\times{\bf P}^{n-1}$ embedded with $\O(1,2)$. We prove that its $(n+1)$-secant variety $\sigma_{n+1}(X)$ is a hypersurface, while it is expected that it fills the ambient space. The equation of $\sigma_{n+1}(X)$ is the…

Let $X$ be a smooth projective curve over a field of characteristic zero and let $D$ be a non-empty set of rational points of $X$. We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on $(X,D)$…

Let X be an irreducible smooth complex projective curve of genus at least 3. Fix a line bundle L on X. Let M_{Sp}(L) be the moduli space of symplectic bundles (E, ExE ---> L) on X, with the symplectic form taking values in L. We show that…

An orthogonal bundle over a curve has an isotropic Segre invariant determined by the maximal degree of a Lagrangian subbundle. This invariant, and the induced stratifications on moduli spaces of orthogonal bundles, were studied for bundles…

Let X be a geometrically connected smooth projective curve of genus one, defined over the field of real numbers, such that X does not have any real points. We classify the isomorphism classes of all stable real algebraic vector bundles over…

We study the linearization of line bundles and the local structure of actions of connected linear algebraic groups, in the setting of seminormal varieties. We show that several classical results about normal varieties extend to that…

Let $\Delta$ be a Delzant polytope in ${\mathbb R}^n$ and ${\bf b}\in{\mathbb Z}^n$. Let $E$ denote the symplectic fibration over $S^2$ determined by the pair $(\Delta, {\bf b})$. We prove the equivalence between the fact that $(\Delta,…

We propose, motivate and give evidence for a relation between the $\mathcal D$-modules of the quantum cohomology of a smooth complex projective manifold $X$ and a projective bundle $\PP(\oplus L_i)$ over $X$.

Let $\mathscr{L} \rightarrow X$ be an ample line bundle over a nonsingular complex projective variety $X$. We construct an admissable flag $X_0 \subseteq X_1 \subseteq...\subseteq X_n=X$ of subvarieties for which the associated Okounkov…

Let $X$ be a perfectoid space with tilt $X^\flat$. We construct a canonical map $\theta:\operatorname{Pic} X^\flat\to\lim\operatorname{Pic} X$ where the (inverse) limit is taken over the $p$-power map, and show that $\theta$ is an…

Let G/Q be an homogeneous variety embedded in a projective space P thanks to an ample line bundle L. Take a projective space containing P and form the cone X over G/Q, we call this a cone over an homogeneous variety. Let $\alpha$ a class of…

The moduli stack M_X(E_8) of principal E_8-bundles over a smooth projective curve X carries a natural divisor Delta. We study the pull-back of the divisor Delta to the moduli stack M_X(P), where P is a semi-simple and simply connected group…

A map Y -> P^n is determined by a line bundle quotient of (O_Y)^{n+1}. In this paper, we generalize this description to the case of maps from Y to an arbitrary smooth toric variety. The data needed to determine such a map consists of a…