Abstract:
Let X be a projective irreducible symplectic manifold and L a non trivial nef divisor on X. Assume that the nef dimension of L is strictly less than the dimension of X. We prove that L is semiample

Abstract:
Given a projective irreducible symplectic manifold $M$ of dimension $2n$, a projective manifold $X$ and a surjective holomorphic map $f:M \to X$ with connected fibers of positive dimension, we prove that $X$ is biholomorphic to the projective space of dimension $n$. The proof is obtained by exploiting two geometric structures at general points of $X$: the affine structure arising from the action variables of the Lagrangian fibration $f$ and the structure defined by the variety of minimal rational tangents on the Fano manifold $X$.

Abstract:
We use the fact that a projective half-spin representation of $Spin_{12}$ has an open orbit to generalize Pfister's result on quadratic forms of dimension 12 in $I^3$ to orthogonal involutions.

Abstract:
In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of corresponding Lie algebras.

Abstract:
Let X be a smooth complex projective curve and S a finite subset of X. We show that an orthogonal or symplectic parabolic Higgs bundle on X with parabolic structure over S admits a Hermitian-Einstein connection if and only if it is polystable.

Abstract:
We classify the irreducible complex characters of the symplectic groups $Sp_{2n}(q)$ and the orthogonal groups $Spin_{2n}^\pm(q)$, $Spin_{2n+1}(q)$ of degrees up to the bound D, where $D=(q^n-1)q^{4n-10}/2$ for symplectic groups, $D=q^{4n-8}$ for orthogonal groups in odd dimension, and $D=q^{4n-10}$ for orthogonal groups in even dimension.

Abstract:
A symplectic or orthogonal bundle $V$ of rank $2n$ over a curve has an invariant $t(V)$ which measures the maximal degree of its isotropic subbundles of rank $n$. This invariant $t$ defines stratifications on moduli spaces of symplectic and orthogonal bundles. We study this stratification by relating it to another one given by secant varieties in certain extension spaces. We give a sharp upper bound on $t(V)$, which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces. In particular, we compute the dimension of each stratum. We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite. We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case.

Abstract:
Let A be an n by d matrix having full rank n. An orthogonal dual A^{\perp} of A is a (d-n) by d matrix of rank (d-n) such that every row of A^{\perp} is orthogonal (under the usual dot product) to every row of A. We define the orthogonal dual for arrangements by identifying an essential (central) arrangement of d hyperplanes in n-dimensional space with the n by d matrix of coefficients of the homogeneous linear forms for which the hyperplanes are kernels. If n is at least 5, we show that if the matroid (or the intersection lattice) of an n-dimensional essential arrangement A contains a modular copoint whose complement spans, then the derivation module of the orthogonally dual arrangement \A^{\perp} has projective dimension at least [n(n+2)/4] - 3,([ ] denotes ceiling).

Abstract:
We observe that, in dimension four, there is a correspondence between symplectic and Lorentzian geometry. The guiding observation is that on a Lorentzian 4-manifold $(M,g)$, null vector fields can give rise to exact symplectic forms. That a null vector field is nowhere vanishing yet orthogonal to itself is essential to this construction. Specifically, we show that if ${\boldsymbol k}$ is a complete null vector field on $M$ with geodesic flow along which $\text{Ric}({\boldsymbol k},{\boldsymbol k}) > 0$, and if $f$ is any function on $M$ with ${\boldsymbol k}(f)$ nowhere vanishing, then $dg(e^f{\boldsymbol k},\cdot)$ is a symplectic form and ${\boldsymbol k}/{\boldsymbol k}(f)$ is a Liouville vector field; any null surface to which ${\boldsymbol k}$ is tangent is then a Lagrangian submanifold. Even if the Ricci curvature condition is not satisfied, one can still construct such symplectic forms with additional information from ${\boldsymbol k}$; we give an example of this, with ${\boldsymbol k}$ a complete Liouville vector field, on the maximally extended "rapidly rotating" Kerr spacetime. We also discuss applications to Weinstein structures: if $(M,g)$ contains a compact Cauchy hypersurface, then the symplectic manifold above yields a trivial Weinstein cobordism.