Relations among smooth integral models associated to quadratic, symplectic and hermitian lattices

$J$-invariant of hermitian forms over quadratic extensions

We develop the version of the $J$-invariant for hermitian forms over quadratic extensions in a similar way Alexander Vishik did it for quadratic forms. This discrete invariant contains informations about rationality of algebraic cycles on the maximal unitary grassmannian associated with a hermitan form over a quadratic extension. The computation of the canonical $2$-dimension of this grassmannian in terms of the $J$-invariant is provided, as well as a complete motivic decomposition.

Hermitian对称空间上的双$\overline{\mathbb{Q}}$-结构与CM周期的二次关系

Bi-$\overline{\mathbb{Q}}$-structures on Hermitian symmetric spaces and quadratic relations between CM periods

本文引入CM点切空间上的双$\overline{\mathbb{Q}}$-结构，证明其可分解为一维子空间直和，并提出解析子空间猜想，以推导CM abelian簇全纯周期的二次关系。

Symplectic and Hamiltonian properties of holomorphic coadjoint orbits

This thesis studies the symplectic structure of holomorphic coadjoint orbits, and their projections. A holomorphic coadjoint orbit O is an elliptic coadjoint orbit which is endowed with a natural invariant Kählerian structure. These coadjoint orbits are defined for a real semi-simple connected non-compact Lie group G with finite center, such that G/K is a Hermitian symmetric space, where K is a maximal compact subgroup of G. Holomorphic coadjoint orbits are a generalization of the Hermitian symmetric space G/K. In this thesis, we prove that the McDuff's symplectomorphism on Hermitian symmetric…

A properness result for degenerate Quadratic and Symplectic Bundles on a smooth projective curve

Let $(V,q)$ be a vector bundle on a smooth projective curve $X$ together with a quadratic form $q: \mathrm{Sym}^2(V) \ra \mathcal{O}_X$ (respectively symplectic form $q: Λ^2V \ra \mathcal{O}_X$). Fixing the degeneracy locus of the quadratic form induced on $V/\ker(q)$, we construct a coarse moduli of such objects. Further, we prove semi-stable reduction theorem for equivalence classes of such objects. In particular, the case when degeneracies of $q$ are higher than one is that of principal interest. We also provide a proof of properness of polystable orthogonal bundles without appealing to Bru…

Kobayashi-Hitchin correspondence of generalized holomorphic vector bundles over generalized Kahler manifolds of symplectic type

In the previous paper \cite{Goto_2017}, the notion of an Einstein-Hermitian metric of a generalized holomorphic vector bundle over a generalized Kahler manifold of symplectic type was introduced from the moment map framework. In this paper we establish a Kobayashi-Hitchin correspondence, that is, the equivalence of the existence of an Einstein-Hermitian metric and $ψ$-polystability of a generalized holomorphic vector bundle over a compact generalized Kahler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles and we obtain $ψ$-stable Poisson mo…

Slices of hermitian K-theory and Milnor's conjecture on quadratic forms

We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice spectral sequence for higher Witt-theory, we prove Milnor's conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian K-groups in terms of motivic cohomology.

Hermitian forms of K3 type

We classify quadratic spaces over endomorphism fields of K3 surfaces. We consider both totally real and CM cases.

Deformation Stability of p-SKT and p-HS manifolds

In this paper, we introduce the notions of $p$-Hermitian-symplectic and $p$-pluriclosed compact complex manifolds as generalisations for an arbitrary positive integer $p$ not exceeding the complex dimension of the manifold of the standard notions of Hermitian-symplectic and SKT manifolds that correspond to the case $p=1$. We then notice that these two properties are equivalent on $\partial\bar{\partial}$-manifolds and go on to prove that in (smooth) complex analytic families of $\partial\bar{\partial}$-manifolds, they are deformation open. Concerning closedness results, we prove that the cones…

Fujiki relation on symplectic varieties

We generalize Fujiki relation of Beauville-Bogomolov quadratic form on a projective symplectic variety. As an application, we study a fibre space structure of a projective symplectic variety.

Subtle characteristic classes and Hermitian forms

Following [14], we compute the motivic cohomology ring of the Nisnevich classifying space of the unitary group associated to the standard split hermitian form of a quadratic extension. This provides us with subtle characteristic classes which take value in the motivic cohomology of the Čech simplicial scheme associated to a hermitian form. Comparing these new classes with subtle Stiefel-Whitney classes arising in the orthogonal case, we obtain relations among the latter ones holding in the motivic cohomology of the Čech simplicial scheme associated to a quadratic form divisible by a 1-fold Pfi…