Geometry and Automorphisms of Non-Kähler Holomorphic Symplectic Manifolds

We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works by D. Guan and the 1st author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kähler manifold $W_F$, which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction and prove that the automorphism group of $Q$ satisfies the Jordan property.

Blow-ups and infinitesimal automorphisms of CR-manifolds

For a real-analytic connected CR-hypersurface M of CR-dimension $$n\geqslant 1$$ n ⩾ 1 having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra $$\mathfrak {s}={\mathfrak {s}}(M)$$ s = s ( M ) : (i) either $$\dim {\mathfrak {s}}=n^2+4n+3$$ dim s = n 2 + 4 n + 3 and M is spherical everywhere; (ii) or $$\dim {\mathfrak {s}}\leqslant n^2+2n+2+\delta _{2,n}$$ dim s ⩽ n 2 + 2 n + 2 + δ 2 , n and in the case of equality M is spherical and has fixed signature of the Levi form in the complement to its Levi-degeneracy locus. A version of this result is proved for the Lie group of global automorphisms of M. Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (ii) are constructed. We provide many other models with large symmetry using the technique of blow-up, in particular we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as a symmetry.

Coisotropic Submanifolds in b-symplectic Geometry

We study coisotropic submanifolds of b-symplectic manifolds. We prove that b-coisotropic submanifolds (those transverse to the degeneracy locus) determine the b-symplectic structure in a neighborhood, and provide a normal form theorem. This extends Gotay’s theorem in symplectic geometry. Further, we introduce strong b-coisotropic submanifolds and show that their coisotropic quotient, which locally is always smooth, inherits a reduced b-symplectic structure.

A BOGOMOLOV UNOBSTRUCTEDNESS THEOREM FOR LOG-SYMPLECTIC MANIFOLDS IN GENERAL POSITION

We consider compact Kählerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure $\unicode[STIX]{x1D6F1}$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor $D(\unicode[STIX]{x1D6F1})$. We prove that $(X,\unicode[STIX]{x1D6F1})$ has unobstructed deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on $H^{2}$ of the open symplectic manifold $X\setminus D(\unicode[STIX]{x1D6F1})$, and in fact coincides with this $H^{2}$ provided the Hodge number $h_{X}^{2,0}=0$, and finally that the degeneracy locus $D(\unicode[STIX]{x1D6F1})$ deforms locally trivially under deformations of $(X,\unicode[STIX]{x1D6F1})$.

Diagrams and Essential Sets for Signed Permutations

We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations. In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation. Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.

Schubert complexes and degeneracy loci

International audience The classical Thom―Porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of Schur polynomials. A proof of this formula was given by Pragacz by expressing this alternating sum as the Euler characteristic of a Schur complex, which gives an explanation for the signs. Fulton later generalized this formula to the situation of flags of vector bundles by using alternating sums of Schubert polynomials. Building on the Schubert functors of Kraśkiewicz and Pragacz, we introduce Schubert complexes and show that Fulton's alternating sum can be realized as the Euler characteristic of this complex, thereby providing a conceptual proof for why an alternating sum appears. \par La formule classique de Thom―Porteous exprime la classe d'homologie du locus de la dégénérescence d'une fonction générique entre deux fibrés vectoriels comme une somme alternée des polynômes de Schur. Un preuve de cette formule a été donnée par Pragacz en exprimant ce alternant somme comme la caractéristique d'Euler d'un complexe de Schur, ce qui donne une explication pour les signes. Fulton puis généralisée cette formule à la situation des drapeaux de fibrés vectoriels à l'aide alternant des sommes de polynômes de Schubert. S'appuyant sur le Schubert foncteurs de Kraśkiewicz et Pragacz, nous introduisons les complexes de Schubert et montrent que la somme alternée de Fulton peuvent être réalisées en tant que Euler caractéristique de ce complexe, fournissant ainsi une preuve conceptuelle pour lesquelles une somme alternée appara\^ıt.