On complex solvmanifolds and affine structures

Summary We study the problem of the existence of affine structures on compact complex solvmanifolds, in particular without the Kähler hypothesis. The main theorem gives a positive answer under a mild condition, which is true for low dimensions. A standard way to describe every solvmanifold as a quotient of (C ⁿ , *),where *is a certain Lie group structure, is also given.     

On almost complex structures which are not compatible with symplectic forms

In this Note we prove that the underlying almost complex structure to a non-Kähler almost Hermitian structure admitting a compatible connection with skew-symmetric torsion cannot be calibrated by a symplectic form even locally.     

Cohomologies of locally conformally symplectic manifolds and solvmanifolds

We study the Morse–Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz condition. We consider solvmanifolds and Oeljeklaus–Toma manifolds. In particular, we prove that Oeljeklaus–Toma manifolds with precisely one complex place, and under an additional arithmetic condition, satisfy the Mostow property. This holds in particular for the Inoue surface of type \(S^0\).     

Six-dimensional product Lie algebras admitting integrable complex structures

We classify the 6-dimensional Lie algebras of the form $\mathfrak{g}\times \mathfrak{g}$ that admit an integrable complex structure. We also endow a Lie algebra of the kind $\mathfrak{o}(n)\times \mathfrak{o}(n)$ ($n\ge 2$) with such a complex structure. The motivation comes from geometric structures à la Sasaki on $\mathfrak{g}$-manifolds.     

A note on complex projective threefolds admitting holomorphic contact structures

Oblatum 20-VI-1992 & 20-IV-1993     

Compact simple Lie groups admitting left-invariant Einstein metrics that are not geodesic orbit

In this article, we prove that the compact simple Lie groups $\mathrm{S}\mathrm{U}(n)$ for $n\ge 6$, $\mathrm{S}\mathrm{O}(n)$ for $n\ge 7$, $\mathrm{S}\mathrm{p}(n)$ for $n\ge 3$, ${\mathrm{E}}_6,{\mathrm{E}}_7,{\mathrm{E}}_8$, and ${\mathrm{F}}_4$ admit left-invariant Einstein metrics that are not geodesic orbit. This gives a positive answer to an open problem recently posed by Nikonorov.     

Compact symplectic nilmanifolds associated with graphs

In this paper we consider 2-step nilpotent Lie algebras, Lie groups and nilmanifolds associated with graphs. We present a combinatorial construction of the second cohomology group for these Lie algebras. This enables us to characterize those graphs giving rise to symplectic or contact nilmanifolds.     

Shifted symplectic structures

This is the first of a series of papers about quantization in the context of derived algebraic geometry. In this first part, we introduce the notion of n-shifted symplectic structures (n-symplectic structures for short), a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks (see Toën and Vezzosi in Mem. Am. Math. Soc. 193, 2008 and Toën in Proc. Symp. Pure Math. 80:435–487, 2009). We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-symplectic structures. Our main existence theorem states that for any derived Artin stack F equipped with an n-symplectic structure, the derived mapping stack Map(X,F) is equipped with a canonical (n?d)-symplectic structure as soon a X satisfies a Calabi-Yau condition in dimension d. These two results imply the existence of many examples of derived moduli stacks equipped with n-symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We explain how the known symplectic structures on smooth moduli spaces of simple objects (e.g. simple sheaves on Calabi-Yau surfaces, or simple representations of π ₁ of compact Riemann surfaces) can be recovered from our results, and that they extend canonically as 0-symplectic structures outside of the smooth locus of simple objects. We also deduce new existence statements, such as the existence of a natural (?1)-symplectic structure (whose formal counterpart has been previously constructed in (Costello, arXiv:1111.4234, 2001) and (Costello and Gwilliam, 2011) on the derived mapping scheme Map(E,T ^? X), for E an elliptic curve and T ^? X is the total space of the cotangent bundle of a smooth scheme X. Canonical (?1)-symplectic structures are also shown to exist on Lagrangian intersections, on moduli of sheaves (or complexes of sheaves) on Calabi-Yau 3-folds, and on moduli of representations of π ₁ of compact topological 3-manifolds. More generally, the moduli sheaves on higher dimensional varieties are shown to carry canonical shifted symplectic structures (with a shift depending on the dimension).     

On complex structures on two-dimensional tori admitting metrics with nontrivial quadratic integral

In this paper the set of complex structures on a torus admitting Riemannian metrics (consistent with these complex structures) with nontrivial quadratic integrals is described. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 643–652, November, 1995. This work was partially supported by the Volkswagen Fund and was mainly carried out at Bochum University (Germany). The authors are grateful to Bochum University for its hospitality.     

Conformally parallel G 2 structures on a class of solvmanifolds

Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G₂-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G₂ structure. By suitably deforming the SU(3) structures obtained, we are able to describe the corresponding non-homogeneous Ricci-flat metrics with holonomy contained in G₂. In the process we also find a new metric with exceptional holonomy. Received: 20 September     

Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology

We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology is canonically isomorphic to the -cohomology of the bigraded complex of complex valued left invariant differential forms on the nilpotent Lie group .

     

Examples of symplectic structures

Summary In this paper we construct symplectic forms , on a compact manifold which have the same homotopy theoretic invariants, but which are not diffeomorphic.Research partially supported by NSF grant no. DMS 8504355     

A new linear quotient of C 4 admitting a symplectic resolution

We show that the quotient C ⁴/G admits a symplectic resolution for ${G = Q_8 \times_{{\bf Z}/2} D_8 < {\sf Sp}_4({\bf C})}$ . Here Q ₈ is the quaternionic group of order eight and D ₈ is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements ?Id of each. It is equipped with the tensor product representation ${{\bf C}^2 \boxtimes {\bf C}^2 \cong {\bf C}^4}$ . This group is also naturally a subgroup of the wreath product group ${Q_8^2 \rtimes S_2 < {\sf Sp}_4({\bf C})}$ . We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C ⁴/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V/G admitting symplectic resolutions.     

Differentiable structures on symplectic reductions

LetG be a complex reductive Lie group with maximal compact subgroupK andG×X X a holomorphic action on a Stein manifoldX. LetR _o andR ₁ be two Kempf-Ness sets arising from moment maps induced by strictly plurisubharmonic,K-invariant, proper functions. Then there is a globalK-equivariant diffeomorphism :XX with (R ₀)=R ₁. In particular, the induced differentiable structures on the categorical quotientX G are diffeomorphic. The proof is based on a variant of Moser's method using time-dependent vector fields. An example shows that the differentiable structures can indeed be different, even though they are isomorphic.     

Pseudotoric structures on toric symplectic manifolds

We prove the existence of a rank-one pseudotoric structure on an arbitrary smooth toric symplectic manifold. As a consequence, we propose a method for constructing Chekanov-type nonstandard Lagrangian tori on arbitrary toric manifolds.