Stefan Ivanov

Vanishing theorems and string backgrounds

We show various vanishing theorems for the cohomology groups of compact Hermitian manifolds for which the Bismut connection has a (restricted) holonomy contained in SU(n) and classify all such manifolds of dimension four. In this way we provide necessary conditions for the existence of such structures on compact Hermitian manifolds with vanishing first Chern class of non-Kahler type. Then we apply our results to solutions of the string equations and show that such solutions admit various cohomological restrictions such as, for example, that under certain natural assumptions the plurigenera vanish. We also find that under some assumptions the string equations are equivalent to the condition that a certain vector is parallel with respect to the Bismut connection.

Killing spinor equations in dimension 7 and geometry of integrable G2-manifolds

Abstract We compute the scalar curvature of seven-dimensional G 2 -manifolds admitting a G 2 -connection with totally skew-symmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3-form field. In dimension n =7 the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable G 2 -structure into a cocalibrated one of pure type W 3 .

A no-go theorem for string warped compactifications

Abstract We give necessary conditions for the existence of perturbative heterotic and common sector type II string warped compactifications preserving four and eight supersymmetries to four spacetime dimensions, respectively. In particular, we find that the only compactifications of heterotic string with the spin connection embedded in the gauge connection and type II strings are those on Calabi–Yau manifolds with constant dilaton. We obtain similar results for compactifications to six and to two dimensions.

Vanishing theorems on Hermitian manifolds

Abstract We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with dd c -harmonic Kahler form and positive (1,1) -part of the Ricci form of the Bismut connection. This implies the vanishing of the Dolbeault cohomology groups on complex surfaces which admit a conformal class of Hermitian metrics, such that the Ricci tensor of the canonical Weyl structure is positive. As a corollary we obtain that any such surface must be rational with c 1 2 >0 . As an application, the p th Dolbeault cohomology groups of a left-invariant complex structure compatible with bi-invariant metric on a compact even dimensional Lie group are computed.

Heterotic supersymmetry, anomaly cancellation and equations of motion

Abstract We show that the heterotic supersymmetry (Killing spinor equations) and the anomaly cancellation imply the heterotic equations of motion in dimensions five, six, seven, eight if and only if the connection on the tangent bundle is an instanton. For heterotic compactifications in dimension six this reduces the choice of that connection to the unique SU ( 3 ) instanton on a manifold with stable tangent bundle of degree zero.

Non-Kaehler Heterotic String Compactifications with Non-Zero Fluxes and Constant Dilaton

We construct new explicit compact supersymmetric valid solutions with non-zero field strength, non-flat instanton and constant dilaton to the heterotic equations of motion in dimension six. We present balanced Hermitian structures on compact nilmanifolds in dimension six satisfying the heterotic supersymmetry equations with non-zero flux, non-flat instanton and constant dilaton which obey the three-form Bianchi identity with curvature term taken with respect to either the Levi-Civita, the (+)-connection or the Chern connection. Among them, all our solutions with respect to the (+)-connection on the compact nilmanifold M3 satisfy the heterotic equations of motion.

Curvature properties of twistor spaces of quaternionic Kähler manifolds

We present a study of natural almost Hermitian structures on twistor spaces of quaternionic Kahler manifolds. This is used to supply (4n + 2)-dimensional examples (n > 1) of symplec tic non-Kähler manifolds. Studying their curvature properties we give a negative answer to the questions raised by D.Blair-S.Ianus and A.Gray, respectively, of whether a compact almost Kähler manifold with Hermitian Ricci tensor or whose curvature tensor belongs to the class AH2 is Kähler.

Riemannian Manifold in Which the Skew-Symmetric Curvature Operator has Pointwise Constant Eigenvalues

We study manifolds where the natural skew-symmetric curvature operator has pointwise constant eigenvalues. We give a local classification (up to isometry) of such manifolds in dimension 4. In dimension 3, we describe such manifolds up to a classification of three - dimensional Riemannian manifolds with principal Ricci curvatures r1 = r2 = 0, r3- arbitrary. We give examples of such manifolds in all dimensions which do not have constant sectional curvature; these manifolds are not pointwise Osserman manifolds in general.

Nearly hypo structures and compact nearly Kähler 6-manifolds with conical singularities

We prove that any totally geodesic hypersurface N5 of a 6-dimensional nearly K¨ahler manifold M6 is a Sasaki–Einstein manifold, and so it has a hypo structure in the sense of Conti and Salamon [Trans. Amer. Math. Soc. 359 (2007) 5319–5343]. We show that any Sasaki–Einstein 5-manifold defines a nearly K¨ahler structure on the sin-cone N5 × R, and a compact nearly Kahler structure with conical singularities on N5 × [0, π] when N5 is compact, thus providing a link between the Calabi–Yau structure on the cone N5 × [0, π] and the nearly K¨ahler structure on the sin-cone N5 × [0, π]. We define the notion of nearly hypo structure, which leads to a general construction of nearly K¨ahler structure on N5 × R. We characterize double hypo structure as the intersection of hypo and nearly hypo structures and classify double hypo structures on 5-dimensional Lie algebras with non-zero first Betti number. An extension of the concept of nearly Kahler structure is introduced, which we refer to as nearly half-flat SU(3)-structure,and which leads us to generalize the construction of nearly parallel G2-structures on M6 × R given by Bilal and Metzger [Nuclear Phys. B 663 (2003) 343–364]. For N5 = S5 ⊂ S6 and for N5 = S2 × S3 ⊂ S3 × S3, we describe explicitly a Sasaki–Einstein hypo structure as well as the corresponding nearly K¨ahler structures on N5 × R and N5 × [0, π], and the nearly parallel G2-structures on N5 × R2 and (N5 × [0, π]) × [0, π].

Extremals for the Sobolev inequality on the seven-dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem

A complete solution to the quaternionic contact Yamabe problem on the seven-dimen- sional sphere is given. Extremals for the Sobolev inequality on the seven-dimensional Heisenberg group are explicitly described and the best constant in theL 2 Folland-Stein embedding theorem is determined.