Narrow Results

In this paper we treat in details a Siegel modular variety that has a Calabi–Yau model, . We shall describe the structure of the ring of modular forms and its geometry. We shall illustrate two different methods of producing the Hodge numbers. The first uses the definition of as the quotient of another known Calabi–Yau variety . In this case we will get the Hodge numbers considering the action of the group on a crepant resolution of . The second, purely algebraic geometric, uses the equations derived from the ring of modular forms and is based on determining explicitly the Calabi–Yau model and computing the Picard group and the Euler characteristic.

According to the open-closed mirror symmetry and generalized Gel'fand–Kapranov–Zelerinski (GKZ) system, we calculate off-shell D-brane/F-theory effective superpotentials of four compact Calabi–Yau (CY) manifolds by integrating special periods of subsystem, and extracting the open Ooguri–Vafa invariants from the expansion of off-shell D-brane superpotentials.

This paper is a sequel to [Continuity of extremal transitions and flops for Calabi–Yau manifolds, J. Differential Geom.89 (2011) 233–270]. We further investigate the Gromov–Hausdorff convergence of Ricci-flat Kähler metrics under degenerations of Calabi–Yau manifolds. We extend Theorem 1.1 in [Continuity of extremal transitions and flops for Calabi–Yau manifolds, J. Differential Geom.89 (2011) 233–270] by removing the condition on existence of crepant resolutions for Calabi–Yau varieties.

We prove a Bochner-type vanishing theorem for compact complex manifolds $Y$ in Fujiki class ${𝒞}$, with vanishing first Chern class, that admit a cohomology class $[\alpha ]\in H^{1,1}(Y,ℝ)$ which is numerically effective (nef) and has positive self-intersection (meaning ${\int }_Y{\alpha }^n>0$, where $n={\mathrm{\text{dim}}}_{ℂ}Y$). Using it, we prove that all holomorphic geometric structures of affine type on such a manifold $Y$ are locally homogeneous on a non-empty Zariski open subset. Consequently, if the geometric structure is rigid in the sense of Gromov, then the fundamental group of $Y$ must be infinite. In the particular case where the geometric structure is a holomorphic Riemannian metric, we show that the manifold $Y$ admits a finite unramified cover by a complex torus with the property that the pulled back holomorphic Riemannian metric on the torus is translation invariant.

A differential geometric statement of the noncommutative topological index theorem is worked out for covariant star products on noncommutative vector bundles. To start, a noncommutative manifold is considered as a product space $X=Y\times Z$, wherein $Y$ is a closed manifold, and $Z$ is a flat Calabi–Yau $m$-fold. Also, a semi-conformally flat metric is considered for $X$ which leads to a dynamical noncommutative spacetime from the viewpoint of noncommutative gravity. Based on the Kahler form of $Z,$ the noncommutative star product is defined covariantly on vector bundles over $X$. This covariant star product leads to the celebrated Groenewold–Moyal product for trivial vector bundles and their flat connections, such as $C^{\infty }(X)$. Hereby, the noncommutative characteristic classes are defined properly and the noncommutative Chern–Weil theory is established by considering the covariant star product and the superconnection formalism. Finally, the index of the ⋆-noncommutative version of elliptic operators is studied and the noncommutative topological index theorem is stated accordingly.

We analyze the coupled supergravity and Yang–Mills system using holomorphy, near the rigid limit where the former decouples from the latter. We find that there appears generically a new mass scale around gM_pl where g is the gauge coupling constant and M_pl is the Planck scale. This is in accord with the weak-gravity conjecture proposed recently.

The solution of the Strominger system can be viewed as a canonical structure on non-Kähler Calabi-Yau threefolds with balanced metrics. In this talk, we review the existence of balanced metrics on non-Kähler complex manifolds and the existence of solutions to the Strominger system.