Schedule for: 19w5051 - Bridging the Gap between Kahler and non-Kahler Complex Geometry

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will describe a geometric problem on families of elliptic operators, which is solved via a deformation to a family of non self-adjoint Fredholm operators. Let $p:M\to S$ be a proper holomorphic projection of complex manifolds. Let $F$ be a holomorphic vector bundle on $M$. We assume that the $H^{(0,p)}(Xs, F|X_s)$ have locally constant dimension. They are the fibers of a holomorphic vector bundle on $S$. The problem we will address is the computation of characteristic classes associated with the above vector bundle in a refinement of the ordinary de Rham cohomology of $S$, its Bott-Chern cohomology, and the proof of a corresponding theorem of Riemann-Roch-Grothendieck. When $M$ is not K\"ahler, none of the existing techniques to prove such a result using the fiberwise Dolbeault Laplacians can be used. The solution is obtained via a proper deformation of the corresponding Dolbeault Laplacians to a family of hypoelliptic Laplacians, for which the corresponding result can be proved. This deformation is made to destroy the geometric obstructions which exist in the elliptic theory, like the fact the metric should be K\"ahler, or the K\"ahler form to be $\bar\partial\partial$-closed.

In this talk I will overview a new gauge-theoretical approach to the Hull-Strominger system using holomorphic string algebroids. In the smooth setup, a string algebroid provides an infinitesimal version of a principal bundle for the string group. The main focus will be on the consequences of this approach for the existence and uniqueness problem, as well as for the moduli space metric. The discussion will be illustrated by a key example. Joint work with Rubio, Tipler, and Shahbazi, in arXiv:1803.01873, arXiv:1807.10329

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Generalized Kähler-Ricci solitons are canonical geometric structures on complex, non-Kähler manifolds. In this talk I will give a complete classification of such structures on complex surfaces, which is joint work with Y. Ustinovskiy.

We consider normal compact surfaces $Y$ obtained from a minimal class VII surface $X$ by contraction of a cycle $C$ of $r$ rational curves with $c_2<0$. Our main result states that, if the obtained cusp is smoothable, then $Y$ is globally smoothable. The proof is based on a vanishing theorem for $H^2(Y,\Theta)$ where $\Theta$ is the dual the sheaf of K\"ahler forms. If $r\le b_2(X)$ any smooth small deformation of $Y$ is rational, and if $r=b_2(X)$ (i.e. when $X$ is a half-Inoue surface) any smooth small deformation of $Y$ is an Enriques surface.

A locally conformally Kähler (LCK) manifold is a compact Hermitian manifold $(M,g,J)$ whose fundamental 2-form $\omega:=g(J\cdot,\cdot)$ verifies $d\omega=\theta\wedge\omega$ for a certain closed 1-form $\theta$ called the Lee form. We study here LCK manifolds whose Lee vector field (the metric dual of $\theta$) is holomorphic. We will show that if its norm is constant or if its divergence vanishes, then the metric is Vaisman, i.e. the Lee form is parallel with respect to the Levi-Civita connection of $g$. We will then give examples of non-Vaisman LCK manifolds with holomorphic Lee field, and we classify all such structures on manifolds of Vaisman type. These results have been obtained in collaboration with F. Madani, S. Moroianu, L. Ornea and M. Pilca.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will discuss the behavior of the Chern-Ricci flow on compact complex surfaces, emphasizing some open questions and new directions.

The Hull-Strominger system describes the geometry of compactifications of heterotic superstrings with flux, which can be viewed as a generalization of Ricciflat Kahler metrics on non-Kahler Calabi-Yau manifolds. To overcome the difficulty of lacking ddbar-lemma, Phong-Picard-Zhang initiated the program of Anomaly flow to understand the Hull-Strominger system. It has been proved in many cases that the Anomaly flow serves as an effective way to investigate the Hull-Strominger system and in general canonical metrics on complex manifolds, such as giving new proofs of the Calabi-Yau theorem and the existence of Fu-Yau solution. In this talk, we present some new progress on the Anomaly flow, including the behavior of Anomaly flow on generalized Calabi-Gray manifolds and a unification of the Anomaly flow with vanishing slope parameter and the Kahler-Ricci flow, which further allows us to generalize the notion of the Anomaly flow to arbitrary complex manifolds. This talk is based on joint work with Z.-J. Huang, D.H. Phong and S. Picard.

Deligne, Griffiths, Morgan and Sullivan famously characterised the $\partial\bar\partial$-Lemma as by the following property: The double complex of forms decomposes as a direct sum of two kinds of irreducible subcomplexes: 'Squares' and 'dots', where only the latter contribute to cohomology. In this talk, we explore the implications of the following folklore generalisation of this: Every (suitably bounded) double complex decomposes into irreducible complexes and these are 'squares' and 'zigzags', with a dot being a zigzag of length 1. This yields insight into the structure of and relation between the various cohomology groups. Applied to complex manifolds, we obtain, among others, Serre duality for all pages of the FSS, a three space decomposition on the middle cohomology and new bimeromorphic invariants. We end the talk with several open questions.

In the talk will be reviewed the construction of various non-Kähler metrics on toric bundles over Kähler base space. When the base is a K3 orbifold, it provides examples of solutions to Hull-Strominger system on simply connected spaces diffeomorphic to $S^1\times\sharp_k(S^2\times S^3)$ for $13 \leq k \leq 22$ and $\sharp_r (S^2 \times S^4) \sharp_{r+1} (S^3 \times S^3)$ for $14\leq r\leq 22$. The talk is based on the joint work with A. Fino and L. Vezzoni.

The Yamabe invariant is a real-valued diffeomorphism invariant coming from Riemannian geometry. Using Seiberg-Witten theory, LeBrun showed that the sign of the Yamabe invariant of a Kähler surface is determined by its Kodaira dimension. We show that the Yamabe invariant of Inoue surfaces and their blowups is zero which demonstrates that the non-Kähler analogue of LeBrun's theorem does not hold.

In 1995, Dan Guan constructed examples of non-Kähler, simply-connected holomorphically symplectic manifolds. An alternative construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. We prove the local Torelli theorem, showing that holomorphically symplectic deformations of BG-manifolds are unobstructed, and the corresponding period map is locally a diffeomorphism. Using the local Torelli theorem, we prove the Fujiki formula for a BG-manifold, showing that there exists a symmetric form q on the second cohomology with the same properties as the Beauville-Bogomolov-Fujiki form for hyperkahler manifolds. This is a joint work with Nikon Kurnosov.

In this talk we describe Kato manifolds, also known as manifolds with global spherical shell. We revisit Brunella’s proof of the fact that Katosurfaces admit locally conformally K\"ahler metrics, and we show that it holdsfor a large class of higher dimensional complex manifolds containing a globalspherical shell. On the other hand, we construct manifolds containing a globalspherical shell which admit no locally conformally K\"ahler metric. We then con-sider a specific class, which can be seen as a higher dimensional analogue of Inoue-Hirzebruch surfaces, and study several of their analytical properties. Inparticular, we give new examples, in any complex dimension $n\geq 3$, of com-pact non-exact locally conformally K\"ahler manifolds with algebraic dimension $n-2$, algebraic reduction bimeromorphic to $\C\Proj^{n-2}$ and admitting non-trivialholomorphic vector fields. These results are joint work with Nicolina Istrati(University of Tel Aviv) and Massimiliano Pontecorvo (Roma Tre University).

I will discuss the existence of special Lagrangian torus fibrations on log Calabi-Yau manifolds constructed from del Pezzo surfaces, and some progress towards establishing SYZ mirror symmetry for these non-compact Calabi-Yau manifolds. This is joint work with A. Jacob and Y.-S. Lin.

I will introduce a Fr\"olicher-type spectral sequence that is valid for all almost complex manifolds, yielding a natural Dolbeault cohomology theory for non-integrable structures and a rich theory surrounding it. As an application, we will see how Dolbeault cohomology may be used to detect the non-existence of compatible nearly K\"ahler metrics. In fact, for nearly K\"ahler manifolds, the Frölicher spectral sequence always degenerates at $E_2$, and so the second page recovers the Hodge-Verbitsky decomposition. I will end with a list of open questions that would be great to discuss with the participants during the workshop. This is joint work with Scott Wilson.

The cone of nef (1,1) real Bott-Chern cohomology classes on a compact complex manifold is defined in analogy with the Kahler case, as the cone of classes which admit representatives with arbitrarily small negative part. On a K\"ahler manifold this coincides with the closure of the Kahler cone. I will review some basic open problems about these classes, which are either trivial or known in the Kahler case, and I will present partial progress towards some of these (joint with S. Kolodziej and B. Weinkove) and some speculations.

An Hermitian metric $g$ on a complex manifold $M$ is called {\em balanced} if its fundamental form $\omega$ is co-closed. Typically examples of balanced manifolds are given by modifications of K\"ahler manifolds, twistor spaces over anti-self-dual oriented Riemannian $4$-manifolds and nilmanifolds. In the talk it will be discussed a geometric flow of balanced metrics. The flow was introduced in \cite{BFV1} and consists in a generalisation of the Calabi flow to the balanced context. The flow preserves the Bott-Chern cohomology class of the initial metric and in the K\"ahler case reduces to the classical Calabi flow. It will be showed that the flow is well-posedn and stable around Ricci flat K\"ahler metrics. Furthermore, the talk focuses on a rencent problem in balanced geometry proposed in \cite{FV}. \begin{thebibliography}{12} \bibitem{BFV1} {\sc L. Bedulli, L. Vezzoni}, A parabolic flow of balanced metrics. {\em Journal f\"ur die reine und angewandte Mathematik (Crelle)}. {\bf 723} (2017), 79--99. \bibitem{BFV2} {\sc L.~Bedulli and L. Vezzoni}, A scalar Calabi-type flow in Hermitian Geometry, to appear in {\em Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)}. \bibitem{BFV3} {\sc L.~Bedulli and L. Vezzoni}, Stability of geometric flows of closed forms. {\tt\, arXiv:1811.09416}. \bibitem{FGV} {\sc A. Fino, G. Grantcharov, L. Vezzoni}, Astheno-K\"ahler and balanced structures on fibrations. {\tt arXiv:1608.06743}, to appear in {\em IMRN}. \bibitem{FV} {\sc A. Fino, L. Vezzoni,} Special Hermitian metrics on compact solvmanifolds, {\em J. Geom. Phys.} {\bf 91} (Special Issue) (2015), 40--53. \end{thebibliography}

The symmetries of the Hodge diamond of a Kähler manifold provide many non-trivial conditions on both the Hodge numbers and the Betti numbers of the underlying manifold. In this talk I will describe generalizations of these to the setting of Hermitian and almost Kähler manifolds. Both discussions involve zeroeth order terms that vanish in the Kähler case, and yield an interesting representation of $sl(2)$ on a naturally defined subspace of the harmonic forms. I'll explain several topological corollaries involving Betti numbers and the fundamental group, which is joint work with Joana Cirici. The two discussions have some interesting and yet unexplained algebraic mirror-type symmetry between them, which may lead the participants to interesting questions for future research.

We show that any K\"ahler metric pertaining to the Taub-NUT hyperkaehler structure on $\mathbb{R} ^4$ comes naturally coupled with a complete K\"ahler surface isomorphic to a Bryant self-dual K\"ahler metric on $\mathbb{C} ^2$ to form a regular ambitoric structure of parabolic type.

Let X be a compact complex manifold with trivial canonical bundle and satisfying the $\partial\bar\partial$-Lemma. If X is Kähler then, up to a finite cover, X is product of a simply connected manifold and its Albanese Torus $Alb(X)$, be the Beauville-Bogomolov decomposition theorem. We show that in the more general setting, the Albanese map is still a holomorphic submersion but will in general not split after finite pullback. We also show that the Kuranishi space of $X$ is a smooth universal deformation and that small deformations enjoy the same properties as $X$. If, in addition, $X$ admits a complex symplectic form, then the local Torelli theorem holds and we obtain some information about the period map. I will also mention some open question. Based on joint work with B. Anthes, A. Cattaneo, A. Tomassini.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.