Toric geometry and topology provide numerous examples of manifolds with "non-standard" complex structures, i.e., non-Kählerian and not even Moishezonian. One of the main classes of such examples with holomorphic torus symmetry are moment-angle manifolds.

A complex structure on a moment-angle manifold $Z$ is defined by a set of combinatorial geometric data, including a complete simplicial (but not necessarily rational) fan. Examples of complex moment-angle manifolds are Hopf and Calabi-Ekman manifolds, as well as their deformations.

When the fan is rational, the manifold $Z$ is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In this case, invariants of the complex structure of $Z$, such as Dolbeault cohomology and Hodge numbers, can be analysed using the Borel spectral sequence of the holomorphic fibre bundle.

In the general case, the fibers of the holomorphic bundle "open up", and the bundle becomes a canonical holomorphic foliation $F$ on a complex moment-angle manifold $Z$, equivariant under the action of an algebraic torus. The holomorphic foliated manifolds $(Z,F)$ are models for irrational toric varieties.

In general, a complex moment-angle manifold $Z$ has only finitely many complex submanifolds of positive dimension, so there are no non-constant meromorphic functions on such a complex manifold, and its algebraic dimension is zero.

The construction and classification of complex manifolds with a torus action is based on the concept of an exponential action defined by a vector configuration. Exponential actions defined by vector configurations provide a universal framework for several constructions of holomorphic dynamics, non-Kähler complex geometry, toric geometry and topology. These include leaf spaces of holomorphic foliations, intersections of real and Hermitian quadrics, the quotient construction of simplicial toric varieties, $LVM$ and $LVMB$ manifolds, complex-analytic structures on moment-angle manifolds and their partial quotients. In all cases, the geometry and topology of the appropriate quotient object can be described by combinatorial data including a pair of Gale dual vector configurations.

Programme

1. Exponential actions and holomorphic foliations, free orbits (non-degenerate leaves).
2. Linear Gale duality.
3. Fans and triangulated vector configurations.
4. Proper actions.
5. Completeness and compactness of quotient spaces.
6. Polyhedral products and moment-angle manifolds.
7. Convex polytopes and polyhedra, normal fans, and intersections of quadrics.
8. Holomorphic exponential actions and complex structures on moment-angle manifolds.
9. Gale duality for rational configurations.
10. Partial quotients and torus-exponential actions.
11. $LVM$- and $LVMB$-manifolds.
12. Toric varieties and their irrational deformations: divisors, $Nef$- and ample cones, symplectic reduction.
13. Transversely Kähler forms on complex manifolds with a torus action, divisors, and submanifolds.
14. Basic de Rham and Dolbeault cohomology.

Literature
[1] Arzhantsev, Ivan; Derenthal, Ulrich; Hausen, Juergen; Laface, Antonio, Cox Rings. Cambridge Studies in Advanced Mathematics, 144. Cambridge University Press, Cambridge, 2015.
[2] Audin, Michele, The Topology of Torus Actions on Symplectic Manifolds. Progress in Mathematics, 93. Birkhauser, Basel, 1991.
[3] Bosio, Frederic; Meersseman Laurent, Real quadrics in $C^n$, complex manifolds and convex polytopes. Acta Math. 197 (2006), no. 1, 53-127.
[4] Buchstaber, Victor; Panov, Taras, Toric Topology. Math. Surv. and Monogr., 204, Amer. Math. Soc., Providence, RI, 2015.
[5] Cox, David A.; Little John B.; Schenck, Henry K., Toric varieties. Graduate Studies in Mathematics, 124. Amer. Math. Soc., Providence, RI, 2011.
[6] De Loera, Jesus; Rambau, Joerg; Santos, Francisco, Triangulations. Structures for Algorithms and Applications. Algorithms Comput. Math., 25, Springer-Verlag, Berlin, 2010.
[7] Guillemin, Victor, Moment Maps and Combinatorial Invariants of Hamiltonian $T^n$-spaces. Progress in Mathematics, 122. Birkhaeuser Boston, Inc., Boston, MA, 1994.
[8] Ishida, Hiroaki, Complex manifolds with maximal torus actions. J. Reine Angew. Math. 751 (2019), 121-184.
[9] Katzarkov, Ludmil; Lupercio, Ernesto; Meersseman, Laurent; Verjovsky, Alberto, Quantum (non-commutative) toric geometry: foundations. Adv. Math. 391 (2021), Paper No. 107945, 110 pp.
[10] Panov, Taras, Exponential actions defined by vector configurations, Gale duality, and moment-angle manifolds. Bulletin of the London Mathematical Society 57 (2025), no. 9, 2571-2629.
[11] Panov, Taras; Ustinovskiy, Yury; Verbitsky, Misha, Complex geometry of moment-angle manifolds. Math. Z. 284 (2016), no. 1-2, 309-333.

Lecturer
Panov Taras Evgenievich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center