This book collects the proceedings of a series of conferences dedicated to birational geometry of Fano varieties held in Moscow, Shanghai and Pohang

The conferences were focused on the following two related problems:

• existence of KählerEinstein metrics on Fano varieties

• degenerations of Fano varieties

on which two famous conjectures were recently proved. The first is the famous BorisovAlexeevBorisov Conjecture on the boundedness of Fano varieties, proved by Caucher Birkar (for which he was awarded the Fields medal in 2018), and the second one is the (arguably even more famous) TianYauDonaldson Conjecture on the existence of KählerEinstein metrics on (smooth) Fano varieties and K-stability, which was proved by Xiuxiong Chen, Sir Simon Donaldson and Song Sun. The solutions for these longstanding conjectures have opened new directions in birational and Kähler geometries. These research directions generated new interesting mathematical problems, attracting the attention of mathematicians worldwide.

These conferences brought together top researchers in both fields (birational geometry and complex geometry) to solve some of these problems and understand the relations between them. The result of this activity is collected in this book, which contains contributions by sixty nine mathematicians, who contributed forty three research and survey papers to this volume. Many of them were participants of the MoscowShanghaiPohang conferences, while the others helped to expand the research breadth of the volumethe diversity of their contributions reflects the vitality of modern Algebraic Geometry.

Inhalt
T. Abe, Classification of exceptional complements: elliptic curve case.- E. Ballico, E. Gasparim, F. Rubilar, B. Suzuki, LAGRANGIAN SKELETA, COLLARS AND DUALITY.- G. Belousov, CYLINDERS IN DEL PEZZO SURFACES OF DEGREE TWO.- M. Benzerga, FINITENESS OF REAL STRUCTURES ON KLT CALABI-YAU REGULAR SMOOTH PAIRS OF DIMENSION 2.- C. Birkar, ANTICANONICAL VOLUME OF FANO 4-FOLDS.- C. Boyer Christina Tonnesen-Friedman, CONSTANT SCALAR CURVATURE SASAKI METRICS AND PROJECTIVE BUNDLES.- G. Brown, J. Buczynski, A. Kasprzyk, TORIC SARKISOV LINKS.- I. Burban, DU VAL SINGULARITIES.- I. Cheltsov, H. Suess, K-POLYSTABILITY OF TWO SMOOTH FANO THREEFOLDS.- G. Codogni, Z. Patakfalvi, A NOTE ON FAMILIES OF K-SEMISTABLE LOG-FANO PAIRS.- T. Delcroix, THE YAU-TIAN-DONALDSON CONJECTURE FOR COHOMOGENEITY ONE MANIFOLDS.- A. Dubouloz, FIBRATIONS BY AFFINE LINES ON RATIONAL AFFINE SURFACES WITH IRREDUCIBLE BOUNDARIES.- K. Fujita, ON FANO THREEFOLDS OF DEGREE 22 AFTER CHELTSOV AND SHRAMOV.- K. Fujita, Y. Liu, H. Suess,K. Zhang, Z. Zhuang, ON THE CHELTSOV-RUBINSTEIN CONJECTURE.- S. Grishin, Ilya Karzhemanov, Ming-Chang Kang, RATIONALITY OF QUOTIENTS BY FINITE HEISENBERG GROUPS.- Y. Hashimoto.- J. Keller, QUOT-SCHEME LIMIT OF FUBINISTUDY METRICS AND ITS APPLICATIONS TO BALANCED METRICS.- Z. Hu, EXISTENCE OF CANONICAL MODELS FOR KAWAMATA LOG TERMINAL PAIRS.- Y. Imagi, GENERALIZED THOMASYAU UNIQUENESS THEOREMS.- K. Jamieson, BIRATIONALLY RIGID COMPLETE INTERSECTIONS OF CODIMENSION 3.- D. Jeong.- J. Park, SIMPLY CONNECTED SASAKI-EINSTEIN 5-MANIFOLDS: OLD AND NEW.- C. Jiang, CHARACTERIZING Q-FANO THREEFOLDS WITH THE SMALLEST ANTI-CANONICAL VOLUME.- L. Katzarkov, Kyoung-Seog Lee, J. Svoboda, A. Petkov, INTERPRETATIONS OF SPECTRA.- Young-Hoon Kiem, Kyoung-Seog Lee, FANO VISITORS, FANO DIMENSION AND FANO ORBIFOLDS.- In-kyun Kim, N. Viswanathan, J. Won, ON SINGULAR DEL PEZZO HYPERSURFACES OF INDEX 3.- S. Kudryavtsev, Blow-ups of three-dimensional toric singularities.- N. Kurnosov, E. Yasinsky, AUTOMORPHISMS OF HYPERKAHLER MANIFOLDS AND GROUPS ACTING ON CAT(0) SPACES.- A. Laface, R. Quezada, ON GENERALIZED BUCHI SURFACES.- Chi Li, K-STABILITY AND FUJITA APPROXIMATION.- Y. Li, Zhenye Li, ON A CONJECTURE OF FULTON ON ISOTROPIC GRASSMANNIANS.- Y. Maeda, Y. Odaka, FANO SHIMURA VARIETIES WITH MOSTLY BRANCHED CUSP.- L. Makar-Limanov, ON LOCALLY NILPOTENT DERIVATIONS OF DANIELEWSKI DOMAINS.- D. Markouchevitch, A. Moreau, ACTION OF THE AUTOMORPHISM GROUP ON THE JACOBIAN OF KLEIN'S QUARTIC CURVE.- J. Martinez-Garcia, C. Spotti, SOME OBSERVATIONS ON THE DIMENSION OF FANO K-MODULI.- D. Witt Nystrom, OKOUNKOV BODIES AND THE KAHLER GEOMETRY OF PROJECTIVE MANIFOLDS.- J. Park, SINGULARITIES OF PLURI-FUNDAMENTAL DIVISORS ON GORENSTEIN FANO VARIETIES OF COINDEX.- J. Paulhus, A DATABASE OF GROUP ACTIONS ON RIEMANN SURFACES.- A. Petracci, A 1-DIMENSIONAL COMPONENT OF K-MODULI OF DEL PEZZO SURFACES.- T. De Piro, A NON-STANDARD BEZOUT THEOREM FOR CURVES.- Y. Prokhorov, EMBEDDINGS OF THE SYMMETRIC GROUPS TO THE SPACE CREMONA GROUP.- J. Ross, M. Toma, ON HODGE-RIEMANN COHOMOLOGY CLASSES.- Y. Rubinstein, ON LARGE DEVIATION PRINCIPLES AND THE MONGEAMPERE EQUATION (FOLLOWING BERMAN, HULTGREN).- T. Sano, ON BIRATIONAL BOUNDEDNESS OF SOME CALABI-YAU HYPERSURFACES.- Y. Zarhin, ABELIAN VARIETIES, QUATERNION TRICK AND ENDOMORPHISMS. <p