Ensemble of Algebra and Geometry

Abstract: Projective manifolds with positive tangent bundles have been studied in the past decades by many mathematicians. While there are many works on the projective manifolds with strongly positive tangent bundles (e.g. nefness and ampleness), the projective manifolds with weakly positive tangent bundles are less understood (e.g. bigness and pseudo-effectivity). In general, bigness of tangent bundle may be very pathological and it is very difficult to determine whether the tangent bundle of a given projective manifold is big. 

In this talk I will start with some basic facts about projective manifolds with big tangent bundles by introducing some interesting examples. Then I will focus on quasi-homogeneous varieties and explain that how one can use the moment map to prove/disprove the bigness of certain quasi-homogeneous varieties. In the end, I will discuss possible open questions for spherical varieties.

Spherical varieties form a remarkable class of algebraic varieties equipped with an action of an algebraic group, which contains several classes of interest: toric varieties, projective homogeneous varieties, symmetric spaces, wonderful varieties. Toric varieties are classified by fans, which provide a well-developed dictionary between their geometry and combinatorics. This makes toric varieties an excellent testing ground for algebro-geometric questions, even if they form a very special class. Spherical varieties are much more general, and include many examples from classical projective geometry. They also admit a combinatorial classification, whose relation to geometry is less understood. The lectures will present basic results on spherical varieties, together with the relevant background on the structure, actions and representations of algebraic groups. They will conclude with open questions. 

Lecture 1: Actions and representations of algebraic groups, December 14 (Tuesday) 19:00-21:00 (KST, GMT +9). Lecture Note: https://www-fourier.univ-grenoble-alpes.fr/~mbrion/SV1.pdf  

- Overview; basic notions and results on actions and representations of algebraic groups; toric varieties.

Lecture 2: Properties and examples of spherical varieties, December 15 (Wednesday) 16:00-18:00 (KST, GMT +9). Lecture Note: www-fourier.univ-grenoble-alpes.fr/~mbrion/SV2.pdf  

- Further background on the structure and representations of linear algebraic groups; projective homogeneous varieties; spherical varieties: definition, first properties, local structure. 

Lecture 3: Embeddings of spherical homogeneous spaces, December 16 (Thursday) 16:00-18:00 (KST, GMT +9). Lecture Note: www-fourier.univ-grenoble-alpes.fr/~mbrion/SV3.pdf 

- Embeddings of spherical homogeneous spaces; wonderful varieties; open questions.

• November 5 (Friday)   Lecture 1. 10:00-11:00,   Lecture 2. 11:10-12:10 (KST, GMT +9)    Lecture Note 1

• November 8 (Monday)   Lecture 3. 10:00-11:00,   Lecture 4. 11:10-12:10,   Lecture 5. 14:00-15:00 (KST, GMT +9)   Lecture Note 2

Talk 1. November 12 (Friday) 10:00-11:00 (KST, GMT +9) = November 11 (Thursday) 19:00-20:00 (CST)

Title: Lattice Polarizations on K3 Surfaces 

Abstract: This talk provides an introduction to the basic theory of lattice polarized K3 surfaces. We shall discuss the general structure of the period moduli spaces, as well as specific examples.     

Talk 2. November 12 (Friday) 11:10-12:10 (KST, GMT +9) = November 11 (Thursday) 20:10-21:10 (CST)

Title: On K3 Surfaces of High Picard Rank 

Abstract: This talk will focus on several families of K3 surfaces of high Picard rank. We shall discuss specific geometric features, period moduli spaces, as well as explicit classifications in terms of modular forms of appropriate type. This is joint work with A. Malmendier.

Talk 1. November 19 (Friday) 10:20-11:20 (KST, GMT +9) 

Title: On lattice polarized K3 surfaces of Picard rank $\ge 14$

Abstract: A smooth K3 surface obtained as the minimal resolution of the quotient of an abelian surface by the involution automorphism is called a Kummer surface. However, for Kummer surfaces, their Picard rank must always be greater than or equal to 17. In this talk I will present recent results where we obtained an explicit description for a family of lattice polarized K3 surfaces of Picard ranks 14 and 16 associated with the double covering of the projective plane branched along certain sextics, together with a notion of geometric two-isogeny that generalizes the Shioda-Inose construction for Kummer surfaces. This is joint work with A. Clingher.     

Talk 2. November 19 (Friday) 11:30-12:30 (KST, GMT +9)

Title: On K3 surfaces of high Picard rank and string dualities  

Abstract: Kummer surfaces provide a purely geometric interpretation for a certain duality in string theory. Building on this foundation, I will explain how algebraic K3 surfaces obtained from abelian varieties provide a fascinating arena for string compactification and string dualities as they are not-trivial spaces, but are sufficiently simple to analyze most of their properties in detail. I will then describe recent results where we used families of lattice polarized K3 surfaces of Picard rank 10, 14, and 16 to provide a geometric interpretation, called geometric two-isogeny, for the so-called F-theory/heterotic string duality in eight dimensions with up to 4 Wilson lines. This is joint work with A. Clingher.

Abstract: Existence of canonical Kähler metrics on complex manifolds is one of the major problems in analytic and algebraic complex geometry. In this series of lectures, I will present various results regarding this question, on the class of spherical varieties. I will explain why and how these varieties are more tractable both from the differential geometry and the algebraic geometry viewpoint. A short and biased historical review of examples will serve as a motivation to study this particular class, but I will also try to highlight how this study may yield useful insight for the study of more general varieties, and hint at various significant obstacles still to be lifted for spherical varieties. 

The plan of the lecture will roughly be as follows: 

Lecture 1 - Overview of canonical metrics on spherical varieties (short historical review, combinatorial criterions, examples and counter-examples)   January 5 (Wednesday), 19:00-21:00 (KST)     Lecture Note 1

main references: 

• Examples of K-unstable Fano manifolds https://arxiv.org/abs/1911.08300

• The Yau–Tian–Donaldson conjecture for cohomogeneity one manifolds https://arxiv.org/abs/2011.07135

Lecture 2 - Differential geometric aspects: horosymmetric varieties (complex Monge–Ampère equations and variational approach on horosymmetric varieties)   January 6 (Thursday) 19:00-21:00 (KST)     Lecture Note 2

main references: 

• Kähler geometry of horosymmetric varieties, and application to Mabuchi's K-energy functional https://doi.org/10.1515/crelle-2018-0040  

• Coupled complex Monge–Ampère equations on Fano horosymmetric manifolds, with Jakob Hultgren https://doi.org/10.1016/j.matpur.2020.12.002  

Lecture 3 - Algebro-geometric aspects (test configurations and non-Archimedean functionals on spherical varieties)   January 7 (Friday) 19:00-21:00 (KST)     Lecture Note 3

main references: 

• K-Stability of Fano spherical varieties https://doi.org/10.24033/asens.2430  

• Uniform K-stability of polarized spherical varieties, with an appendix by Yuji Odaka https://arxiv.org/abs/2009.06463 

• Lecture 1. May 27th (Friday) 10:00-11:00  (KST, GMT +9)   Lecture 1

Title: The Isoparametric Story, a Heritage of Élie Cartan, I, The Works of Cartan and Muenzner 

Abstract:  We survey the fundamental work, from 1938 to 1940, of Catan and the groundbreaking work, in 1970, of Muenzner on the classification of isoparametric hypersurfaces in the sphere.

• Lecture 2. May 27th (Friday) 11:15-12:15 (KST, GMT +9)   Lecture 2

Title: The Isoparametric Story, a Heritage of Élie Cartan, II, The Works of Ozeki-Takeuchi, Ferus-Karcher-Muenzner, and Stolz 

Abstract:  We survey the second wave of groundbreaking works, from 1975 to 1999, by Ozeki-Takeuchi, Ferus-Karcher-Muenzner, and Stolz on the classification of isoparametric hypersurfaces with four principal curvatures in the sphere.

• Lecture 3. May 30th (Monday)  10:00-11:00 (KST, GMT +9)   Lecture 3

Title: The Isoparametric Story, a Heritage of Élie Cartan, III, The Classification, Part I 

Abstract: We outline the groundwork for the classification of isoparametric hypersurfaces with four principal curvatures in the sphere, that leads to the classification when the larger multiplicity is no smaller than twice the smaller multiplicity minus one. Serre's criterion of normal varieties is instrumental.

• Lecture 4. May 30th (Monday)  11:15-12:15 (KST, GMT +9)   Lecture 4

Title:  The Isoparametric Story, a Heritage of Élie Cartan, IV, The Classification, Part II 

Abstract: We outline the classification of isoparametric hypersurfaces with four principal curvatures in the sphere for the four exceptional cases when the two multiplicities are {3, 4}, {4, 5}, {6, 9}, or {7, 8}. Condition A of Ozeki and Takeuchi plays a pivotal role.

Abstract: Cox rings are important invariants of algebraic varieties, which encode much information and are generally hard to determine. The lectures will present results and questions on the Cox rings of certain classes of almost homogeneous varieties, which can be handled via methods of equivariant algebraic geometry: toric varieties (these started the theory of Cox rings), wonderful varieties, spherical varieties, and some varieties of complexity one. Of special interest are the Cox rings of wonderful varieties: they turn out to be closely related to representation theory, deformations of affine spherical varieties, and a remarkable class of algebraic monoids constructed by Vinberg. 

Lecture 1. Introduction to Cox rings: definitions, first classes of examples (toric varieties, homogeneous spaces). Relation to birational geometry via Mori dream spaces. Equivariant Cox rings: construction and first properties. June 20 (Monday) 16:00-18:00 (KST, GMT +9)

Main references: 

I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface: Cox rings. Cambridge University Press, 2015. 

A. Vézier: Equivariant Cox ring. Transformation Groups (2022).

Lecture 2. Cox rings of wonderful varieties. June 24 (Friday) 16:00-18:00 (KST, GMT +9)

Wonderful varieties, their Picard group and cone of effective divisor. Descriptions of their Cox ring via representation theory and commutative algebra. Cox rings of wonderful group compactifications and their relation to the Vinberg monoid. 

Main references: 

M. Brion: The total coordinate ring of a wonderful variety. J. Algebra 313, 61-99 (2007). 

E. B. Vinberg: On reductive algebraic semigroups. In: American Mathematical Society. Transl., Ser. 169, 145-182 (1995).

Lecture 3. Cox rings of spherical varieties. June 29 (Wednesday) 16:00-18:00 (KST, GMT +9)

Description of these Cox rings by reduction to the wonderful case. Cox rings of horospherical varieties. If time allows: further developments for Cox rings of almost homogeneous varieties of complexity one, e.g. almost homogeneous SL(2)-threefolds. 

Main references: 

G. Gagliardi: The Cox ring of a spherical embedding. J. Algebra 397, 548-569 (2014). 

A. Vézier: Cox rings of almost homogeneous SL(2)-threefolds, arXiv:2009.08676.

Abstract: The Tits-Freudenthal magic square was discovered in the 1950's, as a way to construct the exceptional complex simple Lie algebras from a pair of normed algebras (typically, the octonions). It concentrates a whole range of enigmatic phenomena, some of which are still not completely understood, and have connections with important questions in representation theory, topology and geometry. 

The goal of the lectures will be to explain the main ideas of the construction of the magic square, and discuss a geometric version that was first devised by Freudenthal. We will discover a rich zoo of amazing algebraic varieties, having intricate connections with one another. I will discuss how and why these varieties are important for various problems of general interest in complex algebraic geometry. 

Lecture 1: December 7 (Wednesday) 17:00-18:00 (KST, GMT +9), 2022     Lecture Note 1

Lecture 2: December 8 (Thursday) 17:00-18:30 (KST, GMT +9), 2022     Lecture Note 2

Lecture 3: December 9 (Friday) 17:00-18:30 (KST, GMT +9), 2022     Lecture Note 3

Abstract: For a complex connected semisimple linear algebraic group G of adjoint type, De Concini and Procesi constructed its wonderful compactification, which is a smooth Fano equivariant embedding of G enjoying many interesting properties. In these talks, we will discuss on the properties of wonderful group compactifications, especially the deformation rigidity of them. The talks are based on a joint work with Baohua Fu. 

Lecture 1. Wonderful group compactifications and the properties. January 10 (Tuesday) 10:00-11:30 (KST, GMT +9), 2023 

Lecture 2. The application of VMRT theory to deformation rigidity problem. January 11 (Wednesday) 10:00-11:30 (KST, GMT +9), 2023 

Abstract: Let M be a Fano manifold. Consider the Kahler–Ricci flow (KRF) on M : ∂/∂t ω(t) = −Ric(ω(t)) + ω(t), ω(0) = ω_0 ∈ c_1(M), here by ω_0 and ω(t) we denote the initial metric and the solution of KRF respectively. We will study the limiting behavior of ω(t) as t→+∞. Tian–Zhu [7, 8] proved that if M admits a Kahler–Ricci soliton ω_{KRS}, then ω(t) converges to ω_{KRS}. But in general ω(t) has no limits on M. According to the Hamilton–Tian conjecture, proved by [6, 1, 2], any sequence {(M, ω(t_i))} for t_i→+∞ includes a subsequence converging to a Q-Fano variety (M_∞, ω_∞) in the Gromov–Hausdorff topology. Furthermore, ω_∞ is a weak Kahler–Ricci soliton. Thus if M itself has no Kahler–Ricci solitons, then the complex structures of M and M_∞ can not be the same. Note that in this case M is K-unstable. 

Recently Li–Han [3] proved that the limit M_∞ can be derived by two steps: through semistable and polystable degenerations. They also proved that the semistable degeneration is precisely the minimizer of the H-invariant among all R-test configurations. In general the limit of semistable degeneration is not M_∞. However, it equals to M_∞ if one can prove that the limit is K-stable. In this case the polystable degeneration will be trivial. 

Now we state our main results. Let M be a (Q-Fano) compactification of a reductive Lie group. We classify all its equivariant normal R-test configurations. Then we compute the corresponding H-invariants and find the semistable degeneration of M. Furthermore, for the two smooth K-unstable Fano SO_4(C)-compactifications, we prove that the semistable limits are indeed K-stable. Finally we find the limits of KRF on them. These are the limiting space of the first Type II solutions of KRF in [5]. Details can be found in [4].

References

[1] R. Bamler, Convergence of Ricci flows with bounded scalar curvature, Ann. Math., 188 (2018), 753-831.

[2] X.-X. Chen, and B. Wang, Space of Ricci flows (II) Part B: Weak compactness of the flows, J. Differential Geom. 116 (2020), 1-123.

[3] J.-Y. Han and Ch. Li, Algebraic uniqueness of Kahler–Ricci flow limits and optimal degenerations of Fano varieties, arXiv:2009.01010.

[4] Y. Li and Zh.-Y. Li, Equivariant R-test congurations and semistable limits of Q-Fano group compactifications, arXiv:2103.06439.

[5] Y. Li, G. Tian and X.-H. Zhu, Singular limits of Kahler–Ricci flow on Fano G-manifolds, arXiv:1807.09167.

[6] G. Tian and Zh.-L. Zhang, Regularity of Kahler–Ricci flows on Fano manifolds, Acta Math. 216 (2016), 127-176.

[7] G. Tian and X.-H. Zhu, Convergence of the Kahler–Ricci flow, J. Amer Math. Sci. 17 (2006), 675-699.

[8] G. Tian and X.-H. Zhu, Convergence of the Kahler–Ricci flow on Fano manifolds, J. Reine Angew Math. 678 (2013), 223-245.

Abstract: A toric degeneration is a flat degeneration from a projective variety to a toric variety, which can be used to apply the theory of toric varieties to other projective varieties. In the case of a flag variety, its toric degeneration with desirable properties induces degenerations of Richardson varieties to unions of irreducible toric subvarieties, called semi-toric degenerations. Semi-toric degenerations are closely related to Schubert calculus. For instance, Kogan–Miller constructed semi-toric degenerations of Schubert varieties from Knutson–Miller's semi-toric degenerations of matrix Schubert varieties which give a geometric proof of the pipe dream formula of Schubert polynomials. In the 1st talk, we review the notion of extended g-vectors in cluster theory, which induces Newton–Okounkov bodies and toric degenerations of a compactified cluster variety. In the 2nd talk, we discuss such Newton–Okounkov bodies and toric degenerations for a flag variety. We see that these induce semi-toric degenerations of Richardson varieties, which can be regarded as generalizations of Kogan–Miller's semi-toric degeneration. This series of talks is partly based on a joint work with Hironori Oya. 

Abstract: The first part of this talk will give an introduction to dimers and their connection to non-compact toric Calabi-Yau 3-folds. Dimers appear in many different areas of mathematics and theoretical physics, and the talk will give a brief overview on how dimer models appear as gauge theories realized in string theory. 

The second part of this talk will focus on more recent developments, where in string theory a higher-dimensional object known as the brane brick model was discovered in relation to non-compact toric Calabi-Yau 4-folds. The talk will focus on complex cones over Fano 3-folds and will make use of Calabi-Yau mirror symmetry and tropical geometry in order outline the construction of brane brick models from toric Calabi-Yau 4-folds. 

Abstract: Let $\mathbb{P}(a,b,c)$ be a weighted projective space, where $a,b,c$ are pairwise coprime positive integers. Let $X(a,b,c)$ be a blowup of $\mathbb{P}(a,b,c)$ at a general nonsingular point. Recently there has been raised interest in finding out whether $X(a,b,c)$ is a Mori dream space for given $a,b,c$, due to its connection to the Mori dreamness of the moduli space of $\overline{M}_{0,n}$. Much progress has been made in this direction, but there are still unanswered questions. One of the unanswered questions is whether every $X(a,b,c)$ contains a negative curve apart from the exceptional curve. In this talk, I will present a sufficient condition for finding negative curves on $X(a,b,c)$. Applying this method, we find some new cases where $X(a,b,c)$ are Mori dream spaces.

Abstract: To each complex composition algebra A, there associates a projective symmetric manifold X(A) of Picard number one, which is just a smooth hyperplane section of the varieties Lag(3, 6), Gr(3, 6), S_6, E7/P7. In this paper, it is proven that these varieties are rigid, namely for any smooth family of projective manifolds over a connected base, if one fiber is isomorphic to X(A), then every fiber is isomorphic to X(A).

Abstract: In this talk, we will first introduce a classification theorem of equivariant normal $R$-test configurations of a polarized spherical variety. The $Q$-Fano case of this classification result will be used to study the limit problem of Kahler-Ricci flow on these varieties. Also we will discuss weighted K-stability of $Q$-Fano spherical varieties which is related to the existence of generalized (or weighted) Kahler-Ricci solitons.

Abstract: Projective manifolds with positive tangent bundles have been studied in the past decades by many mathematicians. While there are many works on the projective manifolds with strongly positive tangent bundles (e.g. nefness and ampleness), the projective manifolds with weakly positive tangent bundles are less understood (e.g. bigness and pseudo-effectivity). In general, bigness of tangent bundle may be very pathological and it is very difficult to determine whether the tangent bundle of a given projective manifold is big. 

In this talk I will start with some basic facts about projective manifolds with big tangent bundles by introducing some interesting examples. Then I will focus on quasi-homogeneous varieties and explain that how one can use the moment map to prove/disprove the bigness of certain quasi-homogeneous varieties. In the end, I will discuss possible open questions for spherical varieties.

Abstract: In this seminar, we shall study how to solve Monge–Ampère type equations using a variational method by Berman and others. This gives a way to obtain singular Kähler–Einstein metrics on Q-Fano varieties. Especially, we will focus on the setting of group compactifications, based on the recent paper by Li–Tian–Zhu 'Singular Kähler–Einstein metrics on Q-Fano compactifications of Lie groups'. 

Abstract: This is a learning seminar and we follow the article "Smooth projective horospherical varieties of Picard group Z^2" by Boris Pasquier.