Title: Continuity of singular Kähler-Einstein potentials
Abstract: Eyssidieux-Guedj-Zeriahi showed in 2009 that any compact normal Kähler variety with klt singularities admits a singular Kähler-Einstein (SKE) metric if it has a trivial or ample canonical ℚ -line bundle. This in particular generalized the works of Aubin and Yau. A SKE potential generating the SKE metric is known to be locally Hölder continuous on the regular locus of the variety. But understanding the regularity of the potential on the singular locus still remains to be a major open problem. In this presentation, we show that any SKE potential on a compact normal Kähler variety is continuous on the whole variety. As an application, we also prove the continuity of the potential of the Kähler-Ricci flow on a compact Kähler variety X constructed by Guedj-Lu-Zeriahi (2020). This is joint work with Y.-J. Choi in Pusan National University.
Title: The Weak Eisenbud-Goto Inequality for Monomial Ideals
Abstract: We explore the Castelnuovo-Mumford regularity, a crucial algebraic invariant, for varieties defined by monomial ideals. Eisenbud and Goto conjectured that the regularity is bounded by the difference between degree and codimension of a variety. While this holds for certain classes of varieties, it may not apply to algebraic sets with multiple connected components. However, Terei showed that the Eisenbud-Goto inequality applies to special monomial ideals, connected to Stanley-Reisner complexes.
In this poster, we introduce a function on the simplicial complex that bounds the regularity and investigate some complexes where the equality holds. We establish the Weak Eisenbud-Goto inequality, a generalization of Terai's result, applicable to various complexes. Additionally, we present a bound based on Weighted Euler Characteristic. This is joint work with Jinha Kim, Minki Kim, and Yeongrak Kim.
Title: Bigness of the tangent bundles of projective bundles over curves
Abstract: The positivity of the tangent bundle imposes restrictions on the geometry of the underlying variety. For instance, projective spaces are the only projective manifolds with ample tangent bundles, and rational homogeneous manifolds are conjectured to be the only Fano manifolds with nef tangent bundles. In this poster, we focus on the case of projective bundles X=ℙ(E) over curves, exploring the relationship between the stability of E and the bigness of the tangent bundle of X.
Title: A result on the classification of polarized varieties with high nefvalue
Abstract: Let (X,L) be a polarized variety of dimension n. Beltrametti and Sommese, and later Andreatta classified birationally these varieties when X has klt singularities and the nefvalue of L is larger than n-1. We extend their results to the case when X is log canonical and deduce that when X is semi-log canonical, the polarized variety (X,L) is birationally equivalent to a projective bundle over a nodal curve.
Title : On the rank index of some quadratic varieties
Abstract : A projective variety X is said to satisfy property QR(k) if its homogeneous ideal can be generated by quadratic polynomials of rank at most k. We define the rank index of X to be the smallest integer k such that X satisfies property QR(k). Many classical varieties, such as Segre-Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property QR(4). Recently, we show that every Veronese embedding has rank index 3 if the base field has characteristic ̸= 2, 3. In this talk, we explain the rank index of X when it is some other classical projective variety such as rational normal scrolls, Segre varieties, Plucker embeddings of the Grassmannians of lines and del Pezzo varieties.
Title: Higgs bundles with a fixed determinant on an irreducible nodal curve
Abstract: We introduce a construction of the moduli space of Higgs bundles with a fixed determinant on an irreducible nodal curve. To construct the moduli space, we modify the construction of the moduli space of vector bundles with a fixed determinant on an irreducible nodal curve given by U. Bhosle. Then we define the Hitchin map on the moduli space and describe their fibers. This work is in progress.
Title: Statistical Bergman Geometry
Abstract: For a bounded domain Ω in ℂn, let P(Ω) be the set of all (real) probability distributions on Ω. Then, in general, P(Ω) becomes an infinite-dimensional smooth manifold and it always admit a natural Riemannian pseudo-metric, called the Fisher information metric, on P(Ω). Information geometry studies a finite-dimensional submanifold M, which is called a statistical manifold, in P(Ω) using geometric concepts such as Riemannian metric, distance, connection, and curvature, to better understand the properties of statistical models M and provide insights into the behavior of learning algorithms and optimization methods.
In this talk, we first introduce a map Φ : Ω → P(Ω) and prove that the pull-back of the Fisher information metric on P(Ω) is exactly same as the Bergman metric of Ω. This map provides a completely new perspective that allows us to view Bergman geometry from a statistical viewpoint. We will discuss several results in this framework. This is a joint work with Gunhee Cho at UC Santa Barbara University.
Title: Non-isomorphic endomorphisms of Fano threefolds
Abstract: Non-isomorphic endomorphisms of smooth projective varieties are expected to impose rather restrictive geometry. Conjecturally, the fundamental building blocks of such varieties are abelian varieties and toric varieties. In this poster, I would like to summarize our recent results on Fano threefolds which admit non-isomorphic endomorphisms, and then sketch the main ideas of the proofs via the equivariant minimal model program. This poster is based on the joint work with Sheng Meng and De-Qi Zhang.
Title : Characterizations of Fano type varieties and projective spaces via absolute complexity
Abstract : We first recollect the definitions of potential pairs and absolute complexity, and present characterizations of varieties of Fano type and projective spaces via absolute complexity. Also, we show that if the absolute complexity of a given pair (X,Δ) is negative, then the pair (X,Δ) does not admit any -(KX+Δ) -minimal models.
Title: Automorphisms and deformations of Hessenberg varieties
Abstract: In 1977, Demazure provided a complete classification of automorphisms and deformations of flag varieties. We initiate the study of those for Hessenberg varieties, which are certain subvarieties in flag varieties having interesting properties with connection to many other areas in mathematics including representation theory and combinatorics. In this work, we focus on Hessenberg varieties of codimension one in flag varieties, which are Fano manifolds in regular semisimple cases in type A. In type A, we provide a complete answer and using this, relate the moduli of Hessenberg varieties to the moduli of pointed smooth rational curves. In other types, we compute the dimensions of the automorphism groups and the deformation spaces. As a byproduct, we prove that every deformation of a regular semisimple Hessenberg variety of codimension one is a regular semisimple Hessenberg variety. This is a joint work in progress with P. Brosnan, L. Escobar, J. Hong, E. Lee, A. Mellit and E. Sommers.
Title: Symmetric differentials on complex hyperbolic space forms and its L2 holomorphic jet extension
Abstract: Let Σ be a compact complex ball quotient Σ under the action of a cocompact and torsion-free lattice Γ ⊂ Aut ( 𝔹n ). Since the complex unit ball has a Kähler metric induced by the Bergman metric on 𝔹n , it becomes a complex hyperbolic space form. Moreover, Σ has a holomorphic ball fiber bundle Ω, which is induced by the diagonal action of Γ for 𝔹n × 𝔹n , and Ω has the maximal compact analytic subvariety D which is biholomorphic to Σ .
When Σ is a compact Riemann surface, M. Adachi (2017) proved that a given holomorphic jet on D induces a weighted L2 holomorphic function on Ω using some recursive formula for $\bar \partial$-operator. Applying this result, he showed that Ω has no any non-constant bounded holomorphic function. The author extend his result for n-dimensional compact complex hyperbolic forms by developing a Hodge type identity for symmetric powers of the holomorphic cotangent bundle of Σ and a suitable recursive formula.
Furthermore, we prove that these phenomena are still valid for some compact submanifolds of finite volume ball quotients. This is joint work with Aeryeong Seo of Kyungpook National University.
Title: Diagrams for LVMRT (subtitle: Diagrams for variety of minimal rational tangents of wonderful symmetric varieties)
Abstract: We describe varieties of minimal rational tangents on the wonderful symmetric varieties. An irreducible component of a variety of minimal rational tangents is a rational homogeneous space, and hence, we have a corresponding Dynkin diagram expression. On the other hand, we have diagrams from the marked Kac diagram of a symmetric space by marking adjacent nodes to the marked node, similar to the case of rational homogeneous spaces. We note that the above two diagrams coincide when the restricted root system is not of type A.
Title: Cylindrical approaches to unipotent group actions on some affine varieties
Abstract: Cylinder is an 𝔸1-ruled Zariski open subset in a normal projective variety over some affine variety. If the boundary of cylinder is defined by an effective member in numerical class of given divisor, then it is called polar for the divisor. The existence of such a structure has deep connections to unipotent group actions on the corresponding affine cone. From this point of view, the polar cylinders for ample divisors on del Pezzo surfaces has been extensively studied in many areas. We present some results of them.
Title : Ulrich bundles on special cubic fourfolds
Abstract : An Ulrich bundle E on a polarized pair (X, O(1)) is a coherent sheaf whose twisted cohomology groups are exactly same as the structure sheaf of the projective space of dimension same as X. It has various connections to many other topics in algebra and geometry. It is tempting to ask whether there is an Ulrich bundle of given rank over a given projective variety (X, O(1)). For smooth cubic fourfolds, the answer may vary on the existence of surfaces contained in cubic fourfolds which are not homologous to complete intersections. We discuss computational and deformation theoretic approaches to find Ulrich bundles of unexpected ranks over special cubic fourfolds.
Title: Spherical Geometry of Spaces of Conics in Adjoint Varieties
Abstract: Adjoint varieties are rational homogeneous spaces defined by the adjoint representations of complex simple Lie algebras, and their spaces of lines are well-understood. In this work, for each adjoint variety not isomorphic to the projective space or its projectivized cotangent bundle, we study two natural compactifications of the space of smooth conics: the Hilbert compactification and the Chow compactification. Namely, we show that the normalizations of the two compactifications are algebraic symmetric varieties, and then describe their spherical geometry by computing the colored fans. As a corollary, we describe the conjugacy classes of conics in adjoint varieties and determine the singular loci of the normalizations of the compactifications.
Title: Legendre transformation related to toric plurisubharmonic functions and its application
Abstract: We briefly recall the notion of Legendre transformation of convex functions and characterize the condition for toric plurisubharmonic functions to have analytic singularities. Using convex analysis, we reveal a crucial relation between the epigraphs of a convex function corresponding a toric plurisubharmonic functions with analytic singularities and that of its Legendre transformation. As its application, for an arbitrary toric plurisubharmonic function, we give a criterion of admitting a `nice’ approximation. Our results generalize a recent result of Guan for toric plurisubharmonic functions of the diagonal type.