Title: Generalized sheaf counting
Abstract: It is well known that many problems in algebraic geometry are reduced to finding vector bundles or sheaves with desired properties. To enumerate them, we construct their moduli spaces and apply intersection theory. To get a moduli space with an intersection theory, we have to pick a stability condition and delete unstable objects. In good circumstances where there are no strictly semistable sheaves, integrating cohomology classes against the (virtual) fundamental class gives us enumerative invariants like Donaldson invariant, Seiberg-Witten invariant and Donaldson-Thomas invariant. However when there are strictly semistable sheaves, the moduli space is an Artin stack on which integration doesn't make sense under current technology. Generalized sheaf counting is about finding a way to modify the moduli space of semistable sheaves to get a Deligne-Mumford stack by which an enumerative invariant can be defined. In this talk, I will report recent progresses on generalized sheaf counting on curves, surfaces and 3-folds.
Title: Compactifications of the moduli space of Higgs pairs on an elliptic curve
Abstract: It is known that the moduli space of Higgs pairs on a projective scheme is a quasi-projective by C.T. Simpson in 1994. In 2000, A.H.W Schmitt constructed two kinds of compactifications of the moduli space of Higgs pairs on a smooth projective scheme. One is the moduli space of framed Hitchin pairs and another is the moduli space of oriented framed Hitchin pairs as a master space. He showed that the moduli space of framed Hitchin pairs varies depending on variation of stabilities and then there exists a chain of flips connecting to the moduli space of Higgs pairs. He also showed that this chain of flips is indeed a chain of C*-flips via the moduli space of oriented framed Hitchin pairs.
The purpose of this work is to find concrete descriptions of the moduli spaces of framed Hitchin pairs and of oriented framed Hitchin pairs on an elliptic curve.
In the first talk, we introduce the moduli spaces of Higgs pairs, framed Hitchin pairs and oriented framed Hitchin pairs on a smooth curve. We will give their GIT constructions briefly and then explain how to get a chain of C*-flips connecting to the moduli space of Higgs pairs.
In the second talk, we consider the same moduli spaces on an elliptic curve. Further we consider a closed subscheme of the moduli space of framed Hitchin pairs with a forgetful morphism to the moduli space of framed modules. We will give concrete descriptions of these moduli spaces and the chain of C*-flips.
This is based on a joint work in progress with Dr. Yongjoo Shin.
Title: Representations on the cohomology of the moduli space of pointed rational curves
Abstract: The moduli space of pointed rational curves carries a natural action of the symmetric group permuting the marked points. In this talk, I will present combinatorial and recursive formulas for the induced representation on its cohomology, part of which generalize the well-known recursive formula for the Poincare polynomial originally observed by Manin.
These formulas allow us to refine and extend two previously known results: first, that the ungraded cohomology group is a permutation representation, as shown by Castravet and Tevelev; and second, that the coefficients of the Poincare polynomial satisfy asymptotic log-concavity, as proved by Aluffi, Chen and Marcolli. As a refinement of the latter, we propose the conjecture that the graded cohomology exhibits equivariant log-concavity, supported by numerical evidence.
Title: Characteristic polynomial of the moduli space of pointed rational curves and log-concavity
Abstract: Motivated by Stanley’s generalization of the chromatic polynomial of a graph to the chromatic symmetric function, we introduce the characteristic polynomial of a symmetric function, or equivalently, of a representation of the symmetric group. This polynomial appears to share one of the most interesting features of the chromatic polynomial, log-concavity, especially when it arises from the cohomology of a variety. Interesting examples include the n-fold products of the projective space, the GIT moduli spaces of n points on the projective line, and regular semisimple Hessenberg varieties.
For the moduli space of pointed rational curves, the bivariate characteristic polynomial associated with its graded cohomology appears to exhibit several log-concavity properties. I will present these observations and conjectures, supported by numerical evidence and asymptotic formulas.
Based on joint works with Prof. Jinwon Choi and Young-Hoon Kiem.
Title: The Chern filtration of moduli spaces of sheaves
Abstract: The cohomology ring of moduli spaces of sheaves has been a central topic in enumerative geometry due to its rich additional structures, such as tautological generators and relations, Hall algebra, perverse filtration, etc. In this lecture, I will introduce a relatively new structure, called the Chern filtration, and explain why it is interesting and how it interacts with existing structures. The main focus will be on moduli spaces of one-dimensional sheaves on surfaces and moduli spaces of bundles on curves. This is based on joint works with Y. Kononov, M. Moreira, W. Pi.
Title: Simply connected positive Sasakian 5-manifolds
Abstract: A Sasakian manifold is an odd dimensional analogue of a Kähler manifold. On the other hand, simply connected 5-manifolds are classified by Smale and Barden. In this talk, I will explain how to understand a Sasakian structure from an algebro-geometric perspective. Then, I will present a list of certain Smale-Barden 5-manifolds that admit Sasakian structures. This is a joint work with Jihun Park and Joonyeong Won.
Title: Ulrich bundles on the smooth intersection of two quadrics
Abstract: Ulrich bundles are vector bundles exhibiting the simplest possible cohomological behavior, which makes them central objects in algebraic geometry and homological algebra. While various results suggest that every projective variety admits an Ulrich bundle, existing constructions remain largely confined to low-dimensional cases. Semiorthogonal decomposition, however, offers a broader framework for constructing Ulrich bundles such as on smooth intersections of two quadrics. In this talk, I will describe joint work in progress with Kyoung-Seog Lee and Han-Bom Moon.
Title: Moduli space of Ulrich bundles on Fano threefolds V22
Abstract: Quiver representation attract a lot of interest as they provide convenient tool for studying the moduli problems. In this talk, I will introduce an explicit description of the moduli space of stable Ulrich bundles on Fano threefolds V22 (Picard number 1), obtained by identifying it with a suitable moduli space of representations of quiver with relations. This talk is based on joint work with Kyoung-Seog Lee and Kyeong-Dong Park.