Algebraic Geometry Seminar

Abstract:  I will report on a joint work with Hamid Abban, Elena Denisova, Erroxe Etxabarri-Alberdi, Anne-Sophie Kaloghiros, Dongchen Jiao, Jesus Martinez-Garcia, Theodoros Papazachariou about the K-moduli of smooth Fano threefolds in deformation families 2-22, 2-24, 2-25, 3-12, 3-13, 4-13 (see https://www.fanography.info/ for the description of these families). These are all one-dimensional families of smooth Fano threefolds for which K-moduli exist. We know all smooth K-polystable Fano threefolds in these families. I will explain how to find their K-polystable singular limits.

Abstract: The large genus asymptotic behaviour of intersection numbers on the moduli space of stable curves has been studied for several years. A recent milestone has been achieved with the proof of their conjectural asymptotic behaviour, uniformly in all powers of psi classes, by Aggarwal, for the Airy case (i.e. the cohomological field theory of the identity). We propose a method employing the techniques of topological recursion and resurgence. This method is applied again to the Airy case as well as to two other cohomological field theories: the r-spin case and the Theta class case.

Abstract:  A complex algebraic variety is said to be Brody hyperbolic if it contains no entire curves, which are non-constant holomorphic images of the complex line. The Green-Griffiths-Lang conjecture predicts that varieties of (log) general type are hyperbolic outside of a proper subvariety called an exceptional locus. We prove an algebraic version of this Conjecture, with respect to Demailly’s algebraic version of hyperbolicity, for the complement of a very general degree 2n hypersurface in Pn. Moreover, for the complement of a very general quartic plane curve, we completely characterize the exceptional locus as the union of the flex and bitangent lines. Based on joint work with Xi Chen and Eric Riedl. 

Abstract: The theories of KSBA stability and K-stability furnish compact moduli spaces of general type pairs and Fano pairs respectively. However, much less is known about the moduli theory of Calabi-Yau pairs. In this talk I will present an approach to constructing a moduli space of Calabi-Yau pairs which should interpolate between KSBA and K-stable moduli via wall-crossing.  I will explain how this approach can be used to construct projective moduli spaces of plane curve pairs. This is based on joint work with K. Ascher, H. Blum, K. DeVleming, G. Inchiostro, Y. Liu, X. Wang.

Abstract: This talk presents upcoming joint work with Trent Lucas in which we investigate complex surfaces that fiber over Teichmüller curves, isometrically immersed curves in the moduli spaces of curves. The generic fibers of these vibrations are Veech surfaces, a class of highly symmetric translation surfaces. In most cases, the vibrations are minimal surfaces of general type. We will describe how we compute topological and complex-geometric invariants of these surfaces in terms of a monodromy action on the mod $m$ homology of the fiber. For the class of algebraically primitive Veech surfaces, we describe how to compute exact values of the invariants.

Abstract: The perverse filtration captures interesting homological information of algebraic maps. Recent studies of Hitchin systems, affine Springer fibers, compactified Jacobians, and enumerative geometry suggest two mysterious features of the perverse filtration of an abelian fibration. They are (1) the multiplicativity of the perverse filtration with respect to the cup product, and (2) the perversity of a tautological class is given by its Chern degree. In this talk, I will discuss these features and a uniform method proving them. For the approach, we establish a theory of Fourier transform extending the Beauville decomposition from abelian varieties/schemes to certain abelian fibrations with singular fibers. Based on joint work with Davesh Maulik and Qizheng Yin.

Abstract: The Hamilton-Tian Conjecture implies that any smooth complex Fano variety admits a degeneration to a possibly singular Fano variety admitting a Kahler-Ricci soliton, which is a type of canonical Kahler metric. In this talk, I will discuss an algebraic analogue of this statement phrased in the language of K-stability. This result has applications to both the study of Kahler-Ricci soliton in differential geometry and to the moduli theory of unstable Fano varieties. This talk is based on joint work with Yuchen Liu, Chenyang Xu, and Ziquan Zhuang.

Abstract: Recently I was informed that some results of mine are essentially the same as the results recently obtained by Prof. Chen and Prado, though their motivations, formulations and proofs are quite different from mine.  In my formulation, I consider the problem of counting the number of affine conjugacy classes of polynomial maps of one complex variable with fixed degree whose set of fixed-point multipliers is the given one.  For this problem, I succeeded completely for polynomials having no multiple fixed points, and partly for polynomials having multiple fixed points.  On the other hand, Prof. Chen and Prado consider meromorphic 1-forms on the complex projective line (modulo automorphisms) which have unique zero and whose orders of all the poles are fixed.  In this setting, they succeeded completely in counting the number of such 1-forms whose set of residues of poles is the given one.  In this talk, I will present my results first.  Then we will see that these two are essentially the same, and compare the results obtained.

One possible interpretation of the SYZ conjecture is that for a polarized family of CY manifolds near the large complex structure limit, there is a special Lagrangian fibration on the generic region of the CY manifold. Generic here means a set with a large percentage of the CY measure, and the percentage tends to 100% in the limit. I will discuss my recent progress on this version of the SYZ conjecture, with some emphasis on the approach involving nonarchimedean geometry.

Abstract: In this talk, I will introduce a recent research progress on the following conjecture: any 3-fold of general type has the volume lower bound 1/420. I will also report some new results on the classification of some varieties attaining minimal volumes.

Gauged Linear Sigma Models (GLSM) are Gromov-Witten curve counting theories capturing geometry of critical loci of regular functions on a GIT quotients of vector spaces. GLSM both provide insights about the known enumerative theories and extend them to a larger class of models. In the talk I will explain what are GLSM invariants and how variation of GIT stability parameter leads to the relation of GLSM invariants for different targets.

Abstract: The Fujita freeness conjecture predicts that for an ample line bundle L on a smooth projective variety X of dimension n, the adjoint bundle K+mL is basepoint free for m at least n+1.  While this lower bound is sharp, as can be seen from the hyperplane bundle on projective space, there are examples (e.g. abelian varieties) for which the lower bound is much smaller.  In this talk, I will discuss some results motivated by the goal of characterizing both the varieties where this "Fujita number" is very small and the varieties for which it is n+1.  This is joint work with Jiaming Chen, Alex Kuronya, and Jakob Stix.    

Abstract: K-stability provides a powerful tool for constructing compact moduli spaces, known as K-moduli spaces, for Fano varieties. However, determining the K-moduli space for specific Fano varieties, such as Fano hypersurfaces, can be a challenging problem. Previously, K-moduli space for cubic hypersurfaces was shown to be the same as GIT up to dimension 4. In this talk, I’ll discuss some recent progress on the K-moduli space of quartic threefolds where K-moduli and GIT differ significantly. We find a new codimension 3 locus in the K-moduli space that parametrizes certain weighted complete intersections. Moreover, we show that this locus is closed by relating the K-stability of such complete intersections to certain del Pezzo surface pairs. This is joint work with Hamid Abban, Ivan Cheltsov, Alexander Kasprzyk, and Andrea Petracci.

Abstract: We shall first explain the relation between a family of deformations of genus zero p-adic string worldsheet action and Tate's thesis. We then propose a genus one p-adic string worldsheet action. The key is the definition of a p-adic Laplacian operator on the Tate curve. We show that the genus one p-adic Green's function exists, is unique under some obvious constraints, is locally constant off the diagonal, and has a reflection symmetry. We shall then work out the Green's function as an explicit infinite series. In particular, it turns out that, near the diagonal, the asymptotic behavior of the Green's function is a direct p-adic counterpart of the familiar Archimedean case, although the p-adic Laplacian is not a local operator. Relevant history of p-adic string theory shall be mentioned. Joint work in progress with Rebecca Rohrlich.