Jie Liu

Let \(C\) be a curve of genus \(g\) , and \(J\) its Jacobian; let \(G\) be a finite group acting on \(C\), hence also on \(J\). A result of Kollár-Larsen implies that the quotient \(J/G\) either has Kodaira dimension zero, or is uniruled. I will show that the latter case can occur only for \(g \le 5\) , and give examples in this range. The question is motivated by a classical problem about the algebraic cycle \([C]-[(-1)^*C]\) on \(J\).

The minimal log discrepancy is an invariant of singularities related to termination of flips. The ACC for minimal log discrepancies is still unknown in dimension three, and it is one of the most important remaining problems in the minimal model theory of threefolds. In the talk, I will introduce a proof of the ACC for minimal log discrepancies on smooth threefolds. More details will be explained in my seminar talk at Peking University on the following day, 16 Oct.

We present an adjunction formula for foliations on varieties and we consider applications of the adjunction formula to the cone theorem for rank one foliations and the study of foliation singularities. Joint work with C. Spicer.

The previous talk is about Higgs fields and flat connections on vector bundles. In the first half of this talk, we consider how the picture changes when we add log poles. It turns out that the de Rham moduli space is, up to a twist and over an open of the Hitchin base, a Galois cover of the Dolbeault moduli space. This happens due to a curious appearance of the Artin-Schreier map on the residues. In the second half of this talk, we consider the case with G-bundles and no poles. We show that the analogous results on semistability and cohomology also hold. This relies on a detailed study of Ngo’s regular centralizers, and the theory of Theta-semistability. Based on joint works with Mark de Cataldo and Andres Fernandez Herrero.

The Non Abelian Hodge Theory for a curve in characteristic p relates the moduli of flat connections on the curve and the moduli of Higgs bundles on the Frobenius twist of the curve. Previously, it is known that both moduli stacks sit over a Hitchin base, and that they differ by a twist of a torsor under a Picard stack over the Hitchin base. Now we know that, by shrinking the Picard stack and its torsor simultaneously, we can make the twist respect semistability. Consequently, the semistable Dolbeault and de Rham moduli spaces differ by a twist of a torsor under a connected group scheme over the Hitchin base. We then deduce some results on the cohomologies of the moduli spaces, some of which entail new results in characteristic zero. Based on joint works with Mark de Cataldo and Michael Groechenig.

We give an overview on recent progress on the numerical characterization of the hard Lefschetz classes. During the lectures, the following topics will be discussed: positivity in algebraic geometry, the extremals of geometric inequalities, Lorentzian polynomials and combinatorial poset inequalities. Based on joint works with J. Hu and S. Shang.

The wild Riemann-Hilbert correspondence relates irregular singular connections with Stokes local systems. As a direct generalization, the correspondence also holds for reductive groups G. Based on the wild Riemann-Hilbert correspondence, I will discuss Stokes G-local systems (or Stokes G-representations) together with its moduli space in the viewpoint of nonabelian Hodge correspondence. This is joint work with Pengfei Huang.

The celebrated Simpson’s tame nonabelian Hodge correspondence shows that the correct objects from the Betti side are the so-called filtered local systems. However, such nonabelian Hodge correspondence was only known at the level of categories until recently, due to the lack of a construction of the moduli spaces of filtered local systems. In this talk, we will introduce filtered local systems and demonstrate the construction of their moduli spaces. This method is applicable to general reductive groups. Based on joint work with Hao Sun.

Graded sequences of ideals, or filtrations on local rings, play an important role in commutative algebra, the theory of singularities and birational geometry. In this talk, I will introduce the saturation of a filtration and discuss several results about multiplicities of filtrations. The first is a theorem generalizing Rees' theorem on Hilbert-Samuel multiplicities of ideals. Then I will talk about convexity of multiplicities along geodesics, with applications in valuation theory and K-stability. I will also introduce a metric on the space of filtrations as an analogue of the Darvas metric in complex geometry, and prove the semiconuity of log canonical threshold functions in the spirit of Kollár-Demailly. This talk is partially based on joint work with Harold Blum and Yuchen Liu.

A del Pezzo surface is a smooth projective surface with ample anticanonical divisor. Over an algebraically closed field, any surface like this is rational. However, without this assumption del Pezzo surfaces exhibit very interesting birational properties. I will survey some old and new results about birational geometry of del Pezzo surfaces over arbitrary fields, mostly focusing on del Pezzo surfaces of degree 4. These can be realized as intersections of two quadrics. It is known that minimal del Pezzo surfaces of degree 4 over a perfect field are never rational. I will provide a detailed description of birational models for such surfaces, and will discuss the properties of their birational automorphism groups. The talk is based on a joint work in progress with A.Trepalin.