This is a joint effort of many algebraic geometers in East Asia. We aim to create a platform for algebraic geometers and students for further interaction and cooperation.

The seminar will take place on Friday at GMT 1:00 or GMT 7:00, unless otherwise specified, in order to accommodate most participants in East Asia.

From March to May 2021, this seminar is run by Morningside Center of Mathematics.

For previous talks, please visit the following pages:

Yujiro Kawamata, Xiaotao Sun, JongHae Keum, Jungkai Chen, Conan Leung,Phung Ho Hai, De-Qi Zhang, Yusuke Nakamura, Baohua Fu, Kiryong Chung, Hsueh-Yung Lin.

Supported by MCM and NCTS

**Speaker:** Junyi Xie (CNRS Rennes)

**Time:** 15:00–16:00 3/05/2021

**Title:** Some boundedness problems in Cremona group

**Abstract:** This talk is based on a work with Cantat and Deserti. According to the degree sequence, there are 4 types (elliptic, Jonquieres, Halphen and Loxodromic) of elements f in Bir(P^2). For a fixed degree d>=1, we study the set of these 4 types of elements of degree d. We show that for Halphen twists and Loxodromic transformations, such sets are constructible. This statement is not true for elliptic and Jonquieres elements.We also show that for a Jonquieres or Halphen twist f of degree d, the degree of the unique f-invariant pencil is bounded by a constant depending on d. This result may be considered as a positive answer to the Poincare problem of bounding the degree of first integrals,but for birational twists instead of algebraic foliations. As a consequence of this, we show that for two Halphen twists f and g, if they are conjugate in Bir(f), then they are conjugate by some element of degree bounded by a constant depending on deg(f)+deg(g). This statement is not true for Jonquieres twists.

**Speaker:** Guolei Zhong (National University of Singapore)

**Time: **16:15–17:15 3/05/2021

**Title:** Fano threefolds and fourfolds admitting non-isomorphic endomorphisms.

**Abstract:** In this talk, we first show that a smooth Fano threefold X admits a non-isomorphic surjective endomorphism if and only if X is either toric or a product of a smooth rational curve and a del Pezzo surface. Second, we show that a smooth Fano fourfold Y with a conic bundle structure is toric if and only if Y admits an amplified endomorphism. The first part is a joint work with Sheng Meng and De-Qi Zhang, and the second part is a joint work with Jia jia.

**Speaker:** Zhiyuan Li (Shanghai Center for Mathematical Sciences)

**Time: **9:00–10:00 3/19/2021

**Title: **Twisted derived equivalence for abelian surfaces

**Abstract:** Over complex numbers, the famous global Torelli theorem for K3 surfaces says that two integral Hodge isometric K3 surfaces are isomorphic. Recently, Huybrechts has shown that two rational Hodge isometric K3 surfaces are twisted derived equivalent. This is called the twisted derived Torelli theorem for K3. Natural questions arise for abelian varieties. In this talk, I will talk about the twisted derived equivalence for abelian surfaces, including the twisted derived Torelli theorem for abelian surfaces (over all fields) and its applications. This is a joint work with Haitao Zou.

**Speaker:** Michael Kemeny (University of Wisconsin-Madison)

**Time: **10:15–11:15 3/19/2021

**Title: **Universal Secant Bundles and Syzygies

**Abstract: **We describe a universal approach to the secant bundle construction of syzygies provided by Ein and Lazarsfeld. As an application, we obtain a quick proof of Green's Conjecture on the shape of the equations of general canonical curves. Furthermore, we will explain how the same technique resolves a conjecture of von Bothmer and Schreyer on Geometric Syzygies of canonical curves.

**Speaker:** Weizhe Zheng (Morningside Center of Mathematics)

**Time: **15:00–16:00 4/02/2021

**Title: **Ultraproduct cohomology and the decomposition theorem

**Abstract:** Ultraproducts of étale cohomology provide a large family of Weil cohomology theories for algebraic varieties. Their properties are closely related to questions of l-independence and torsion-freeness of l-adic cohomology. I will present recent progress in ultraproduct cohomology with coefficients, such as the decomposition theorem. This talk is based on joint work with Anna Cadoret.

**Speaker:** Kestutis Cesnavicius (U. Paris Sud)

**Time: **16:15–17:15 4/02/2021

**Title: **Grothendieck--Serre in the quasi--split unramified case

**Abstract:** The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. To overcome obstacles that have so far kept the mixed characteristic case out of reach, we adapt Artin's construction of "good neighborhoods" to the setting where the base is a discrete valuation ring, build equivariant compactifications of tori over higher dimensional bases, and study the geometry of the affine Grassmannian in bad characteristics.

**Speaker:** Kuan-Wen Lai (University of Massachusetts Amherst)

**Time: **9:00–10:00 4/16/2021

**Title: **On the irrationality of moduli spaces of K3 surfaces

**Abstract:** As for moduli spaces of curves, the moduli space of polarized K3 surfaces of genus g is of general type and thus is irrational for g sufficiently large. In this work, we estimate how the irrationality grows with g in terms of the measure introduced by Moh and Heinzer. We proved that the growth is bounded by a polynomial in g of degree 15 and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree 10. These results are built upon the modularity of the generating series of these moduli spaces in certain ambient spaces, and also built upon the existence of Hodge theoretically associated cubic fourfolds, Gushel–Mukai fourfolds, and hyperkähler fourfolds. This is a collaboration with Daniele Agostini and Ignacio Barros (arXiv:2011.11025).

**Speaker:** Yu-Shen Lin (Boston University)

**Time: **10:15–11:15 4/16/2021

**Title:** Special Lagrangian Fibrations in Log Calabi-Yau Surfaces and Mirror Symmetry

**Abstract:** Strominger-Yau-Zaslow conjecture predicts that the Calabi-Yau manifolds admit special Lagrangian fibrations and the mirror can be constructed via the dual torus fibration. The conjecture has been the guiding principle for mirror symmetry while the original conjecture has little progress. In this talk, I will prove that the SYZ fibration exists in certain log Calabi-Yau surfaces and their mirrors indeed admit the dual torus fibration under suitable mirror maps. The result is an interplay between geometric analysis and complex algebraic geometry. The talk is based on joint works with T. Collins and A. Jacob.

**Speaker:** Yi Gu (Suzhou University)

**Time: **15:00–16:00 ** **4/30/2021

**Title: **On the equivariant automorphism group of surface fibrations

**Abstract:** Let f:X→C be a relatively minimal surface fibration with smooth generic fibre. We will discuss the finiteness of its equivariant automorphism group, which is the group of pairs {ÎAut(X)´Aut(C)|} with natural group law. We will give a complete classification of those surface fibrations with infinite equivariant automorphism group in any characteristic. As an application, we will show how this classification can be used to study the bounded subgroup property and the Jordan property for automorphism group of algebraic surfaces.

**Speaker:** Takehiko Yasuda (Osaka University)

**Time: **16:15–17:15 4/30/2021

**Title:** On the isomorphism problem of projective schemes

**Abstract:** I will talk about the isomorphism problem of projective schemes; is it algorithmically decidable whether or not two given projective (or, more generally, quasi-projective) schemes, say over an algebraic closure of Q, are isomorphic? I will explain that it is indeed decidable for the following classes of schemes: (1) one-dimensional projective schemes, (2) one-dimensional reduced quasi-projective schemes, (3) smooth projective varieties with either the canonical divisor or the anti-canonical divisor being big, and (4) K3 surfaces with finite automorphism group. Our main strategy is to compute Iso schemes for finitely many Hilbert polynomials. I will also discuss related decidability problems concerning positivity properties (such as ample, nef and big) of line bundles.

**Speaker: **Yuji Odaka (Kyoto University)

**Time: **14:00–15:00 5/14/2021

**Title:** On (various) geometric compactifications of moduli of K3 surfaces

**Abstract:** What we mean by “geometric compactifications” in the title is it still parametrizes “geometric objects” at the boundary. In algebraic geometry, it is natural to expect degenerate varieties as such objects. For the moduli of polarized K3 surfaces (or K-trivial varieties in general) case, it is natural to expect slc and K-trivial degenerations, but there are many such compactifications for a fixed moduli component, showing flexibility / ambiguity / difficulty of the problem. This talk is planned to mainly focus the following. In K3 surfaces (and hyperKahler varieties), there is a canonical geometric compactification whose boundary and parametrized objects are Not varieties but tropical geometric or with more PL flavor. This is ongoing joint work with Y.Oshima (cf., arXiv:1810.07685, 2010.00416).

In general, there is a canonical PARTIAL compactification (quasi-projective variety) of moduli of polarized K-trivial varieties (essentially due to Birkar and Zhang), as a completion with respect to the Weil-Petersson metric. This is characterized by K-stability.

**Speaker:** YongJoo Shin (Chungnam National University)

**Time: **15:15–16:15 5/14/2021

**Title:** Complex minimal surfaces of general type with p_{g}= 0 and K^{2 }= 7 via bidouble covers

**Abstract:** Let S be a minimal surface of general type with p_{g}(S) = 0 and K^{2}_{S} = 7 over the field of complex numbers. Inoue firstly constructed such surfaces S described as Galois Z_{2}×Z_{2}-covers over the four-noda cubic surface. Chen later found different surfaces S constructed as Galois Z_{2}×Z_{2}-covers over six nodal del Pezzo surfaces of degree one. In this talk we construct a two-dimensional family of surfaces S different from ones by Inoue and Chen. The construction uses Galois Z_{2}×Z_{2}-covers over rational surfaces with Picard number three, with eight nodes and with two elliptic fibrations. This is a joint work with Yifan Chen.

**Speaker:** Zhiyu Tian (BICMR-Beijing University)

**Time: **15:00–16:00 5/28/2021

**Title:** Some conjectures about Kato homology of rationally connected varieties and KLT singularities

**Abstract:** A natural question about zero cycles on a variety defied over an arithmetically interesting field is the injectivity/surjectivity of the cycle class map. This leads to the study of a Gersten type complex defined by Bloch-Ogus and Kato. I will present some conjectures about this complex for rationally connected varieties and Kawamata log terminal (KLT) singularities. I will also present some evidence for the conjectures, and explain how they fit into a variety of conjectures about the stability phenomenon observed in topology and number theory.

**Speaker:** Joao Pedro dos Santos (Universite de Paris)

**Time: **16:15–17:15 5/28/2021

**Title:** Group schemes from ODEs defined over a discrete valuation ring.

**Abstract:** Differential Galois theory has the objective to study linear ODEs (or connections) with the help of algebraic groups. Roughly and explicitly, to a matrix $A\in \mathrm{Mat}_n( \mathbb C(x) )$ and a differential system $y'=Ay$, we associate a subgroup of $GL_n(\mathbb C)$, the differential Galois group, whose function is to measure the complexity of the solutions. There are three paths to this theory: Picard-Vessiot extensions, monodromy representations and Tannakian categories.

If instead of working with complex coefficients we deal with a discrete valuation ring $R$, the construction of the differential Galois groups are less obvious and the theory of groups gives place to that of group schemes. This puts forward the Tannakian approach and relevant concepts from algebraic geometry like formal group schemes and blowups. In this talk, I shall explain how to associate to these differential equations certain flat $R$-group schemes, what properties these may have--what to expect from a group having a generically faithful representation which becomes trivial under specialisation?--and how to compute with the help of the analytic method of monodromy. The talk is a horizontal report on several works done in collaboration with P.H.Hai and his students N.D.Duong and P.T.Tam over the past years.

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