Classification of log Enriques surfaces with delta=1

Calabi-Yau广义费马流形的商与对数Enriques簇

Quotients of Calabi-Yau Generalized Fermat manifolds and Log Enriques varieties

本文通过刻画Calabi-Yau型广义费马流形的无不动点自同构商，构造了对数Enriques簇与Enriques流形，证明其为已知两类Calabi-Yau型Enriques流形族中一类的子族，且与另一类的第二Betti数不同。

On the complete classification of extremal log Enriques surfaces

We show that there are exactly, up to isomorphisms, seven extremal log Enriques surfaces Z and construct all of them; among them types D_{19} and A_{19} have been shown of certain uniqueness by M. Reid. We also prove that the (degree 3 or 2) canonical covering of each of these seven Z has either X_3 or X_4 as its minimal resolution. Here X_3 (resp. X_4) is the unique K3 surface with Picard number 20 and discriminant 3 (resp. 4), which are called the most algebraic K3 surfaces by Vinberg and have infinite automorphism groups (by Shioda-Inose and Vinberg).

基变换型Enriques曲面及有理椭圆曲面上的Severi簇

Severi varieties on Enriques surfaces of base change type and on rational elliptic surfaces

本文证明了基变换型Enriques曲面上任意亏格曲线的Severi簇可数族非空，并构造了有理椭圆曲面上特殊超丰对数Severi簇的例子。

Fundamental groups of open K3 surfaces, Enriques surfaces and Fano 3-folds

We investigate when the fundamental group of the smooth part of a K3 surface or Enriques surface with Du Val singularities, is finite. As a corollary we give an effective upper bound for the order of the fundamental group of the smooth part of a certain Fano 3-fold. This result supports Conjecture A below, while Conjecture A (or alternatively the rational connectedness conjecture in [KoMiMo] which is still open when the dimension is at least 4) would imply that every log terminal Fano variety has a finite fundamental group (now a Theorem of S. Takayama).

Analytic torsion for log-Enriques surfaces and Borcherds product

We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type $\frac{1}{4}(1,1)$. The moduli space of such log-Enriques surfaces with $k$ singular points is a modular variety of orthogonal type %of dimension $10-k$ associated with a unimodular lattice of signature $(2,10-k)$. We prove that the invariant, viewed as a function on the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension $2$. As a consequence, th…

Log-canonical threshold for curves on a smooth surface

It is shown that the log-canonical threshold of a curve with an isolated singularity is computed by the term ideal of the curve in a suitable system of local parameters at the singularity. The proof uses the Enriques diagram of the singularity and shows that the log-canonical threshold depends only on a non-degenerate path of that diagram.

Automorphisms of K3 surfaces and their applications

This paper is a survey about $K3$ surfaces with an automorphism and log rational surfaces, in particular, log del Pezzo surfaces and log Enriques surfaces. It is also a reproduction on my talk at "Mathematical structures of integrable systems and their applications" held at Research Institute for Mathematical Sciences in September 2018.

Classification of exceptional log Del Pezzo surfaces with delta=1

The exceptional log Del Pezzo surfaces with delta=1 are classified.

Approximating delta invariants in the sense of complements of plt type

In this note, we will show that delta invariant of a log Fano pair can be approximated by lc places of complements of plt type if it is no greater than one. Give a log Fano pair with delta invariant no greater than one, under the assumption that delta invariant of the log Fano pair can be approximated by lc places of bounded complements of plt type, we show the existence of the divisorial valuation computing delta invariant of this log Fano pair.

Algebraic surfaces with quotient singularities - including some discussion on automorphisms and fundamental groups

We survey some recent progress in the study of algebraic varieties X with log terminal singularities, especially, the uni-ruledness of the smooth locus X^0 of X, the fundamental group of X^0 and the automorphisms group on (smooth or singular) X when dim X = 2. The full automorphism groups of a few interesting types of K3 surfaces are described, mainly by Keum-Kondo. We conjecture that when X is Q-Fano then X^0 has a finite fundamental group, which had been proved if either dim X < 3 or the Fano index is bigger than dim X - 2. We also conjecture that when X is a log Enriques (e.g. a normal K3 o…

On the Log Discrepancies in Toric Mori Contractions

It was conjectured by McKernan and Shokurov that for all Mori contractions from X to Y of given dimensions, for any positive epsilon there is a positive delta, such that if X is epsilon-log terminal, then Y is delta-log terminal. We prove this conjecture in the toric case and discuss the dependence of delta on epsilon, which seems mysterious.

Quotients of K3 Surfaces Modulo Involutions

Let X be a K3 surface with an involution g which has non-empty fixed locus X^g and acts non-trivially on a non-zero holomorphic 2-form. We shall construct all such pairs (X, g) in a canonical way, from some better known double coverings of log del Pezzo surfaces of index at most 2 or rational elliptic surfaces, and construct the only family of each of the three extremal cases where X^g contains 10 (maximum possible) curves. We also classify rational log Enriques surfaces of index 2. Our approach is more geometrical rather than lattice-theoretical (see Nikulin's paper for the latter approach).

反典范度 $\geq 5$ 的弱 del Pezzo 曲面的局部 delta 不变量

Local delta invariants of weak del Pezzo surfaces with the anti-canonical degree $\geq 5$

本文基于delta不变量对K-稳定性的判别准则，确定了反典范度 $\geq 5$ 的所有弱 del Pezzo 曲面的整个局部 delta 不变量。

带符号排列多面体、delta-拟阵及其他

Signed permutohedra, delta-matroids, and beyond

本文建立了B型排列多面体环面簇的代数几何与delta-拟阵组合学的联系，计算了B型广义排列多面体的体积和格点计数，并利用热带Hodge理论确立了包含可实现delta-拟阵在内的该类拟阵的类Tutte不变量的对数凹性。