We give a brief survey of three dimensional divisorial contractions and provide many explicit examples.

These are the lecture notes based on earlier papers with some additional new results. New and simple proofs are given for local freeness theorem and the semipositivity theorem. A decomposition theorem for higher direct images of dualizing sheaves of Kollár is extended to the sheaves of differential forms of arbitrary degrees in the case of a well prepared birational model. We will also prove the log versions of some of the results, i.e., the case where we allow horizontal boundary components.

In this note, we relate the equivariant GW invariants of the symmetric product stacks of any nonsingular toric surface $X$ in genus zero to the equivariant relative GW invariants of the threefold $X \times \mathbb{P}^1$ in all genera. We give an example for which an equivalence between these two theories exists.

In this article we survey the recent developments in ADHM sheaf theory on a smooth projective variety $X$. When $X$ is a curve the theory is an alternative construction of stable pair theory of Pandharipande and Thomas or Gromov–Witten theory on local curve geometries. The construction relies on relative Beilinson spectral sequence and Fourier–Mukai transformation. We will present some applications of the theory, including the derivations of the wallcrossing formulas, higher rank Donaldson–Thomas invariants on local curves, and the coholomogies of the moduli of stable Hitchin pairs.

We discuss a difference between the rational and the real nonvanishing conjecture for pseudo-effective log canonical divisors of log canonical pairs. We also show the log non-vanishing theorem for rationally connected varieties under assuming Shokurov's ACC conjectures.

We show that if $(X, \Theta)$ is a PPAV over an algebraically closed field of characteristic $p \gt 0$ and $D \in |m\Theta |$, then $(X, \frac 1m D)$ is a limit of strongly $F$-regular pairs and in particular $\mathrm{mult}_x (D) \leq m \cdot \dim X$ for any $x \in X$.

The varieties of minimal rational tangents play an important role in the geometry of uniruled projective manifolds. The goal of this paper is to exhibit their role in the symplectic geometry of the cotangent bundles of uniruled projective manifolds. More precisely, let $X$ be a uniruled projective manifold satisfying the assumption that the VMRT at a general point is smooth. We show that the total family of dual cones of the varieties of minimal rational tangents is a coisotropic subvariety in $T^* (X)$. Furthermore, the closure of a general leaf of the null foliation of this coisotropic subvariety is an immersed projective space of dimension $\delta + 1$ where $\delta$ is the dual defect of the variety of minimal rational tangents at a general point. When $\delta = 0$, the symplectic reduction of the coisotropic variety can be realized as a subbundle of the cotangent bundle $T^* (\mathcal{K})$ of the parameter space $\mathcal{K}$ of the rational curves.

We classify all $\mathbb{Q}$-homology projective planes with $A_1$- or $A_2$-singularities (and with no worse singularities). It turns out that such a surface is isomorphic to a global quotient $X/G$, where $X$ is a fake projective plane or the complex projective plane and $G$ a finite abelian group of bi-holomorphic automorphisms. There are only finitely many such surfaces.

We consider the deformation spaces of some singular product-quotient surfaces $X= (C_1 \times C_2)/G$, where the curves $C_i$ have genus 3 and the group $G$ is isomorphic to $\mathbb{Z}_4$. As a by-product, we give a new construction of Todorov surfaces with $p_g = 1$, $q = 0$ and $2 \leq K^2 \leq 8$ by using $\mathbb{Q}$-Gorenstein smoothings.

We prove that there is a projective K3 surface admitting a (fixed point) free automorphism of positive entropy and that no smooth compact Kähler surface other than projective K3 surfaces and their blow up admits such an automorphism.

I describe the projective resolution of a codimension 4 Gorenstein ideal, aiming to extend Buchsbaum and Eisenbud's famous result in codimension 3. The main result is a structure theorem stating that the ideal is determined by its $(k + 1) \times 2k$ matrix of first syzygies, viewed as a morphism from the ambient regular space to the Spin-Hom variety $\mathrm{SpH}_k \subset \mathrm{Mat}(k + 1, 2k)$. This is a general result encapsulating some theoretical aspects of the problem, but, as it stands, is still some way from tractable applications.