We study perverse coherent sheaves on the resolution of rational double points. As examples, we consider rational double points on 2-dimensional moduli spaces of stable sheaves on $K3$ and elliptic surfaces. Then we show that perverse coherent sheaves appear in the theory of Fourier–Mukai transforms. As an application, we generalize the Fourier–Mukai duality for $K3$ surfaces to our situation.

Let $X$ be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of $X$ is a lattice with respect to the Beauville–Bogomolov pairing. A divisor $E$ on $X$ is called a prime exceptional divisor if $E$ is reduced and irreducible and of negative Beauville–Bogomolov degree.

Let $E$ be a prime exceptional divisor on $X$. We first observe that associated to $E$ is a monodromy involution of the integral cohomology $H^{\ast }(X,\mathbb{Z})$, which acts on the second cohomology lattice as the reflection by the cohomology class $\left[E\right]$ of $E$.

We then specialize to the case where $X$ is deformation equivalent to the Hilbert scheme of length $n$ zero-dimensional subschemes of a $K3$ surface, $n\ge 2$. We determine the set of classes of exceptional divisors on $X$. This leads to a determination of the closure of the movable cone of $X$.

We determine the automorphism groups of torus invariant self-dual structures defined by Joyce on the connected sum of copies of the complex projective plane. We determine, actually, the automorphism groups of the associated twistor spaces by using the results of our previous work. When the self-dual structures of Joyce and LeBrun coincide, our results recover the recent results of Honda and Viaclovsky on the automorphism groups of LeBrun’s self-dual structures.

In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover, we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links, and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann–Hilbert correspondence from the moduli space of the connections to the moduli space of the associated generalized monodromy data gives an analytic isomorphism. This shows that differential systems arising from (generalized) isomonodromic deformations of corresponding unramified irregular singular parabolic connections admit the geometric Painlevé property as in the regular singular cases proved generally.

After introducing an equivalence problem for symplectic singularities, we formulate an algebraic version of such a problem. Let $X$ be an affine normal variety with a ${\mathbf{C}}^{\ast }$-action having only positive weights. Assume that the regular part $X_{reg}$ of $X$ admits an algebraic symplectic $2$-form $\omega$ with weight $l$. Our main theorem asserts that any algebraic symplectic $2$-form $\omega \text{'}$ on $X_{reg}$ of weight $l$ is equivalent to $\omega$ up to a ${\mathbf{C}}^{\ast }$-equivariant automorphism of $X$ if $l\ne 0$. When $l=0$ we have a counterexample to this statement. In the latter half of the article, we discuss the equivalence problem up to constant. We associate to $X$ a projective variety $\mathbf{P}\left(X\right)$ and prove that $\mathbf{P}\left(X\right)$ has a contact orbifold structure. Moreover, when $X$ has canonical singularities, the contact orbifold structure is rigid under a small deformation. The equivalence problem is then reduced to the uniqueness of the contact structures. In most examples the symplectic structures turn out to be unique up to constant with very few exceptions. In the final section we pose a splitting conjecture for symplectic singularities.

We generalize Tango’s theorem on the Frobenius map of the first cohomology groups to higher-dimensional algebraic varieties in characteristic $p>0$. As an application we construct counterexamples to Kodaira vanishing in higher dimension and prove the Ramanujam-type vanishing on surfaces which are not of general type when $p\ge 5$.