Related papers: Elementary counterexamples to Kodaira vanishing in…

We present the extended Kuranishi space for Kodaira surface as a non-trivial example to Kontsevich and Barannikov's extended deformation theory. We provide a non-trivial example of Hertling-Manin's weak Frobenius manifold. In addition, we…

We present counterexamples to Fujita's conjecture in positive characteristics. Precisely, we show that over any algebraically closed field $k$ of characteristic $p>0$ and for any positive integer $m$, there exists a smooth projective…

We consider compactifications of the space of triples of distinct points in projective $n$-space. One such space is a singular variety of configurations of points and lines; another is the smooth compactification of Fulton and MacPherson;…

We consider the following conjecture: if X is a smooth projective variety over a field of characteristic zero, then there is a dense set of reductions X_s of X to positive characteristic such that the action of the Frobenius morphism on the…

For any n>3, we provide examples of curves lying on K3 surfaces and vector bundles on those curves which invalidate Mercat's conjecture for rank n bundles.

When $I$ is the radical homogeneous ideal of a finite set of points in projective $N$-space, ${\bf P}^N$, over a field $K$, it has been conjectured that $I^{(rN-N+1)}$ should be contained in $I^r$ for all $r\geq 1$. Recent counterexamples…

In this paper, we study an algebraic fiber space in positive characteristic whose generic fiber $F$ has finitely generated canonical ring and sufficiently large Frobenius stable canonical ring. An example of such a case is when $F$ is…

We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a…

We establish strong vanishing theorems for line bundles on wonderful varieties of hyperplane arrangements, and we show that the resulting positivity properties of Euler characteristics extend to all matroids. We achieve this by showing that…

We prove that a smooth projective variety of dimension n is isomorphic to projective n-space iff the canonical class is -(n+1)-times an ample divisor. In characteristic zero this was proved by Kobayashi-Ochiai. We also extend the second…

We give a new proof that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point, first proven by Popa and Schnell using generic vanishing theorems for Hodge modules. Our proof relies on…

In the first part of this paper we prove a vanishing criterion for higher direct images of projective families of line bundles on a Cohen-Macaulay variety X. The result involves certain first-order deformations of certain curves on X, and…

We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective n-dimensional varieties, n\geq 4, onto smooth projective varieties, arising from classical projective geometry and…

Let X be a smooth projective curve of genus g>1 defined over an algebraically closed field k of characteristic p>0. Let M_X(r) be the moduli space of semi-stable rank r vector bundles with fixed trivial determinant. The relative Frobenius…

In this article we prove a relative Kawamata-Viehweg vanishing-type theorem for PLT $3$-folds in characteristic $p>5$. We use this to prove the normality of minimal log canonical centers and the adjunction formula for codimension $2$…

Let $\mathcal{O}_K$ be a discrete valuation ring with fraction field $K$ of characteristic $0$ and algebraically closed residue field $k$ of characteristic $p > 0$. Let $A/K$ be an abelian variety of dimension $g$ with a $K$-rational point…

We construct explicit counterexamples that show that it is impossible to get any remainder, other than the classical ones, in the Wiener-Ikehara theorem and the Ingham-Karamata theorem under just an additional analytic continuation…

Assuming the Morrison-Kawamata cone conjecture for the generic fiber of a Calabi-Yau fibration and the abundance conjecture, we show (1) the finiteness of minimal models, (2) the existence of a weak rational polyhedral fundamental domain…

This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and…

We show the Frobenius pullback of a general semi-stable vector bundle in the moduli space of vector bundles with fixed rank and degree is still semi-stable by deformation trick. We then present several applications of the main theorem.