Related papers: A Witt Nadel vanishing theorem for threefolds

We show that if $\mathcal{F}$ is an algebraically integrable foliation on a $\mathbb{Q}$-factorial normal projective variety $X$, $ A, B \geq 0$ are $\mathbb{Q}$-divisors on $X$ with $A$ ample such that $(\mathcal{F}, B)$ is foliated dlt…

We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to -X whose class numbers are indivisible by a given prime and whose…

This is a short report on our new vanishing theorems for projective morphisms between complex analytic spaces. We established a complex analytic generalization of Koll\'ar's torsion-freeness and vanishing theorem for analytic simple normal…

The year 2017 marks the 80th anniversary of Witt's famous paper containing key results, including the Witt cancellation theorem, which form the foundation for the algebraic theory of quadratic forms. We pay homage to this paper by…

The first aim of this note is to give a concise, but complete and self-contained, presentation of the fundamental theorems of Mori theory - the nonvanishing, base point free, rationality and cone theorems - using modern methods of…

For a square-free integer $t$, Byeon \cite{byeon} proved the existence of infinitely many pairs of quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{tD})$ with $D > 0$ such that the class numbers of all of them are indivisible…

We show that the Cappell-Shaneson version of Pick's theorem for simple lattice polytopes is a consequence of a general relation between characteristic numbers of virtual submanifolds dual to the characteristic classes of a stably almost…

We provide a detailed proof of the validity of the Minimal Model Program for threefolds over excellent Dedekind separated schemes whose residue fields do not have characteristic 2 or 3.

Let $\mathcal Q_D$ be the set of moduli points on Siegel's modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order $\mathcal O$ in an indefinite quaternion algebra of…

In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we provide $L^2$-Dolbeault fine resolutions and isomorphisms, and $L^2$-estimates, for holomorphic line bundles on…

We establish the equivalence of three notions of $\mathbb{F}_q$-rational points on weighted projective spaces $\mathbb{P}_{\mathbf{w}}^n$ and derive explicit combinatorial formulas for their enumeration, leveraging Burnside's lemma and gcd…

We show that there exists a positive real number $\delta>0$ such that for any normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less…

Let $X$ be a smooth complex projective rationally connected threefold with $-K_X$ nef and not semi-ample. In our previous work, we classified all such threefolds when $|{-}K_X|$ has no fixed divisor. In this paper, we continue our…

A proof based on reduction to finite fields of Esnault-Viehweg's stronger version of Sommese Vanishing Theorem for $k$-ample line bundles is given. This result is used to give different proofs of isotriviality results of A. Parshin and L.…

Let $L$ be a line bundle on a K3 or Enriques surface. We give a vanishing theorem for $H^1(L)$ that, unlike most vanishing theorems, gives necessary and sufficient geometrical conditions for the vanishing. This result is essential in our…

We prove existence of flips for log canonical foliated pairs of rank one on a Q-factorial projective klt threefold. This, in particular, provides a proof of the existence of a minimal model for a rank one foliation on a threefold for a…

We prove the following results for projective klt pairs of dimension $3$ over an algebraically closed field of char $p>5$: the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal models, termination…

A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed characteristic.) It…

We show that if a universal quadratic form exists over an infinite degree, totally real extension of the field of rationals $\mathbb{Q}$, then the set of totally positive integers in the extension does not have the Northcott property. In…

As a special case of a conjecture by Schwede and Smith, we prove that a smooth complex projective threefold with nef anti-canonical divisor is weak Fano if it is of globally $F$-regular type.