Related papers: On the relative Minimal Model Program for fourfold…
We refine Theorem A due to Gursky \cite{G3}. As applications, we give some rigidity theorems on four-manifolds with postive Yamabe constant. In particular, these rigidity theorems are sharp for our conditions have the additional properties…
We discuss aspects of the algebraic geometry of compact non-commutative Calabi-Yau manifolds. In this setting, it is appropriate to consider local holomorphic algebras which can be glued together into a compact Calabi-Yau algebra. We…
A stable generalized complex structure is one that is generically symplectic but degenerates along a real codimension two submanifold, where it defines a generalized Calabi-Yau structure. We introduce a Lie algebroid which allows us to view…
We study compactifications of F-theory on certain Calabi--Yau threefolds. We find that $N=2$ dualities of type II/heterotic strings in 4 dimensions get promoted to $N=1$ dualities between heterotic string and F-theory in 6 dimensions. The…
We completely determine, up to homeomorphism, which simply connected compact oriented 4-manifolds admit scalar-flat, anti-self-dual Riemannian metrics. The key new ingredient is a proof that the connected sum of five reverse-oriented…
We give upper bounds on the size of the gap between the constant term and the next non-zero Fourier coefficient of an entire modular form of given weight for \Gamma_0(2). Numerical evidence indicates that a sharper bound holds for the…
This paper is motivated by the question of how motivic Donaldson--Thomas invariants behave in families. We compute the invariants for some simple families of noncommutative Calabi--Yau threefolds, defined by quivers with homogeneous…
We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich-Polishchuk, Kuznetsov, Lieblich, and…
We develop new techniques to study regularity questions for moduli spaces of pseudoholomorphic curves that are multiply covered. Among the main results, we show that unbranched multiple covers of closed holomorphic curves are generically…
This paper presents a rigidity property of the exceptional locus of some kind of small birational contractions. An application in the context of geometric transitions and Calabi-Yau threefolds moduli space is then given, with some physical…
We show that, in any dimension greater than one, there are an abelian variety, a smooth rational variety and a Calabi-Yau manifold, with primitive birational automorphisms of first dynamical degree $>1$. We also show that there are smooth…
In this note we prove the semiampleness conjecture for klt Calabi--Yau surface pairs over an excellent base ring. As applications we deduce that generalised abundance and Serrano's conjecture hold for surfaces. Finally, we study the…
We give a simplified and more algebraic proof of the finiteness of the families of Calabi-Yau n-folds with non-vanishing of Yukawa-coupling over a fixed base curve and with fixed degeneration locus. We also give a generalization of this…
Complex structure moduli of a Calabi-Yau threefold in $N=1$ supersymmetric heterotic compactifications can be stabilized by holomorphic vector bundles. The stabilized moduli are determined by a computation of Atiyah class. In this paper, we…
Let K be the function field of a connected regular scheme S of dimension 1, and let f : X -> Y be a finite cover of projective smooth and geometrically connected curves over K with g(X) greater or equal to 2. Suppose that f can be extended…
We perform a Kaluza-Klein reduction of IIB supergravity including purely gravitational $\alpha '^3$-corrections on a Calabi-Yau threefold, and perform the orientifold projection accounting for the presence of $O3/O7$-planes. We consider…
We introduce nonlocal minimal surfaces on closed manifolds and establish a far-reaching Yau-type result: in every closed, $n$-dimensional Riemannian manifold we construct infinitely many nonlocal $s$-minimal surfaces. We prove that, when…
We study type III contractions of Calabi-Yau threefolds containing a ruled surface over a smooth curve. We discuss the conditions necessary for the image threefold to by smoothable. We describe the change in Hodge numbers caused by this…
We extend two results on Chow (semi-)stability to positive characteristics. One is on the stability of non-singular projective hypersurfaces of degree greater than 2, and the other is the criterion by Y. Lee in terms of log canonical…
By the results of the author and Chiantini in Math.AG/0110102, on a general quintic threefold $X \subset {\mathbf P}^4$ the minimum integer $p$ for which there exists a positive dimensional family of irreducible rank $p$ vector bundles on…