Automorphism groups of positive entropy on minimal projective varieties

We determine the geometric structure of a minimal projective threefold having two ‘independent and commutative’ automorphisms of positive topological entropy, and generalize this result to higher-dimensional smooth minimal pairs (X,G). As a consequence, we give an effective lower bound for the first dynamical degree of these automorphisms of X fitting the ‘boundary case’.     

Covers of algebraic varieties V: Examples of covers of degree 8 and 9 as catalecticant loci

Let Y be an irreducible, noetherian, separated scheme over an algebraically closed field k. A cover is flat and finite morphism, its degree being d?[k(X):k(Y)]. In this paper we give some new methods for constructing covers of degree d=8,9.     

Lengths of chains of minimal rational curves on Fano manifolds

In this paper, we consider a natural question how many minimal rational curves are needed to join two general points on a Fano manifold X of Picard number 1. In particular, we study the minimal length of such chains in the cases where the dimension of X is at most 5, the coindex of X is at most 3 and X equips with a structure of a double cover. As an application, we give a better bound on the degree of Fano 5-folds of Picard number 1.     

Convex Pencils of Real Quadratic Forms

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space ?P ⁿ (e.g. X is the intersection of two real quadrics). We give explicit formulas for its Betti numbers and for those of its double cover in the sphere S ⁿ ; we also give similar formulas for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X)??2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular?X. In the nondegenerate case we also prove the bound on each specific Betti number b _k (X)??2(k+2).     

Singularities of the Modular Curve

Let X₀(ℓ) be the modular curve, parameterizing cyclic isogenies of degree ℓ, and Z₀(ℓ) be its plane model, given by the classical modular equation Φ_ℓ(X, Y)=0. We prove that all singularities of Z₀(ℓ), except two cusps, are intersections of smooth branches, and evaluate the order of contact of these branches.     

A conforming decomposition theorem,a piecewise linear theorem of the alternative,and scalings of matrices satisfying lower and upper bounds

A scaling of a nonnegative matrixA is a matrixXAY ^?1, whereX andY are nonsingular, nonnegative diagonal matrices. Some condition may be imposed on the scaling, for example, whenA is square,X=Y or detX=detY. We characterize matrices for which there exists a scaling that satisfies predetermined upper and lower bound. Our principal tools are a piecewise linear theorem of the alternative and a theorem decomposing a solution of a system of equations as a sum of minimal support solutions which conform with the given solutions.     

Frobenius pull backs of vector bundles in higher dimensions

We prove that for a smooth projective variety X of arbitrary dimension and for a vector bundle E over X, the Harder?CNarasimhan filtration of a Frobenius pull back of E is a refinement of the Frobenius pull back of the Harder?CNarasimhan filtration of E, provided there is a lower bound on the characteristic p (in terms of rank of E and the slope of the destabilizing sheaf of the cotangent bundle of X). We also recall some examples, due to Raynaud and Monsky, to show that some lower bound on p is necessary. We also give a bound on the instability degree of the Frobenius pull back of E in terms of the instability degree of E and well defined invariants of X.     

On a question of Sean Keel

Let X be a smooth projective surface defined over , and let L be a line bundle over X such that for every complete curve Y contained in X. A question of Keel asks whether L is ample. If X is a P¹-bundle over a curve, we prove that this question has an affirmative answer.     

A criterion for ample vector bundles over a curve in positive characteristic

Let X be a smooth projective curve defined over an algebraically closed field of positive characteristic. We give a necessary and sufficient condition for a vector bundle over X to be ample. This generalizes a criterion given by Lange in [Math. Ann. 238 (1978) 193-202] for a rank two vector bundle over X to be ample.     

On additive polynomials and certain maximal curves

We show that a maximal curve over F_q² given by an equation A(X)=F(Y), where A(X)∈F_q²[X] is additive and separable and where F(Y)∈F_q²[Y] has degree m prime to the characteristic p, is such that all roots of A(X) belong to F_q². In the particular case where F(Y)=Y^m, we show that the degree m is a divisor of q+1.     

Linearizability and integrability of vector fields via commutation

In this paper, we consider complex smooth and analytic vector fields X in a neighborhood of a nondegenerate singular point. It is proved the equivalence between linearizability and commutation, i.e., the existence of a commuting vector field Y such that the Lie brackets [X,Y]≡0. For complex smooth and analytic vector fields in the plane and in a neighborhood of a nondegenerate singular point, it is also proved the equivalence between integrability and the existence of a smooth vector field Y, such that Y is a normalizer of X, i.e., [X,Y]=μX.     

Second order homological obstructions on real algebraic manifolds

Let Y be a compact nonsingular real algebraic set whose homology classes (over Z/2) are represented by Zariski closed subsets. It is well known that every smooth map from a compact smooth manifold to Y is unoriented bordant to a regular map. In this paper, we show how to construct smooth maps from compact nonsingular real algebraic sets to Y not homotopic to any regular map starting from a nonzero homology class of Y of positive degree. We use these maps to obtain obstructions to the existence of local algebraic tubular neighborhoods of algebraic submanifolds of Rn and to study some algebro-homological properties of rational real algebraic manifolds.     

A remark on Fano 4-folds having (3,1)-type extremal contractions

Let π : X → Y be the blow-up of a four-dimensional complex manifold Y along a smooth curve C. Assume that X is a Fano manifold and has another (3,1)-type extremal contraction ${\varphi : X \to Z}$ whose exceptional divisor meet that of the blow-up π : X → Y. We show that if the exceptional divisor of ${\varphi}$ is smooth, then Y is isomorphic to four-dimensional projective space and C is an elliptic curve of degree 4.     

Motivation for Hodge cycles

Given two smooth projective varieties X and Y, X is defined to motivate Y if the motive of Y is contained in the tensor category generated by X. Some techniques are given for checking this condition. It is shown that in a number of cases moduli spaces of sheaves over curves or surfaces are motivated by the underlying curve or surface. This is used to check the Hodge and related conjectures for some of these examples.     

Basic deformation theory of smooth formal schemes

We provide the main results of a deformation theory of smooth formal schemes as defined in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Algebra 35 (2007) 1341-1367]. Smoothness is defined by the local existence of infinitesimal liftings. Our first result is the existence of an obstruction in a certain Ext¹ group whose vanishing guarantees the existence of global liftings of morphisms. Next, given a smooth morphism f₀:X₀→Y₀ of noetherian formal schemes and a closed immersion Y₀?Y given by a square zero ideal I, we prove that the set of isomorphism classes of smooth formal schemes lifting X₀ over Y is classified by and that there exists an element in which vanishes if and only if there exists a smooth formal scheme lifting X₀ over Y.     

Flows on open manifolds with positively Lagrange stable trajectories

We give the conditions for a flow generated by a smooth vector field X which guarantee that every smooth vectorfield Y in some C⁰-neighborhood of X defines a flow with positively Lagrange stable trajectories.     

Relative Gromov-Witten invariants and the mirror formula

Let X be a smooth complex projective variety, and let be a smooth very ample hypersurface such that is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry. Received: 11 July 2001 / Published online: 4 February 2003 Funded by the DFG scholarships Ga 636/1–1 and Ga 636/1–2.     

Optimal designs for weighted approximation and integration of stochastic processes on [0,∞)

We study minimal errors and optimal designs for weighted L₂-approximation and weighted integration of Gaussian stochastic processes X defined on the half-line [0,∞). Under some regularity conditions, we obtain sharp bounds for the minimal errors for approximation and upper bounds for the minimal errors for integration. The upper bounds are proven constructively for approximation and non-constructively for integration. For integration of the r-fold integrated Brownian motion, the upper bound is proven constructively and we have a matching lower bound.     

On line bundles over real algebraic curves

Let X be a geometrically irreducible smooth projective curve defined over the real numbers. Let nX be the number of connected components of the locus of real points of X. Let x₁,…,x? be real points from ? distinct components, with ?<nX. We prove that the divisor x₁+?+x? is rigid. We also give a very simple proof of the Harnack's inequality.