Degree of the divisor of solutions of a differential equation on a projective variety

Using the data schemes from [1] we give a rigorous definition of algebraic differential equations on the complex projective space Pⁿ. For an algebraic subvariety S?Pⁿ, we present an explicit formula for the degree of the divisor of solutions of a differential equation on S and give some examples of applications. We extend the technique and result to the real case.     

Kodaira fibrations and beyond: methods for moduli theory

Kodaira fibred surfaces are remarkable examples of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. Our tour through algebraic surfaces and their moduli (with results valid also for higher dimensional varieties) deals with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, especially a recent answer given (in joint work with Dettweiler) to a long standing question posed by Fujita. In the landscape of our tour, Galois coverings, deformations and rigid manifolds (there are by the way rigid Kodaira fibrations), projective classifying spaces, the action of the absolute Galois group on moduli spaces, stand also in the forefront. These questions lead to interesting algebraic surfaces, for instance remarkable surfaces constructed from VHS, surfaces isogenous to a product with automorphisms acting trivially on cohomology, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces.     

The Hodge theory of algebraic maps

We give a geometric proof of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber for the direct image of the intersection cohomology complex under a proper map of complex algebraic varieties. The method rests on new Hodge-theoretic results on the cohomology of projective varieties which extend naturally the classical theory and provide new applications.     

On K-stability of reductive varieties

G. Tian and S.K. Donaldson formulated a conjecture relating GIT stability of a polarized algebraic variety to the existence of a Kähler metric of constant scalar curvature. In [D3] Donaldson partially confirmed it in the case of projective toric varieties. In this paper we extend Donaldson’s results and computations to a new case, that of reductive varieties.Received: November 2003 Revision: January 2004 Accepted: January 2004     

THE GROWTH OF SOLUTIONS OF SYSTEMS OF COMPLEX NONLINEAR ALGEBRAIC DIFFERENTIAL EQUATIONS

We investigate the problem of growth order of solutions of a type of systems of non-linear algebraic differential equations, and extend some results of the growth order of solutions of algebraic differential equations to systems of algebraic differential equations.     

On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety

We extend the lower bounds on the complexity of computing Betti numbers proved in [P. Bürgisser, F. Cucker, Counting complexity classes for numeric computations II: algebraic and semialgebraic sets, J. Complexity 22 (2006) 147–191] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACE-hard. Then we prove PSPACE-hardness for the more general problem of deciding whether the Betti number of fixed order of a complex affine or projective variety is at most some given integer.     

Homological projective duality

We introduce a notion of homological projective duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are homologically projectively dual, then we prove that the orthogonal linear sections of X and Y admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate homological projective duality for projectivizations of vector bundles.     

The problem of lacunas and analysis on root systems

A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens' principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and Gårding (1970-73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-Gårding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.

     

微分周形式与稀疏微分结式

代数周(Chow)形式和代数结式是代数几何的基本概念,同时还是消去理论的强大工具.一个自然的想法是在微分代数几何中发展相应的周形式和结式理论.但是由于微分结构的复杂性,在本文的研究工作之前,微分结式只有部分结果,而微分周形式与稀疏微分结式理论一直没有得到发展.本文的主要结果包括:第一,发展一般(generic)情形的微分相交理论,作为应用,证明一般情形的微分维数猜想.第二,初步建立微分周形式理论.对不可约微分代数簇定义微分周形式并证明其基本性质,特别地,给出微分周形式的Poisson分解公式,引入微分代数簇的主微分次数这一不变量并证明一类微分代数闭链的周簇和周坐标的存在性.作为应用,首次严格定义微分结式,证明其基本性质.第三,初步建立稀疏微分结式理论.引入Laurent微分本性系统的概念,定义稀疏微分结式,证明其基本性质,特别地,引入微分环面簇的概念,给出稀疏微分结式阶数和次数界的估计,并基于此给出计算稀疏微分结式的单指数时间算法.     

A formal proof of the projective Eisenbud–Evans–Storch theorem

We extend a constructive proof of the Eisenbud–Evans–Storch theorem, developed in a previous work by Coquand, Schuster, and Lombardi, from the affine to the projective case. The main tool is that of distributive lattices, which allows us to replace the classical topological arguments by more algebraic and constructive ones. Given a suitable graded ring, we work in the distributive lattice in which the prime filters correspond to the homogeneous prime ideals. The proof presented here is one of the first examples of concrete results obtained using this tool.     

Geometric nullstellensatz and symbolic powers on arbitrary varieties

In recent years, a multiplier ideal defined on arbitrary varieties, so called Mather–Jacobian multiplier ideal, has been developed independently by Ein–Ishii–Mustata and de Fernex–Docampo. With this new tool, we have a chance of extending some classical results proved in nonsingular case to arbitrary varieties to establish their general forms. In this paper, we first extend a result of geometric nullstellensatz due to Ein–Lazarsfeld in nonsingular case to any projective varieties. Then we prove a result on comparison of symbolic powers with ordinary powers on any varieties, which extends results of Ein–Lazarsfeld–Smith and Hochster–Huneke.     

Symmetric Orientations of Dividing T-curves

In 1974, Rokhlim introduced complex orientations for nonsingular real algebraic plane projective curves of type I. Here we give a definition of symmetric orientations and of "type" for T-curves which are PL-curves constructed using a combinatorial method called T-construction. An important aspect of T-construction is that, under particular conditions, the constructed T-curve has the isotopy type of a nonsingular real algebraic plane projective curve. T-construction is in fact a particular case of the method of construction of real algebraic projective varieties due to O. Ya. Viro. We prove that if an algebraic curve is associated to a T-curve by the Viro process, then the type of the T-curve coincides with the type of the algebraic curve and its symmetric orientations are complex orientations as defined by Rokhlin. The main result of this paper is the classification theorem for T-curves of type I.     

Projective Blaschke manifolds

In this paper we define the concept of projective Blaschke manifolds and extend the theory of equiaffine differential geometry to the projective Blaschke manifolds. partially supported by NSFC 10771146 and RFDP(20060610004)     

Hilbert-Kunz functions and multiplicities for full flag varieties and elliptic curves

We compute the Hilbert-Kunz functions and multiplicities for certain projective embeddings of flag varieties G/B and elliptic curves, over algebraically closed fields of positive characteristics. The group theoretic nature of both these classes of examples is used, albeit in different ways, to explicitly describe the cokernels in each degree of the Frobenius twisted multiplication maps for the corresponding graded rings. This detailed information also enables us to extend our results to arbitrary products of such varieties.     

Tate''s Conjecture, Algebraic Cycles and Rational K-Theory in Characteristic p

The purpose of this article is to discuss conjectures on motives, algebraic cycles and K-theory of smooth projective varieties over finite fields. We give a characterization of Tate's conjecture in terms of motives and their Frobenius endomorphism. This is used to prove that if Tate's conjecture holds and rational and numerical equivalence over finite fields agree, then higher rational K-groups of smooth projective varieties over finite fields vanish (Parshin's conjecture). Parshin's conjecture in turn implies a conjecture of Beilinson and Kahn giving bounds on rational K-groups of fields in finite characteristic. We derive further consequences from this result.     

The algebraic degree of semidefinite programming

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.     

Estimates on the Distribution of the Condition Number of Singular Matrices

We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can exhibit an average Bezout Equality for equidimensional varieties. We also state an upper bound for the volume of a tube about a projective variety. As a main outcome, we prove an upper bound estimate for the volume of the intersection of a tube with an equidimensional projective algebraic variety. We apply these techniques to exhibit upper bounds for the probability distribution of the generalized condition number of singular complex matrices.     

An application of the Krasnoselskii theorem to systems of algebraic equations

Based on the Krasnoselskii theorem, we study the existence, multiplicity and nonexistence of positive solutions of general systems of nonlinear algebraic equations under superlinearity and sublinearity conditions. Systems of nonlinear algebraic equations often arise from studies of differential and difference equations. Our results significantly extend and improve those in the literature. A?number of examples and open questions are given to illustrate these results.     

Towards an Algebraic Proof of Deligne’s Regularity Criterion. An Informal Survey of Open Problems

We review the notion of regular singular point of a linear differential equation with meromorphic coefficients, from the viewpoint of algebraic geometry. We give several equivalent definitions of regularity along a divisor for a meromorphic connection on a complex algebraic manifold and discuss the global birational theory of fuchsian differential modules over a field of algebraic functions. We describe the generalized algebraic version of Deligne’s canonical extension, constructed in [1, I.4]. Our main interest lies in the algebraic form of Deligne’s regularity criterion [2, II.4.4 (iii)], asserting that, on a normal compactification, only one codimensional components of the locus at infinity need to be considered. If one considers the purely algebraic nature of the statement, it is surprising that the only existing proof of this criterion is the transcendental argument given by Deligne in his corrigendum to loc. cit. dated April 1971. The algebraic proof given in our book [1, I.5.4] is also incorrect, as J. Bernstein kindly indicated to us.We introduce some notions of logarithmic geometry to let the reader appreciate Bernstein’s (counter)examples to some statements in our book [1]. Standard methods of generic projection in projective spaces reduce the question to a two-dimensional puzzle. We report on ongoing correspondence with Y. André and N. Tsuzuki, leading to partial results and provide examples indicating the subtlety of the problem. Lecture held in the Seminario Matematico e Fisico on January 31, 2005 Received: June 2005