Letters of a Bi-rationalist. VII Ordered termination

To construct a resulting model in the LMMP, it is sufficient to prove the existence of log flips and their termination for some sequences. We prove that the LMMP in dimension d − 1 and the termination of terminal log flips in dimension d imply, for any log pair of dimension d, the existence of a resulting model: a strictly log minimal model or a strictly log terminal Mori log fibration, and imply the existence of log flips in dimension d + 1. As a consequence, we prove the existence of a resulting model of 4-fold log pairs, the existence of log flips in dimension 5, and Geography of log models in dimension 4. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 184–208. To V.A. Iskovskikh, who has greatly shaped my vision of mathematics     

Existence of log canonical flips and a special LMMP

Let (X/Z,B+A) be a Q-factorial dlt pair where B,A??0 are Q-divisors and K _X +B+A?? _Q 0/Z. We prove that any LMMP/Z on K _X +B with scaling of an ample/Z divisor terminates with a good log minimal model or a Mori fibre space. We show that a more general statement follows from the ACC for lc thresholds. An immediate corollary of these results is that log flips exist for log canonical pairs.     

Semistability of Logarithmic Cotangent Bundle on Some Projective Manifolds

In this article, we investigate the semistability of logarithmic de Rham sheaves on a smooth projective variety (X, D), under suitable conditions. This is related to existence of Kähler–Einstein metric on the open variety. We investigate this problem when the Picard number is one. Fix a normal crossing divisor D on X and consider the logarithmic de Rham sheaf Ω _X (log D) on X. We prove semistability of this sheaf, when the log canonical sheaf K _X  + D is ample or trivial, or when ?K _X  ? D is ample, i.e., when X is a log Fano n-fold of dimension n ≤ 6. We also extend the semistability result for Kawamata coverings, and this gives examples whose Picard number can be greater than one.     

Cone theorem via Deligne–Mumford stacks

We prove the cone theorem for varieties with LCIQ singularities using deformation theory of stable maps into Deligne–Mumford stacks. We also obtain a sharper bound on −(K _X + D)-degree of (K _X + D)-negative extremal rays for projective -factorial log terminal threefold pairs (X, D).     

On existence of log minimal models and weak Zariski decompositions

We first introduce a weak type of Zariski decomposition in higher dimensions: an -Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective -Cartier divisor. We then prove that there is a very basic relation between Zariski decompositions and log minimal models which has long been expected: we prove that assuming the log minimal model program in dimension d − 1, a lc pair (X/Z, B) of dimension d has a log minimal model (in our sense) if and only if K _X  + B has a weak Zariski decomposition/Z.     

Affine threefolds whose log canonical bundles are not numerically effective

Let X?(T,D) be a compactification of an affine 3-fold X into a smooth projective 3-fold T such that the (reduced) boundary divisor D is SNC. In this paper, as an affine counterpart to the work due to S. Mori (cf. [S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982) 133-176]), we shall classify (K+D)-negative extremal rays on T. In particular, if such an extremal ray R=R₊[C] intersects K non-negatively, we shall describe the log flips and divisorial contractions appearing explicitly.     

On log canonical divisors that are log quasi-numerically positive

Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK _X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.     

Isonoetherian power series rings

Facchini and Nazemian proved that a valuation domain is isonoetherian if and only if it is discrete of Krull dimension ≤2 and they showed that this cannot be generalized from the local case to the global case: the 2-dimensional generalized Dedekind domain ?+X?[[X]] is not isonoetherian. Let D be an integral domain with quotient field K. We provide necessary and sufficient conditions on D and K, so that the ring D+XK[[X]] is isonoetherian. We deduce that if D is integrally closed, then D+XK[[X]] is isonoetherian if and only if D is a semi-local principal ideal domain.     

Numerical Semigroup Rings and Almost Prüfer v-Multiplication Domains

Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a numerical semigroup with Γ ? ?₀, D[Γ] be the semigroup ring of Γ over D (and hence D ? D[Γ] ? D[X]), and D + X ⁿ K[X] = {a + X ⁿ g∣a ∈ D and g ∈ K[X]}. We show that there exists an order-preserving bijection between Spec(D[X]) and Spec(D[Γ]), which also preserves t-ideals. We also prove that D[Γ] is an APvMD (resp., AGCD-domain) if and only if D[X] is an APvMD (resp., AGCD-domain) and char(D) ≠ 0. We show that if n ≥ 2, then D is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain) and char(D) ≠ 0 if and only if D + X ⁿ K[X] is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain). Finally, we give some examples of APvMDs which are not AGCD-domains by using the constructions D[Γ] and D + X ⁿ K[X].     

Motivic equivalence of quadratic forms. II

Let F be a field of characteristic ≠2 and φ be a quadratic form over F. By X _φ we denote the projective variety given by the equation φ=0. For each positive even integer d≥8 (except for d=12) we construct a field F and a pair φ, ψ of anisotropic d-dimensional forms over F such that the Chow motives of X _φ and X _ψ coincide but . For a pair of anisotropic (2 ^{n -1})-dimensional quadrics X and Y, we prove that existence of a rational morphism Y→X is equivalent to existence of a rational morphism Y→X. Received: 27 September 1999 / Revised version: 27 December 1999     

The Mukai conjecture for log Fano manifolds

For a log Fano manifold (X,D) with D ≠ 0 and of the log Fano pseudoindex ≥2, we prove that the restriction homomorphism Pic(X) → Pic(D ₁) of Picard groups is injective for any irreducible component D ₁ ? D. The strategy of our proof is to run a certain minimal model program and is similar to Casagrande’s argument. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).     

Weak Solutions to Stochastic Wave Equations with Values in Riemannian Manifolds

Let M be a compact Riemannian manifold. We prove existence of a global weak solution of the stochastic wave equation D _t u _t  = D _x u _x  + (X _u  + λ₀(u)u _t  + λ₁(u)u _x )[Wdot] where X is a continuous vector field on M, λ₀ and λ₁ are continuous vector bundles homomorphisms from TM to TM, and W is a spatially homogeneous Wiener process on ? with finite spectral measure. We use recently introduced general method of constructing weak solutions of SPDEs that does not rely on any martingale representation theorem.     

Nonlinear volterra integrodifferential equations in a Banach space

We study the Cauchy problem associated with the Volterra integrodifferential equation u\left( t \right) \in Au\left( t \right) + \int {_0^1 B\left( {t - s} \right)u\left( s \right)ds + f\left( t \right),} u\left( 0 \right) = u_0 \in D\left( A \right), whereA is anm-dissipative non-linear operator (or more generally, anm-D(ω) operator), defined onD(A) ⊂X, whereX is a real reflexive Banach space. We show that ifB is of the formB=FA+K, whereF, K :X →D(D _s), whereD _s is the differentiation operator, withF bounded linear andK andD _sK Lipschitz continuous, then the Cauchy problem is well-posed. In addition we obtain an approximation result for the Cauchy problem.     

Oscillation in the initial segment complexity of random reals

We study oscillation in the prefix-free complexity of initial segments of 1-random reals. For upward oscillations, we prove that _∑n∈ω²−g(n) diverges iff (^∃∞n)K(X?n)>n+g(n) for every 1-random X∈ω². For downward oscillations, we characterize the functions g such that (^∃∞n)K(X?n)<n+g(n) for almost every X∈ω². The proof of this result uses an improvement of Chaitin's counting theorem—we give a tight upper bound on the number of strings σ∈n² such that K(σ)<n+K(n)−m.The work on upward oscillations has applications to the K-degrees. Write XK_?Y to mean that K(X?n)?K(Y?n)+O(1). The induced structure is called the K-degrees. We prove that there are comparable () 1-random K-degrees. We also prove that every lower cone and some upper cones in the 1-random K-degrees have size continuum.Finally, we show that it is independent of ZFC, even assuming that the Continuum Hypothesis fails, whether all chains of 1-random K-degrees of size less than ℵ₀² have a lower bound in the 1-random K-degrees.     

The infinite duration lying oracle game

In the lying oracle game, a Player places bets on the outcomes of a sequence of coin flips. A second player, the Oracle, informs the Player what the outcome of each coin flip is, but may at times lie. We analyse this game when the duration of the game is infinite, where the Oracle's ability to lie or be truthful is specified by a set of lie patterns X, known to both players. By equipping X with Lebesgue measure, we prove that for any ε?>?0, the Player has a strategy that yields an expected fortune of at least 1/λ⁺(X)???ε, and that the Oracle has a strategy that limits the Player's expected fortune to at most 1/λ^?(X), where λ⁺(X) and λ^?(X) are the outer and inner Lebesgue measure of X, respectively.     

Universal space in the Cartesian product of Peano curves without free arcs

We prove that there is the universal space for the class of n-dimensional separable metric spaces in the Cartesian product K₁×?×Kn+1 of Peano curves without free arcs. It is also shown that the set of embeddings of any n-dimensional separable metric space X into this universal space is a residual set in C(X,K₁×?×Kn+1). Other properties of product of Peano curves without free arcs are also proved.     

The multiplier ideal is a universal test ideal

Abstract

For a canonical threefold X, we know that h ⁰(X, 𝒪 _X (nK _X )) ≥ 1 for a sufficiently large n. When χ(𝒪 _X ) > 0, it is not easy to get such an integer n. Fletcher showed that h ⁰(X, 𝒪 _X (12K _X )) ≥ 1 and h ⁰(X, 𝒪 _X (24K _X )) ≥ 2 when χ(𝒪 _X ) = 1. He inquired about existence of a canonical threefold with given conditions which shows the result sharp. We show that such an example does not exist. Using a different technique, we prove h ⁰(X, 𝒪 _X (12K _X )) ≥ 2.     

On Smash Products Of Hopf Algebras

Let H = X? _R A denote an R-smash product of the two bialgebras X and A. We prove that (X,A) is a pair of matched bialgebras, if the R-smash product H has a braiding structure. When X is an associative algebra and A is a Hopf algebra, we investigate the global dimension and the weak dimension of the smash product H = X? _R A and show that lD(H) ≤ rD(A) + lD(X) and wD(H) ≤ wD(A) + wD(X). As an application, we get lD(H ₄) = ∞ for Sweedler's four dimensional Hopf algebra H ₄. We also study the associativity of smash products and the relations between smash products and factorization for algebras.     

Inverse problems and index formulae for Dirac operators

We consider a Dirac-type operator DP on a vector bundle V over a compact Riemannian manifold (M,g) with a non-empty boundary. The operator DP is specified by a boundary condition P(u_|∂M)=0 where P is a projector which may be a non-local, i.e., a pseudodifferential operator. We assume the existence of a chirality operator which decomposes L²(M,V) into two orthogonal subspaces X₊⊕X₋. Under certain conditions, the operator DP restricted to X₊ and X₋ defines a pair of Fredholm operators which maps X₊→X₋ and X₋→X₊ correspondingly, giving rise to a superstructure on V. In this paper we consider the questions of determining the index of DP and the reconstruction of and DP from the boundary data on ∂M. The data used is either the Cauchy data, i.e., the restrictions to ∂M×R₊ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e., the set of the eigenvalues and the boundary values of the eigenfunctions of DP. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in M×C⁴, M⊂R³.     

Note on K-stability of pairs

We prove that a pair (X, D) with X Fano and D an anti-canonical divisor is K-unstable for negative angles, and is K-semistable for zero angle.