singular kahler einstein metrics we study degenerate complex monge ampere equations of the form omega dd c f n e t f mu where omega is a big semi kahler form on a compact kahler manifold x of dimension n t in r and mu f omega n is a positive measure with density f in l p x omega n p 1 we prove the existence and unicity of continuous o plurisubharmonic solutions in case x is projective and omega psi omega where psi x to v is a proper birational morphism to a normal projective variety omega in ns r v is an ample class and mu has only algebraic singularities we prove that the solution is smooth in the regular locus of the equation we use these results to construct singular kahler einstein metrics of non positive curvature on projective klt pairs in particular on canonical models of algebraic varieties of general type

uniqueness and short time regularity of the weak kahler ricci flow let x be a compact kahler manifold we prove that the kahler ricci flow starting from arbitrary closed positive 1 1 currents is smooth outside some analytic subset this regularity result is optimal meaning that the flow has positive lelong numbers for short time if the initial current has we also prove that the flow is unique when starting from currents with zero lelong numbers

some koszul rings from geometry abstract we give examples of koszul rings that arise naturally in algebraic geometry in the first part we prove a general result on koszul property associated to an ample line bundle on a projective variety specifically we show how koszul property of multiples of a base point free ample line bundle depends on its castelnuovo mumford regularity in the second part we give examples of koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type

regularizing properties of complex monge ampere flows ii hermitian manifolds we prove that a general complex monge ampere flow on a hermitian manifold can be run from an arbitrary initial condition with zero lelong number at all points using this property we confirm a conjecture of tosatti weinkove the chern ricci flow performs a canonical surgical contraction finally we study a generalization of the chern ricci flow on compact hermitian manifolds namely the twisted chern ricci flow

resonant semilinear problems with nonlinear term depending on the derivative we study the existence of solution for a boundary value problem at resonance where the nonlinearity depends only on the derivative in a sense we can say that the problem considered is strongly resonant our proofs make use of the lyapunov schmidt reduction in so doing we are led with the asymptotic estimate of the corresponding bifurcation equation

on the weak kahler ricci flow in this paper we define and study the kahler ricci flow with initial data not being smooth and discuss some natural applications

initial ideals veronese subrings and rates of algebras abstract let s be a polynomial ring over an infinite field and let i be a homogeneous ideal of s let t d be a polynomial ring whose variables correspond to the monomials of degree d in s we study the initial ideals of the ideals v d i t d that define the veronese subrings of s i in suitable orders they are easily deduced from the initial ideal of i we show that in v d i is generated in degree max reg i d 2 where reg i is the regularity of the ideal i in other words the d th veronese subrings of any commutative graded ring s i has a grobner basis of degree max i d 2 we also give bounds on the regularity of i in terms of the degrees of the generators of in i and some combinatorial data this implies a version of backelin s theorem that high veronese subrings of any ring are homogeneous koszul algebras in the sense of priddy trans amer math soc 152 1970 39 60 we also give a general obstruction for a homogeneous ideal i s to have an initial ideal in i that is generated by quadrics beyond the obvious requirement that i itself should be generated by quadrics and the stronger statement that s i is koszul we use the obstruction to show that in certain dimensions a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics for the application to backelin s theorem we require a result of backelin whose proof has never appeared we give a simple proof of a sharpened version bounding the rate of growth of the degrees of generators for syzygies of any multihomogenous module over a polynomial ring modulo an ideal generated by monomials following a method of bruns and herzog

valuations and multiplier ideals this article is the third of a series of work on a new approach to the study of singularities of various objects in a local two dimensional setting our focus in the present paper is on multiplier ideals and singularity exponents in the discussion below we fix an equicharacteristic zero two dimensional regular local ring r m with algebraically closed residue field an important example is the ring r do o holomorphic germs at the origin in c2 in fj1 we introduced the space v consisting of all r u 00 valued valuations on r centered at m and normalized by m 1 this space is naturally a tree it is a union of uncountably many real segments patched together in such a way that v remains homotopic to a point it is also an r tree in the classical sense for a natural metric we hence call v the valuative tree it encodes in a natural way all possible blowups of r centered at m and therefore gives a way of measuring quite precisely singularities of different kinds of objects the points in v that are not ends form the subtree vqm of quasimonomial valuations these valuations which can alternatively be characterized as abhyankar valuations of rank 1 or as

the weighted monge ampere energy of quasiplurisubharmonic functions abstract we study degenerate complex monge ampere equations on a compact kahler manifold x we show that the complex monge ampere operator d d c n is well defined on the class e x of plurisubharmonic functions with finite weighted monge ampere energy the class e x is the largest class of psh functions on which the monge ampere operator is well defined and the comparison principle is valid it contains several functions whose gradient is not square integrable we give a complete description of the range of the operator d d c n on e x as well as on some of its subclasses we also study uniqueness properties extending calabi s result to this unbounded and degenerate situation and we give applications to complex dynamics and to the existence of singular kahler einstein metrics

remarks on the range properties of certain semilinear problems of landesman lazer type abstract in the article we consider the resonant dirichlet problem u t u t g u t f t t 0 u 0 u 0 1 where g is a continuous function having finite limits at plus and minus infinity g g imposing g g s g for any s r we formulate a necessary condition our main result provides a characteristic of the set of functions f c 0 t such that 1 has a solution however to prove this we need that g is odd increasing and satisfies a certain asymptotic condition at infinity in addition the main tools used are lyapunov schmidt reduction and certain asymptotical methods

complex monge ampere equations on quasi projective varieties we introduce generalized monge ampere capacities and use these to study complex monge ampere equations whose right hand side is smooth outside a divisor we prove in many cases that there exists a unique normalized solution which is smooth outside the divisor our results still hold if the divisor is replaced by any closed subset

contracting exceptional divisors by the kahler ricci flow we give a criterion under which a solution g t of the kahler ricci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold as t tends to the singular time t from each direction we prove the convergence of g t in the sense of gromov hausdorff and smooth convergence away from the exceptional divisors we call this behavior for the kahler ricci flow a canonical surgical contraction in particular our results show that the kahler ricci flow on a projective algebraic surface will perform a sequence of canonical surgical contractions until in finite time either the minimal model is obtained or the volume of the manifold tends to zero

regularizing properties of complex monge ampere flows we study the regularizing properties of complex monge ampere flows on a kahler manifold x when the initial data are psh functions with zero lelong number at all points we prove that the general monge ampere flow has a solution which is immediately smooth we also prove the uniqueness and stability of solution

the complex monge ampere equation on compact kahler manifolds we consider the complex monge ampere equation on a compact kahler manifold m g when the right hand side f has rather weak regularity in particular we prove that estimate of triangle phi and the gradient estimate hold when f is in w 1 p 0 for any p 0 2n as an application we show that there exists a classical solution in w 3 p 0 for the complex monge ampere equation when f is in w 1 p 0

multigraded regularity and the koszul property abstract we give a criterion for the section ring of an ample line bundle to be koszul in terms of multigraded regularity we discuss applications to adjoint bundles on toric varieties as well as to polytopal semigroup rings

normal presentation on elliptic ruled surfaces from the introduction let x be an irreducible projective variety and l a very ample lline bundle on x whose complete linear series defines l x p h0 l let s 1 m 0 smh0 x l and let r l l1 n 0 h0 x l n be the homogeneous coordinate ring associated to l then r is a finitely generated graded module over s so it has a minimal graded free resolution we say that the line bundle l is normally generated if the natural maps smh0 x l h0 x l m are surjective for all m 2 if l is normally generated then we say that l satisfies property np if the matrices in the free resolution of r over s have linear entries until the p th stage in particular property n1 says that the homogeneous ideal i of x in p h0 l is generated by quadrics a line bundle satisfying property n1 is also called normally presented let r kr1r2 be a graded algebra over a field k the algebra r is a koszul ring iff torri k k has pure degree i for all i in this article we determine exactly theorem 4 2 which line bundles on an elliptic ruled surface x are normally presented as a corollary we show that mukai s conjecture is true for the normal presentation of the adjoint linear series for an elliptic ruled surface in section 5 of this article we show that if l is normally presented on x then the homogeneous coordinate ring associated to l is koszul we also give a new proof of the following result due to butler if deg l 2g 2 on a curve x of genus g then l embeds x with koszul homogeneous coordinate ring

degenerate complex monge ampere flows on strictly pseudoconvex domains we study complex monge ampere flows dot u log det u alpha bar beta f t z u in domains of mathbb c n this equation has a close connection with the kahler ricci flow in this paper we study the smoothing properties of this flow in case the boundary conditions are smooth and the initial conditions are bounded