Boas Assistant Professor
Department of Mathematics, Lunt 303
2033 Sheridan Road, Evanston, IL 60208
Complex algebraic and analytic geometry:
Using the duality of positive cones, we show that applying the polar transform from convex analysis to local positivity invariants for divisors gives interesting and new local positivity invariants for curves. These new invariants have nice properties similar to those for divisors. In particular, this enables us to give a characterization of the divisorial components of the non-Kähler locus of a big class.
Motivated by positivity in complex geometry and complex dynamics of rational self-maps of a projective variety, we study some positivity aspects of translation invariant convex valuations and the dynamics of invertible linear transform group actions on the space of translation invariant convex valuations. In particular, we prove that the positivity of invariant convex valuations is very weak under a natural strict log-concavity assumption on certain dynamical degrees.
We give a Bezout type inequality for mixed volumes, which holds true for any convex bodies. The key ingredient is the reverse Khovanskii-Teissier inequality for convex bodies, which was obtained in our previous work and inspired by its correspondence in complex geometry.
We study positivity in the conjecture proposed by Lejmi and Szekelyhidi on finding effective necessary and sufficient conditions for solvability of the inverse σk equation, or equivalently, for convergence of the inverse σk-flow. In particular, for the inverse σn-1-flow we partially verify their conjecture by obtaining the desired positivity for (n-1, n-1) cohomology classes. As an application, we also partially verify their conjecture for 3-folds.
We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or Kähler manifolds. We study the relation between positive products and mixed volumes. We define and study a Blaschke addition for divisor classes and mixed divisor classes, and prove new geometric inequalities for divisor classes. We also reinterpret several classical convex geometry results in the context of algebraic geometry: the Alexandrov body construction is the convex geometry version of divisorial Zariski decomposition; Minkowski's existence theorem is the convex geometry version of the duality between the pseudo-effective cone of divisors and the movable cone of curves.
This is the second part of our work on Zariski decomposition structures, where we compare two different volume type functions for curve classes. The first function is the polar transform of the volume for ample divisor classes. The second function captures the asymptotic geometry of curves analogously to the volume function for divisors. We prove that the two functions coincide, generalizing Zariski's classical result for surfaces to all varieties. Our result confirms the log concavity conjecture of the first named author for weighted mobility of curve classes in an unexpected way, via Legendre- Fenchel type transforms. We also give a number of applications to birational geometry, including a refined structure theorem for the movable cone of curves.
This is the first part of our work on Zariski decomposition structures, where we study Zariski decompositions using Legendre-Fenchel type transforms. In this way we define a Zariski decomposition for curve classes. This decomposition enables us to develop the theory of the volume function for curves defined by the second named author, yielding some fundamental positivity results for curve classes. For varieties with special structures, the Zariski decomposition for curve classes admits an interesting geometric interpretation.
In this note, we study the positivity of related cohomology classes concerning the convergence problem of the inverse σk-flow in the conjecture proposed by Lejmi and Szekelyhidi.
For divisors over smooth projective varieties, we show that the volume can be characterized by the duality between the pseudo-effective cone of divisors and the movable cone of curves. Inspired by this result, we define and study a natural intersection-theoretic volume functional for 1-cycles over compact Kähler manifolds. In particular, for numerical equivalence classes of curves over projective varieties, it is closely related to the mobility functional studied by Lehmann.
We give a solution of Teissier's proportionality problem for transcendental nef classes over a compact Kähler manifold, which says that the equality in the Khovanskii-Teissier inequalities holds for a pair of big and nef classes if and only if the two classes are proportional. This result recovers the previous one of Boucksom-Favre-Jonsson for the case of big and nef line bundles over a (complex) projective algebraic manifold. Our proof applies degenerate complex Monge-Ampere equations in big classes and basic pluripotential theory.
Based on the method of Chiose, we prove a weak version of Demailly's conjecture on transcendental Morse inequalities on compact Kähler manifolds. Moreover, we note that Chiose's method gives a Morse-type bigness criterion for the differences of certain (k, k) classes.
We study strongly Gauduchon metrics on a compact complex manifold. In particular, we study the positive cone in the de Rham cohomology group generated by all strongly Gauduchon metrics and its direct images under proper modifications. We also observe that a result due to Michelsohn can extend to this setting.
Motivated by form-type Monge-Ampere equations, we consider a natural map from the Kähler cone of a compact Kähler manifold to its balanced cone. We study its injectivity and surjectivity: we show that the map is injective when restricted to the big and nef cone, and for Calabi-Yau manifolds we give a characterization on when a nef class is mapped into the interior of the balanced cone. We also give an analytic characterization theorem on a nef class being Kähler.
We give a Morse-type bigness criterion for the difference of two pseudo-effective (1, 1) classes by using movable intersections. As an application, we give a Morse-type bigness criterion for the difference of two movable (n-1, n-1) classes.
This is a short survey paper on several positivity results in Kähler geometry, dedicated to Professor G.C. Dong on the occasion of his 90th birthday.