Boas Assistant Professor

Department of Mathematics, Lunt 303

Northwestern University

2033 Sheridan Road, Evanston, IL 60208

- 2016 Fall: MATH 224 (Integral Calculus of One Variable Functions)
- 2017 Spring: MATH 240 (Linear Algebra)

Complex algebraic and analytic geometry:

- Analytic methods in complex algebraic geometry.
- Positivity and convexity in algebraic/Kähler geometry.
- Interactions between convex geometry and complex geometry.

**Polar transform and local positivity for curves**(w/ N. McCleerey)

preprint 2017.*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.***Dynamical degrees and positivity of convex valuations**(w/ N.B. Dang)

preprint 2017.*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.***Bezout type inequality in convex geometry**

preprint 2017, submitted.*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.***Positivity in convergence of the inverse σ**_{n-1}-flow

preprint 2016.*We study positivity in the conjecture proposed by Lejmi and Szekelyhidi on finding effective necessary and sufficient conditions for solvability of the inverse σ*_{k}equation, or equivalently, for convergence of the inverse σ_{k}-flow. In particular, for the inverse σ_{n-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.**Correspondences between convex geometry and complex geometry**(w/ B. Lehmann)

Épijournal de Géométrie Algébrique (EPIGA), to appear.*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.***Positivity functions for curves on algebraic varieties**(w/ B. Lehmann)

preprint 2016, submitted.*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.***Convexity and Zariski decomposition structure**(w/ B. Lehmann)

Geometric and Functional Analysis (GAFA) 26 (2016), 1135-1189.*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.***A remark on the convergence of the inverse σ**_{k}-flow

Comptes Rendus Mathematique 354 (2016), 395-399.*In this note, we study the positivity of related cohomology classes concerning the convergence problem of the inverse σ*_{k}-flow in the conjecture proposed by Lejmi and Szekelyhidi.**Characterizing volume via cone duality**

Mathematische Annalen, to appear.*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.***Teissier's problem on proportionality of big and nef classes over a compact Kähler manifold**(w/ J. Fu )

preprint 2014, submitted.*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.***Weak transcendental holomorphic Morse inequalities on compact Kähler manifolds**,

Annales de l'Institut Fourier 65 (2015), 1367-1379.*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.***On strongly Gauduchon metrics of compact complex manifolds**,

Journal of Geometric Analysis 25 (2015), 2011-2027.*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.***Relations between the Kähler cone and the balanced cone of a Kähler manifold**(w/ J. Fu),

Advances in Mathematics 263 (2014), 230-252.*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.*

**Movable intersection and bigness criterion**

preprint 2014.*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.*

**Some positivity results in Kähler geometry**(In Chinese) (w/ J. Fu)

SCIENTIA SINICA Mathematica, to appear.*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.*

**Zariski decomposition of curves on algebraic varieties**(w/ B. Lehmann)

preprint 2015.

- 2018.03: AMS Sectional Meeting, special session ``Recent Development of Nonlinear Geometric PDEs'', The Ohio State University, Columbus.
- 2017.12: The first annual meeting of International Consortium of Chinese Mathematicians (ICCM), Sun Yat-Sen University, Guangzhou.
- 2017.12: Master Lecture Series workshop on Prof. Demailly's works ``Global Aspects of Projective and Kähler Geometry'', Tsinghua Sanya International Mathematics Forum, Sanya.
- 2017.04: Mini workshop on complex geometry, Univ. Notre Dame, Notre Dame.
- 2016.12: Complex Geometry Seminar, Institue of Mathematics, Chinese Academy of Sciences, Beijing.
- 2016.12: Young Mathematician Forum, BICMR, Peking University, Beijing.
- 2016.12: Lectures (Positivity in Kähler geometry), YMSC, Tsinghua University, Beijing.
- 2016.11: Analysis Seminar, Northwestern, Evanston.
- 2016.10: Algebraic Geometry Seminar, UIC, Chicago.
- 2016.10: Algebraic Geometry Seminar, Northwestern, Evanston.
- 2016.10: Harvard Differential Geometry Seminar, Harvard, Boston.
- 2016.06: Recent advances in complex differential geometry, IMT, Toulouse.
- 2016.05: Workshop on algebraic and Kähler geometry, IF, Grenoble.
- 2015.12: Séminaire de géométrie algébrique, ENS, Paris.
- 2015.11: Séminaire de géométrie algébrique, IF, Grenoble.
- 2015.11: Joint Seminar in Algebraic and Complex Geometry (Basel-Freiburg-Nancy-Strasbourg), IRMA, Strasbourg.
- 2015.10: Séminaire d'analyse et géométrie, IMJ, Paris.
- 2015.05: Séminaire de géométrie algébrique, IF, Grenoble.
- 2014.10: Groupes de Travail--Opérateurs de Dirac, Univ. Paris Sud, Orsay.
- 2014.10: Séminaire de géométrie algébrique, IF, Grenoble.
- 2014.05: Several Complex Variables Symposium, Tsinghua Sanya International Mathematics Forum, Sanya.
- 2014.04: Workshop in Several Complex Variables, Institue of Mathematics, Chinese Academy of Sciences, Beijing.
- 2014.01: East Asian Doctorial Forum on Mathematics, Kyoto University, Kyoto.