Conference on Hodge Theory and L-cohomology Titles and Abstracts

I will report on joint work with Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza extending and refining Cheeger’s study of ideal boundary conditions for the de Rham operator from the case of isolated conic singularities to arbitrary stratified pseudomanifolds. This leads to a notion of ‘mezzoperversity’ intermediate between the upper and lower perversities of Goresky-MacPherson. If time permits, I will also discuss a topological version of our theory from joint work with Markus Banagl. Edward Bierstone Hsiang-Pati coordinates Abstract: Given a complex analytic (or algebraic) variety X, can we find a global resolution of singularities p: Y -¿ X such that the pulled-back cotangent sheaf is generated by differential monomials in suitable coordinates at every point of Y (”Hsiang-Pati coordinates”)? The answer is ”yes” in dimension up to 3. It was previously known for surfaces X with isolated singularities (Hsiang-Pati 1985, Pardon-Stern 2001). Consequences include monomialization of the induced Fubini-Study metric on the smooth part of a complex projective variety X (with previous applications to L2 cohomology). Existence of Hsiang-Pati coordinates is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of p. (Joint work with Andre Belotto, Vincent Grandjean and Pierre Milman.) Given a complex analytic (or algebraic) variety X, can we find a global resolution of singularities p: Y -¿ X such that the pulled-back cotangent sheaf is generated by differential monomials in suitable coordinates at every point of Y (”Hsiang-Pati coordinates”)? The answer is ”yes” in dimension up to 3. It was previously known for surfaces X with isolated singularities (Hsiang-Pati 1985, Pardon-Stern 2001). Consequences include monomialization of the induced Fubini-Study metric on the smooth part of a complex projective variety X (with previous applications to L2 cohomology). Existence of Hsiang-Pati coordinates is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of p. (Joint work with Andre Belotto, Vincent Grandjean and Pierre Milman.) Nero Budur Differential graded Lie algebra pairs Abstract: In practice, any deformation problem over fields of characteristic zero is governed by a differential graded Lie algebra (DGLA). Following Deligne, Goldman-Millson and Simpson described via DGLAs the local structure of moduli spaces for various geometric situations. Given an object with a notion of cohomology theory, how can one describe all its deformations subject to cohomology constraints? We will present an approach via DGLA pairs. As applications, we discuss the structure of cohomology jump loci of vector bundles and of local systems. This is joint work with Botong Wang. In practice, any deformation problem over fields of characteristic zero is governed by a differential graded Lie algebra (DGLA). Following Deligne, Goldman-Millson and Simpson described via DGLAs the local structure of moduli spaces for various geometric situations. Given an object with a notion of cohomology theory, how can one describe all its deformations subject to cohomology constraints? We will present an approach via DGLA pairs. As applications, we discuss the structure of cohomology jump loci of vector bundles and of local systems. This is joint work with Botong Wang. Alexandru Dimca On Griffiths’ and Steenbrink’s Theorems for nodal hypersurfaces Abstract: We explore the relation between Hodge filtrations and the pole order of rational differential forms. We explore the relation between Hodge filtrations and the pole order of rational differential forms. Osamu Fujino Direct images of pluricanonical divisors Abstract: I would like to explain the local freeness and the semipositivity of direct images of pluricanonical divisors for surjective morphisms between smooth projective varieties with connected fibers. We can obtain a desirable semipositivity theorem under the assumption that the geometric generic fiber has a good minimal model. I would like to explain the local freeness and the semipositivity of direct images of pluricanonical divisors for surjective morphisms between smooth projective varieties with connected fibers. We can obtain a desirable semipositivity theorem under the assumption that the geometric generic fiber has a good minimal model. Phillip Griffiths Algebro-geometric aspects of limiting mixed Hodge structures ∗ Abstract: This will be a largely informal, descriptive talk about some Hodge-theoretic aspects of two standard algebra-geometric questions. What can a smooth, projective variety degenerate to? What can a singular, projective variety be smoothed to? In the first question we are primarily interested in the extremal cases: the generic and most degenerate specialization. In the second question we shall concentrate on the multi-parameter case. *Based in large part on joint work with Colleen Robles and Mark Green building on earlier work by many people including Friedman, Steenbrink, Zucker, Cattani-Kaplan-Schmid, Kerr-Pearlstein, ... This will be a largely informal, descriptive talk about some Hodge-theoretic aspects of two standard algebra-geometric questions. What can a smooth, projective variety degenerate to? What can a singular, projective variety be smoothed to? In the first question we are primarily interested in the extremal cases: the generic and most degenerate specialization. In the second question we shall concentrate on the multi-parameter case. *Based in large part on joint work with Colleen Robles and Mark Green building on earlier work by many people including Friedman, Steenbrink, Zucker, Cattani-Kaplan-Schmid, Kerr-Pearlstein, ... Richard Hain The Hodge de Rham Theory of Modular Groups Abstract: This talk will be an introduction to the Hodge de Rham theory of modular groups. I will focus on the case of the SL2(Z). In particular, I will give a concrete construction of the mixed Hodge structure on its relative unipotent completion in terms of iterated integrals of classical modular forms. I will explain the relation to Manin’s iterated Shimura integrals and indicate its relevance to my work with Makoto Matsumoto on universal mixed elliptic motives. This talk will be an introduction to the Hodge de Rham theory of modular groups. I will focus on the case of the SL2(Z). In particular, I will give a concrete construction of the mixed Hodge structure on its relative unipotent completion in terms of iterated integrals of classical modular forms. I will explain the relation to Manin’s iterated Shimura integrals and indicate its relevance to my work with Makoto Matsumoto on universal mixed elliptic motives. James Lewis A relative version of the Beilinson-Hodge conjecture Abstract: Let k ⊆ C be an algebraically closed subfield, and X a variety defined over k. One version of the Beilinson-Hodge conjecture that seems to survive scrutiny is the statement that the Betti cycle class map clr,m : H 2r−m M (k(X),Q(r)) → homMHS ( Q(0), H2r−m(k(X)(C),Q(r)) ) is surjective, that being equivalent to the Hodge conjecture in the casem = 0. Now consider a smooth and proper map ρ : X → S of smooth quasi-projective varieties over k, and where η is the generic point of Ss. We anticipate that the corresponding cycle class map is surjective, and provide some evidence in support of this in the case where X = S × X is a product and m = 1. This is based on joint work with Rob de Jeu and Deepam Patel. Let k ⊆ C be an algebraically closed subfield, and X a variety defined over k. One version of the Beilinson-Hodge conjecture that seems to survive scrutiny is the statement that the Betti cycle class map clr,m : H 2r−m M (k(X),Q(r)) → homMHS ( Q(0), H2r−m(k(X)(C),Q(r)) ) is surjective, that being equivalent to the Hodge conjecture in the casem = 0. Now consider a smooth and proper map ρ : X → S of smooth quasi-projective varieties over k, and where η is the generic point of Ss. We anticipate that the corresponding cycle class map is surjective, and provide some evidence in support of this in the case where X = S × X is a product and m = 1. This is based on joint work with Rob de Jeu and Deepam Patel. Zhan Li Boundedness of certain Calabi-Yau varieties and their stringy Hodge numbers Abstract: We prove birational boundedness of certain Calabi-Yau varieties which are complete intersections in (some version of) Fano varieties. Our results imply in particular that Batyrev-Borisov toric construction of Calabi-Yau varieties in mirror symmetry produces only a finite set of stringy Hodge numbers in any given dimension, even as the codimension is allowed to grow. This is a joint work with Lev Borisov. We prove birational boundedness of certain Calabi-Yau varieties which are complete intersections in (some version of) Fano varieties. Our results imply in particular that Batyrev-Borisov toric construction of Calabi-Yau varieties in mirror symmetry produces only a finite set of stringy Hodge numbers in any given dimension, even as the codimension is allowed to grow. This is a joint work with Lev Borisov. Takeo Ohsawa Two extension theorems — effective and noneffective Abstract: Remarkable progress was made on the L extension theorem in 2012 by Z. Blocki, Q.Guan and X.-Y. Zhou. By studying their work I could simplify the proof. This alternate proof will be explained. While I was writing a monograph focused on this extension theorem of Blocki-Guan-Zhou and my new proof, I noticed that Hörmander’s good old method of proving the finiteness theorem of Andreotti-Grauert can be applied to show another extension theorem. Although this result is highly noneffective compared to the former one, I believe it is of some interest. Remarkable progress was made on the L extension theorem in 2012 by Z. Blocki, Q.Guan and X.-Y. Zhou. By studying their work I could simplify the proof. This alternate proof will be explained. While I was writing a monograph focused on this extension theorem of Blocki-Guan-Zhou and my new proof, I noticed that Hörmander’s good old method of proving the finiteness theorem of Andreotti-Grauert can be applied to show another extension theorem. Although this result is highly noneffective compared to the former one, I believe it is of some interest. Gregory Pearlstein Singular hermitian metrics and the Hodge conjecture Abstract: A Hodge class on a smooth complex projective variety gives rise to an associated hermitian line bundle on a Zariski open subset of a complex projective space P . I will discuss recent work with P. Brosnan which shows that the Hodge conjecture is equivalent to the existence of a particular kind of degenerate behavior of this metric near the boundary. A Hodge class on a smooth complex projective variety gives rise to an associated hermitian line bundle on a Zariski open subset of a complex projective space P . I will discuss recent work with P. Brosnan which shows that the Hodge conjecture is equivalent to the existence of a particular kind of degenerate behavior of this metric near the boundary. Colleen Robles Classification of horizontal SL(2)s Abstract: The celebrated Nilpotent Orbit and SL(2)-Orbit Theorems of Schmid and Cattani–Kaplan–Schmid, describe the asymptotic behavior of a variation of Hodge structure and play a fundamental role in the analysis of singularities of the period mapping. Two of the more striking applications of the theorems are the proofs of: (i) the algebraicity of Hodge loci, which provides some of the strongest evidence for the Hodge conjecture; and (ii) Deligne’s conjectured isomorphism between the L and intersection cohomologies. As a consequence it became an important problem to describe the SL(2)s appearing in Schmid’s Theorem. The celebrated Nilpotent Orbit and SL(2)-Orbit Theorems of Schmid and Cattani–Kaplan–Schmid, describe the asymptotic behavior of a variation of Hodge structure and play a fundamental role in the analysis of singularities of the period mapping. Two of the more striking applications of the theorems are the proofs of: (i) the algebraicity of Hodge loci, which provides some of the strongest evidence for the Hodge conjecture; and (ii) Deligne’s conjectured isomorphism between the L and intersection cohomologies. As a consequence it became an important problem to describe the SL(2)s appearing in Schmid’s Theorem. I will give a classification of the horizontal SL(2)s in the setting of Mumford– Tate domains; the latter generalize period domains to include classifying spaces for Hodge structures with nongeneric Hodge tensors (i.e., the Mumford– Tate group of a generic Hodge structure in the domain need not be the full automorphism group of the polarization). From this perspective, the classification describes (as R-split polarized mixed Hodge structures) the degenerations that may arise in a variation of Hodge structure subject to a constraint on the Mumford–Tate group of the generic fibre. Leslie Saper Perverse sheaves and reductive Borel-Serre compactification Wilfried Schmid A new look at mixed Hodge modules Abstract: Shortly after the nilpotent orbit theorem and the SL(2) orbit theorem were proved, Deligne remarked that they could be proved even for variations of Hodge structure without underlying rational structure. At the time, that seemed like a generalization without an obvious application, and no argument – or even detailed statement – was ever published. The orbit theorems and their multi-variable versions have since served as analytic input for Saito’s theory of mixed Hodge modules. Saito’s theory involves an underlying rational structure. It has become apparent in recent years that it should be possible to develop Saito’s theory without a rational structure. I shall outline a proof of the orbit theorems in the absence of a rational structure and discuss the implications for, and applications of, a theory of mixed Hodge modules without a rational structure. Shortly after the nilpotent orbit theorem and the SL(2) orbit theorem were proved, Deligne remarked that they could be proved even for variations of Hodge structure without underlying rational structure. At the time, that seemed like a generalization without an obvious application, and no argument – or even detailed statement – was ever published. The orbit theorems and their multi-variable versions have since served as analytic input for Saito’s theory of mixed Hodge modules. Saito’s theory involves an underlying rational structure. It has become apparent in recent years that it should be possible to develop Saito’s theory without a rational structure. I shall outline a proof of the orbit theorems in the absence of a rational structure and discuss the implications for, and applications of, a theory of mixed Hodge modules without a rational structure. Mark Stern Introduction to nonlinear harmonic forms Abstract: Abstract: We motivate and introduce nonlinear harmonic forms. These are de Rham representatives z of cohomology classes which minimize the energy ∥z∥2L2 subject to a nonlinear constraint. We give basic existence results for quadratic constraints, discuss the rich Euler Lagrange equations, and ask many regularity questions. Abstract: We motivate and introduce nonlinear harmonic forms. These are de Rham representatives z of cohomology classes which minimize the energy ∥z∥2L2 subject to a nonlinear constraint. We give basic existence results for quadratic constraints, discuss the rich Euler Lagrange equations, and ask many regularity questions. Tomohide Terasoma Depth filtration of multiple zeta values and Tate curve. Abstract: On the Q-vector space generated by (motivic) multiple zeta values, there are two filtrations by weight and depth. The depth filtraion is On the Q-vector space generated by (motivic) multiple zeta values, there are two filtrations by weight and depth. The depth filtraion is defined by the numbers of dx/(1-x)’s in the iterated integral expression of the multiple zeta values. The conjecture by Broadhurst-Kreimer, it is strongy expected that this filtration is realted to the space of elliptic modular forms. In this talk, we will try to explain the relation betwee the depth filtraion and the representation of Tannaka fundamental group of mixed elliptic motives on the fundamental group of the Tate curve.