Moduli and Hodge Theory

Schedule

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  • Monday February 1st, 2021

    Semistable Reduction - A Progress Report

    Dr. Dan Abramovich
    Brown University

    9:00am - 10:00am
    Via Zoom
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    Abstract: How well can one resolve the singularities of a family of complex varieties? The problem was raised in 1973 by Mumford on page vii of "Toroidal Embeddings I". A precise conjecture, the semistable reduction conjecture, was made by Karu and me in our 2000 paper, where we prove weak semistable reduction. I will explain recent work with Temkin and Wlodarczyk on functorial weak semistable reduction, and discuss work of Adiprasito, Liu and Temkin which proves the semistable reduction conjecture.


    Compactifications of Moduli — Geometry vs. Hodge Theory

    Dr. Radu Laza
    SUNY Stony Brook

    10:30am - 11:30am
    Via Zoom
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    Abstract: The construction and compactification of moduli spaces is a topic of great interest in algebraic geometry. Typically, there are several options for pursuing this problem. In this talk, I will focus on comparing two approaches to the compactification problem: a geometric approach vs. a Hodge theoretic approach. I will give some examples and illustrate some issues in “the classical case” (corresponding to VHS of abelian variety type and of K3 type respectively). I will then discuss a project (joint with M. Green, P. Griffiths, and C. Robles) to analyze the possibly simplest non-classical case, namely the H and I-surfaces (surfaces of general type with p_g=2).


    Compact Moduli of Polarized K3 Surfaces

    Dr. Valery Alexeev
    University of Gerogia

    1:30pm - 2:30pm
    Via Zoom

    Abstract: We prove that the natural geometrically meaningful compactification of the moduli of lattice polarized K3s corresponding to a canonical polarizing divisor, that is well-behaved under degenerations, is semi toroidal up to normalization. We also show that for ordinary polarized K3s of degree 2d the sum of rational curves in the polarization is such a well-behaved divisor.

     

  • Tuesday February 2nd, 2021

    Logarithmic Resolution of Singularities

    Dr. Michael Temkin
    Hebrew University

    9:00am - 10:00am
    Via Zoom
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    Abstract: I will talk about a recent series of works with Abramovich and Wlodarczyk, where a logarithmic analogue of the classical resolution of singularities of schemes in characteristic zero is constructed. Already for usual schemes, the logarithmic algorithm is faster and more functorial, though as a price one has to work with log smooth ambient orbifolds rather than smooth ambient manifolds. But the main achievement is that essentially the same algorithm resolves log schemes and even morphisms of log schemes, yielding a major generalization of various semistable reduction theorems.


    Completions of Period Maps 

    Dr. Colleen Robles
    Duke University

    10:30am - 11:30am
    Via Zoom
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    Abstract: I will describe (up to finite data) a completion/extension of a period map. The image of the extended period map is a compact Moishezon variety. (The “up to finite data” statement is due to the fact that we work with the Stein factorization of the period map. This allows us to replace the period map with a finite cover that has *connected* fibres.) This is joint work with M. Green and P. Griffiths.

    This result is part of a project (with MG, PG and Radu Laza) to construct completions of period mappings, and to apply those completions to study moduli. In the classical case (including ppav and K3 surfaces) we have “minimal" and “maximal" compactifications of the period space (the work of Satake—Baily—Borel and Ash—Mumford—Rapoport—Tai, respectively). And Borel’s extension theorem yields a completion of the period map to the minimal SBB compactification; in contrast the existence of an extension to the maximal AMRT compactification is a subtle problem, and the extension may not exist.

    The compactified image here has the flavor of the AMRT construction in the sense that it encodes the maximal amount of Hodge theoretical information (similar to the work of Kato—Usui). So it is striking (in comparison with the classical case), that we also obtain an extension.

    The key technical inputs here are new results on the “global structure” of period maps at infinity. Loosely one may think of these results as extending consequences of Schmid's nilpotent orbit theorem from local coordinate neighborhoods at infinity to larger neighborhoods. (These larger neighborhoods contain compact varieties: the period map remains proper when restricted to these sets.) Informally what one obtains are period matrix representations of the period map over these sets. This structure, along with the infinitesimal period relation and the work of Cattani—Deligne—Kaplan, allows us to apply Grauert’s result on the holomorphic equivalence relations to obtain the main result.


    Variation of Extension Data of Limit Mixed Hodge Structures

    Dr. Mark Green
    UCLA

    1:30pm - 2:30pm
    Via Zoom

    Abstract: The extension data in limit mixed Hodge structures carriesinteresting geometric information. I will give some examples of this and discuss restrictions on how this data can vary in a family from joint work with Phillip Griffiths and Colleen Robles. These restrictions go involve a mix of Lie theory and the integral structure.

  • Wednesday February 3rd, 2021

    Stratifications in the Moduli Space of Stable Surfaces

    Dr. Sönke Rollenske
    University of Marburg

    9:00am - 10:00am
    Via Zoom
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    Abstract: The Gieseker moduli space of surfaces of general type admits a modular compactification, the moduli space of stable surfaces. Our knowledge about the "new" surfaces in the boundary is still limited and I will discuss different possibilities to organise them, in particular a Hodge-theoretic approach proposed by Green, Griffiths, Laza, and Robles. Everything will be illustrated with examples and many pictures. Based on joint work with B. Anthes, M. Franciosi, R. Pardini.


    On Wormholes in the Moduli Space of Surfaces

    Dr. Giancarlo Urzúa
    Pontificia Universidad Católica de Chile

    10:30am - 11:30am
    Via Zoom
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    Abstract: I will describe a certain wormholing phenomena that takes place in the Kollár--Shepherd-Barron--Alexeev (KSBA) compactication of the moduli space of surfaces of general type. It occurs because of the appearance of particular extremal P-resolutions in surfaces on the KBSA boundary. I will state a general wormhole conjecture, and I will show we can prove it for a wide range of cases. There will be an emphasis on explaining open questions.


    Divisors in the Moduli Space of Surfaces 

    Dr. Julie Rana
    Lawrence University

    1:30pm - 2:30pm
    Via Zoom

    Abstract: The KSBA moduli space of stable surfaces is a natural compactification of Gieseker's moduli space of surfaces of general type. In contrast with the moduli space of curves, there are very few examples of divisors in KSBA moduli spaces. I will give examples of divisors corresponding to surfaces with cyclic quotient singularities. I will also discuss bounds that help to narrow the search for singular surfaces. Parts of my talk will touch on joint work with Urz\'ua and, separately, with Franciosi, Pardini, and Rollenske. 

     

  • Thursday February 4th, 2021

    I-Surfaces with One T-Singularity 

    Dr. Marco Franciosi
    Universita' di Pisa

    9:00am - 10:00am
    Via Zoom
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    Abstract: I will report on joint work with R.Pardini, J.Rana, S. Rollenske, on I-surfaces, i.e., stable surfaces with K^2=1, p_g=2 and q=0. I will give a review of past results on Gorenstein surfaces and I will consider normal I-surfaces with a T-singular point, giving applications to the analysis of the KSBA compactification of Gieseker's moduli space of canonical surfaces with K^2=1 and Euler characteristic =3.


    Deformations of Semi-Smooth Varieties 

    Dr. Rita Pardini
    Universita' di Pisa

    10:30am - 11:30am
    Via Zoom
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    Abstract: A variety X is semi-smooth if locally in the e'tale topology its singularities are either double crossing points (xy=0) or pinch points (x^2-y^2z=0). Alternatively, X is semi-smooth if it can be obtained from a smooth variety X' by gluing it along a smooth divisor D' via an involution g of D'. We describe explicitly in terms of the triple (X',D', g) the two sheaves on X that control its deformation theory, that is, the tangent sheaf T_X and the sheaf T^1_X:=ext^1(\Omega_X,O_X). As an application, we discuss the smoothability of the semi-smooth Godeaux surfaces (K^2=1, p_g=q=0). This is joint work with Barbara Fantechi and Marco Franciosi.


    Fixed Loci of Anti-Symplectic Involutions 

    Dr. Giulia Sacca
    Columbia University

    1:30pm - 2:30pm
    Via Zoom
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    Abstract: It is known that to some Fano manifolds whose cohomology looks like that of a K3 surface, one can associate via geometric constructions examples of hyperkahler manifolds. In this talk I will report on the first steps of a program whose aim is to reverse this construction: starting from a hyperkahler manifold how to recover geometrically a Fano manifold? This is joint work with L. Flapan, E. Macrì, and K.O. Grady.

     

  • Friday February 5th, 2021

    Top Weight Cohomology of M_g

    Dr. Sam Payne
    University of Texas

    9:00am - 10:00am
    Via Zoom
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    Abstract: I will discuss an approach to studying the top-graded piece of the weight filtration on open moduli spaces with suitable toroidal compactifications, inspired by tropical and nonarchimedean analytic geometry. One application of this approach is the recent proof, joint with Chan and Galatius, that the dimension of H^{4g-6}(M_g, Q) grows exponentially with g. This growth was unexpected and disproves conjectures of Church-Farb-Putman and Kontsevich.


    Deformations of Cusp Singularities of Algebraic Surfaces and Mirror Symmetry 

    Dr. Paul Hacking
    University of Massachusetts Amherst

    10:30am - 11:30am
    Via Zoom
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    Abstract: Cusp singularities arise on the degenerate surfaces at the boundary of the functorial compactification of the moduli space of surfaces of general type. The Milnor fiber of a smoothing of a cusp singularity is mirror to a log Calabi--Yau surface --- a pair (Y,D) consisting of a projective surface Y and a normal crossing divisor D such that K_Y+D=0. This leads to a precise conjectural description of the smoothing components of the deformation space of a cusp singularity in terms of the Kahler cones of the mirror surfaces and their monodromy groups. Based on joint work with Mark Gross and Sean Keel; Ailsa Keating; and graduate students Jennifer Li and Angelica Simonetti.