Christian Schnell

General Information

Research Interests

I study the geometry and topology of complex algebraic varieties. My research focuses on Hodge theory, and on applications of mixed Hodge modules in algebraic geometry. I am a member of the algebraic geometry group.

Spring 2023 Teaching

This semester, I am teaching MAT 512 (Algebra for Teachers).

Mathematical Biography

I received my Ph.D. from Ohio State University in 2008, with Herb Clemens; after that, I was a postdoc at the University of Illinois at Chicago and at Kavli IPMU near Tokyo. Since 2012, I have been working at Stony Brook University. During the academic year 2013/14, I was on leave at the University of Bonn. During the academic year 2015/16, I was supported by a Centennial Fellowship from the American Mathematical Society. During the academic year 2021/22, I was on leave, partially supported by a Simons Fellowship.

Coordinates

Mailing Address

Department of Mathematics
Stony Brook University
Stony Brook, NY 11794-3651

Contact Information

Office: Math Tower 4-110 (map)
Phone: (631) 632-8250
E-Mail: christian.schnell@stonybrook.edu

Preprints

1. Hodge theory and Lagrangian fibrations on holomorphic symplectic manifolds
2. Singular hermitian metrics and the decomposition theorem of Catanese, Fujita, and Kawamata
   [with L. Lombardi]
3. Degenerating complex variations of Hodge structure in dimension one
   [with C. Sabbah]
4. A log resolution for the theta divisor of a hyperelliptic curve
   [with R. Yang]
5. Singular metrics and a conjecture by Campana and Peternell

Published Papers

1. On the behavior of Kodaira dimension under smooth morphisms
   to appear in Algebraic Geometry
   [with M. Popa]
2. Finiteness for self-dual classes in integral variations of Hodge structure
   to appear in Épijournal de Géométrie Algébrique
   [with B. Bakker, Th. Grimm and J. Tsimerman]
3. Correction to "Height bounds and the Siegel property"
   to appear in Algebra Number Theory
   [with M. Orr]
4. Hodge modules and singular hermitian metrics
   to appear in Math. Z.
   [with R. Yang]
5. The Fourier-Mukai transform made easy
   Pure Appl. Math. Quart. 18 (2022), no. 4, 1749–1770
6. Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities
   J. Amer. Math. Soc. 34 (2021), no. 2, 315–368
   [with S. Kebekus]
7. Pushforwards of pluricanonical bundles under morphisms to abelian varieties
   J. Eur. Math. Soc. 22 (2020), 2511–2536
   [with L. Lombardi and M. Popa]
8. On a theorem of Campana and Păun
   Épijournal de Géométrie Algébrique, Sep. 1, 2017, Vol. 1
9. Vanishing theorems for perverse sheaves on abelian varieties, revisited
   Selecta Math. (N.S.) 24 (2018), no. 1, 63–84
   [with B. Bhatt and P. Scholze]
10. Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Păun
    Contemp. Math. 712 (2018), 143–195
    [with Ch. Hacon and M. Popa]
11. Viehweg's hyperbolicity conjecture for families with maximal variation
    Invent. Math. 208 (2017), no. 3, 677–713
    [with M. Popa]
12. Hodge modules on complex tori and generic vanishing for compact Kähler manifolds
    Geometry & Topology 21 (2017), 2419–2460
    [with G. Pareschi and M. Popa]
13. Fields of definition of Hodge loci
    London Math. Soc. Lecture Note Ser. 427 (2016), 275–291
    [with M. Saito]
14. On Saito's vanishing theorem
    Math. Res. Lett. 23, No. 2 (2016), pp. 499–527
15. On direct images of pluricanonical bundles
    Algebra Number Theory 8 (2014), no. 9, 2273–2295
    [with M. Popa]
16. The Cremmer-Scherk Mechanism in F-theory Compactifications on K3 Manifolds
    J. High Energ. Phys. 1405, 135 (2014)
    [with M. Douglas and D. Park]
17. Holonomic D-modules on abelian varieties [Erratum]
    Inst. Hautes Études Sci. Publ. Math. 121 (2015), no. 1, 1–55
18. Kodaira dimension and zeros of holomorphic one-forms
    Ann. of Math. 179 (2014), no. 3, 1109–1120
    [with M. Popa]
19. Torsion points on cohomology support loci: from D-modules to Simpson's theorem
    London Math. Soc. Lecture Note Ser. 417 (2015), 405–421
20. Weak positivity via mixed Hodge modules
    Contemp. Math. 647 (2015), 129–137
21. Generic vanishing theory via mixed Hodge modules
    Forum Math. Sigma 1 (2013), e1, 60pp.
    [with M. Popa]
22. Moduli of products of stable varieties
    Compositio Math. 149 (2013), no. 12, 2036–2070
    [with B. Bhatt, Z. Patakfalvi and W. Ho]
23. The zero locus of the infinitesimal invariant
    Fields Inst. Commun. 67 (2013), 589–602
    [with G. Pearlstein]
24. Residues and filtered D-modules
    Math. Annalen 354 (2012), no. 2, 727–763
25. Complex-analytic Néron models for arbitrary families of intermediate Jacobians
    Invent. Math. 188 (2012), no. 1, 1–81
26. The fundamental group is not a derived invariant
    EMS Ser. Congr. Rep. 8 (2011), 279–285
27. Canonical cohomology as an exterior module
    Pure Appl. Math. Quart. 7 (2011), no. 4, 1529–1542
    [with R. Lazarsfeld and M. Popa]
28. Derived invariance of the number of holomorphic 1-forms and vector fields
    Ann. Sci. Éc. Norm. Supér (4) 44 (2011), no. 3, 527–536
    [with M. Popa]
29. A variant of Néron models over curves
    Manuscripta Math. 134 (2011), no. 34, 359–375
    [with M. Saito]
30. Local duality and polarized Hodge modules
    Publ. Math. Res. Inst. Sci. 47 (2011), no. 3, 705–725
31. Primitive cohomology and the tube mapping
    Math. Z. 268 (2011), no. 3–4, 1069–1089
32. The locus of Hodge classes in an admissible variation of mixed Hodge structure
    C. R. Acad. Sci. Paris, Ser. I 348 (2010), 657–660
    [with P. Brosnan and G. Pearlstein]
33. Two observations about normal functions
    Clay Math. Proc. 9 (2010), 75–79

Unpublished Papers

1. Graded duality for filtered D-modules
   [with M. Saito]
2. The extended locus of Hodge classes
3. Surfaces with big anticanonical class
   [with D. Chen]
4. Canonical extensions of local systems
5. Idempotent ultrafilters and polynomial recurrence

Ph.D. Thesis

• The boundary behavior of cohomology classes and singularities of normal functions
  [Ohio State University, 2008]

Lecture Notes

• A graduate course on Algebraic D-modules (Stony Brook, 2019)
• A graduate course on Basic topology (Stony Brook, 2014 and 2022)
• A talk on Zeros of holomorphic one-forms (Paris, 2014)
• Two lectures on Mixed Hodge modules (Sanya, 2013)
• A graduate course on the Generic vanishing theorem (Stony Brook and Bonn, 2013)
• Five lectures on Absolute Hodge classes (Trieste, 2010)
  [with F. Charles]
• A graduate course on Complex manifolds (Chicago, 2010)
• Two lectures on Mumford-Tate groups (Trento, 2009)
• A seminar on Deformations of complex structure (Columbus, 2005)
• A seminar about Picard-Fuchs equations (Columbus, 2004)
• A talk on the Gauss-Bonnet theorem (Columbus, 2004)
• A lecture on the De Giorgi-Nash-Moser estimates (Columbus, 2002)

Mixed Hodge modules

Claude Sabbah and I are adapting Saito's theory of mixed Hodge modules to the case of complex coefficients. A collection of notes (in progress) can be found on Claude's website.

Miscellaneous

• The set of (-1)-curves on a certain class of elliptic surfaces
• A condition for an ideal in a power series ring to be generated by constants
• The Serre construction in codimension two
• Stable approximations of certain vector bundles
• Hypersurfaces containing a given subvariety
• Computing cohomology of local systems

Upcoming Events

• Complex Analysis
  MFO, Oberwolfach, April 9 to April 14, 2023
• AGNES Stony Brook 2023
  Stony Brook University, April 28 to April 30, 2023
• Alpine meeting on nonpositive curvature in Kähler geometry
  Centre Paul-Langevin, June 5 to June 9, 2023
• Noether-Lefschetz and Hodge loci
  IMPA, Rio de Janeiro, September 8 to January 8, 2025