

2006  PhD, International School for Advanced Studies (SISSA), Trieste, Italy  2006  2007  Postdoc, Max Planck Institute for Mathematics, Bonn  2007  2008  Postdoc, Hausdorff Center for Mathematics, University of Bonn  2008  2010  Assistant Professor, University of Utah, Salt Lake City, UT, USA  2010  2011  Professor (W2, Bonn Junior Fellow), University of Bonn  2011  2014  Assistant Professor (tenuretrack), Ohio State University, Columbus, OH, USA  Since 2015  Associate Professor, Northeastern University, Boston, MA, USA 


My field of interest is algebraic geometry.
In particular, I work on derived categories of coherent sheaves on algebraic varieties.
In my past research, I studied autoequivalence groups of the derived category (in [3,2,4,5]) and stability conditions in the sense of Bridgeland (in [6,7,8]).
Recently, I am studying three dimensional projective varieties. More precisely, I am working on generalizations of the classical BogomolovGieseker inequality for stable sheaves on surfaces to stable complexes on threefolds, and applications of this to birational geometry.
I am also studying deformations of K3 categories embedded in derived categories of cubic fourfolds together with Martí Lahoz (University of Bonn).


DFG Collaborative Research Centre SFB/TR 45 “Periods, moduli spaces, and arithmetic of algebraic varieties”
Member
NSF grant, 2010  2013


Former Research Area E I belong to the research area E (and partly connected with C).
During my first year at HCM I wrote two preprints and completed two papers (one appeared in print and one is going to appear in 2011).
Preprints.
[11] “Bridgeland stability conditions on threefolds II: An application to Fujita's conjecture” (with A. Bayer, A. Bertram and Y. Toda), preprint 2011.
In this paper we show that the conjectural inequality for stable twoterm complexes we formulate in the preprint [12] has applications towards Fujita’s conjecture on threefolds. More precisely, the conjecture implies a Reidertype theorem for threefolds, that is very ample for L ample, and that is very ample when is trivial.
[12] “Bridgeland stability conditions on threefolds I: BogomolovGieseker type inequalities” (with A. Bayer and Y. Toda), preprint 2011. Also arXiv:1103.5010.
In this paper we construct new tstructures on the derived category of coherent sheaves on smooth projective threefolds. We conjecture that they give Bridgeland stability conditions near the large volume limit. We show that this conjecture is equivalent to a BogomolovGieseker type inequality for the third Chern character of certain stable complexes. We also conjecture a stronger inequality, and prove it in the case of projective space, and for various examples.
Finally, we prove a version of the classical BogomolovGieseker inequality, not involving the third Chern character, for stable complexes.
Published papers.
In [1], we study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions, and show that its translates via autoequivalences cover a whole connected component. We prove that this connected component is simplyconnected. We determine the group of autoequivalences preserving this connected component, which turns out to be closely related to .
Finally, we show that there is a submanifold isomorphic to the universal covering of a moduli space of elliptic curves with level structure. The morphism is equivariant, and is given by solutions of PicardFuchs equations. This result is motivated by the notion of stability and by mirror symmetry.
Finally, in [2], we study the general fibre of a formal deformation over the formal disk of a projective variety from the view point of abelian and derived categories. The abelian category of coherent sheaves of the general fibre is constructed directly from the formal deformation and is shown to be linear over the field of Laurent series. The various candidates for the derived category of the general fibre are compared.
If the variety is a surface with trivial canonical bundle, we show that the derived category of the general fibre is again a linear triangulated category with a Serre functor given by the square of the shift functor. 


[ 2] Daniel Huybrechts, Emanuele Macr\`\i, Paolo Stellari
Formal deformations and their categorical general fibre Comment. Math. Helv. , 86: (1): 4171 2011 DOI: 10.4171/CMH/217[ 3] Daniel Huybrechts, Emanuele Macr\`\i, Paolo Stellari
Derived equivalences of K3 surfaces and orientation Duke Math. J. , 149: (3): 461507 2009 DOI: 10.1215/001270942009043[ 4] Emanuele Macr\`\i, Paolo Stellari
Infinitesimal derived Torelli theorem for K3 surfaces With an appendix by Sukhendu Mehrotra Int. Math. Res. Not. IMRN (17): 31903220 2009 DOI: 10.1093/imrn/rnp049[5] Emanuele Macr\`\i, Paolo Stellari
Automorphisms and autoequivalences of generic analytic K3 surfaces J. Geom. Phys. , 58: (1): 133164 2008 DOI: 10.1016/j.geomphys.2007.10.002 [ 6] Emanuele Macr\`\i
Stability conditions on curves Math. Res. Lett. , 14: (4): 657672 2007 DOI: 10.4310/MRL.2007.v14.n4.a10[ 7] Daniel Huybrechts, Emanuele Macr\`\i, Paolo Stellari
Stability conditions for generic K3 categories Compos. Math. , 144: (1): 134162 2008 DOI: 10.1112/S0010437X07003065[ 8] Emanuele Macr\`\i, Sukhendu Mehrotra, Paolo Stellari
Inducing stability conditions J. Algebraic Geom. , 18: (4): 605649 2009 DOI: 10.1090/S1056391109005244[ 9] Claudio Bartocci, Emanuele Macr\`\i
Classification of Poisson surfaces Commun. Contemp. Math. , 7: (1): 8995 2005 DOI: 10.1142/S0219199705001647[ 10] Emanuele Macr\`\i, Marc NieperWiÃ? kirchen, Paolo Stellari
The module structure of Hochschild homology in some examples C. R. Math. Acad. Sci. Paris , 346: (1516): 863866 2008 DOI: 10.1016/j.crma.2008.05.017




2005  “Heterotic Strings, Derived Categories and Stacks”, Oberwolfach Research Institute for Mathematics (MFO), Oberwolfach  2006  “KIAS school on derived categories of coherent sheaves”, Korea Institute for Advanced Study (KIAS), Seoul, South Korea  2006  “Giornate di Geometria Algebrica e Argomenti Correlati VIII”, International School for Advanced Studies (SISSA), Trieste, Italy  2006  “Categorical constructions of primitive forms, II”, Research Institute for Mathematical Sciences (RIMS), Kyoto, Japan  2007  “Stability conditions, derived categories, etc”, Max Planck Institute for Mathematics, Bonn  2007  “Categorical aspects of algebraic geometry in mirror symmetry”, Research Institute for Mathematical Sciences (RIMS), Kyoto, Japan  2007  “Seminario di natale”, University of Milano, Italy  2008  “First CTS conference on vector bundles”, TIFR, Mumbai, India  2008  “Bundles, gerbes, and derived categories in string theory”, University of Salamanca, Spain  2009  “Derived categories and the Langlands programme”, FU Berlin  2009  “BPS state countings, stability structures, and derived algebraic geometry”, University of Hamburg  2009  “WAGS”, University of California, Los Angeles, CA, USA  2010  “School on birational geometry and moduli spaces”, University of Utah, Salt Lake City, UT, USA  2010  “Categorical methods in geometry and gauge theory”, Chern Institute, Tianjin, China  2011  “Derived Categories”, University of Tokyo, Japan 


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