Personale docente

General news
- Born: 01/28/1969, Rome.
- Bachelor degree in Mathematics, received on 06/19/1992 by Università di Firenze.
- PhD in Mathematics, Università di Firenze, 07/1997, advisor Paolo de Bartolomeis.
- 12/30/1996 – 10/31/2002: Assistant Professor, SSD GEOMETRIA, Università di Palermo and Parma.
- 11/2002-06/2011: Associate Professor, Università di Parma, SSD MAT03/GEOMETRIA.
- Current position: Full Professor, SSD MAT03/GEOMETRIA, Università di Parma (since 06/30/2014).
- He spent research periods and has been visiting professor at Universities of Michigan, Minnesota, Stanford, Notre Dame, Florida International University, Bourgogne, Ruhr, Edinburgh, Pais Vasco, Zaragoza, Centre International de Rencontres Mathématiques, Luminy, Grenoble, Trondheim, where he gave talks.
- He attended and was invited to give talks at international scientific meetings held in: Pisa at Centro de Giorgi, Firenze, Perugia, Trento, Kuehlungsborn, Tenerife, Les Rasses Sainte Croix, Bordeaux, Castro Urdiales, Torino, Benasque, Fribourg, Bucharest, Trieste, Bedlewo, Trondheim, Paris, Oberwolfach, Niigata, Pisa Scuola Normale Superiore, Grenoble.
- Local coordinator of the Research Unit in Parma, Project PRIN 2010, 2011 2015: "Varieta' reali e complesse: geometria, topologia e analisi armonica", Scientific coordinator Fulvio Ricci.
- Local coordinator of the Research Unit in Parma, Project PRIN 2017: "Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics", Scientific coordinator Filippo Bracci.
- Member of Collegio dei Docenti of PhD Program in Mathematics at the Universities of Pavia, Milano-Bicocca and INdAM.
- Presidente del Corso di Laurea triennale in Matematica, starting from December 2012 to December 2013.
- Chairman of the Dipartimento di Matematica e Informatica dell'Università di Parma, starting from December 2013 to December 2016.
- Chairman of the Dipartimento di Scienze Matematiche, Fisiche e Informatiche dell’Università di Parma, since 2020.
- President of “Centro Servizi E-Learning e Multimediali di Ateneo” since 2017.
- Chairman of Scuola Matematica Interuniversitaria, SMI, since January 2018.
- Member of the Committee of Abilitazione Scientifica Nazionale, Settore Concorsuale 01/A2 Geometria e Algebra.
- Editor-in-chief of Rivista di Matematica della Università di Parma, from 2009 to 2020.
- Member of the Governing Board of Annali di Matematica Pura ed Applicata, since January 2022.
- Referee for Duke Math. J., Intern. Math. Res. Notices, J. Geom. and Phys., Adv. In Geom., Ann. Mat. Pura Appl., Ann. SNS, J. Geom Anal., Proc. AMS, Inventiones Mathematicae, Ann. Inst. Fourier, Math. Z., Quarterly J. Math., Comm. Anal. Geom., Math. Annalen, Compositio Mathematica, Forum Mathematicum, Annals of Global Analysis and Geometry, Manuscripta Mathematica, Journal für die reine und angewandte Mathematik, Nagoya Math. J., Transactions of AMS, Journal of AMS, Selecta Mathematica, international J. of Math., American J. of Math..

Advisor of the PhD theses of:
Catellani, Firenze, 2008.
Angella, Pisa, 2013.
Rossi. Milano-Bicocca, 2013.
Tardini, Pisa, 2017.
Maschio, Parma, 2018.
Piovani, Pisa, 2021.
Sferruzza, III year of PhD program Parma

Research activity
A.T.’s research activity focuses mainly on two topics, both related to analytic and geometric properties of manifolds endowed with special structures.
The first one deals with the existence problem for special structures on (almost) complex and symplectic manifolds. In particular, he investigates on compact 2n-dimensional symplectic manifolds, endowed with a tamed almost complex structure and a d-bar closed nowhere vanishing (n,0)-form of constant length. These manifolds provide a possible generalization to the non holomorphic case of the notion of Calabi-Yau manifolds. In [de Bartolomeis, T., Ann. Inst. Fourier (Grenoble) 2006, de Bartolomeis, T., Internat. J. Math. 2006, Conti, A.T., Q. J. Math. 2007], the geometry of such manifolds is studied, when the almost compex structure is compatible with the symplectic form: then the notion of special Lagrangian submanifold is introduced, showing that the Maslov class of a special Lagrangian submanifold vanishes and several examples of these structures on non Kähler solvmanifolds are given.
The Tian-Todorov Theorem states that the moduli space of Calabi-Yau is totally unobstructed: in [de Bartolomeis, T., Ann. Inst. Fourier (Grenoble) 2013], an appropriated moduli space of almost complex deformations of a Calabi-Yau manifold is defined and it is proved that, under a suitable cohomological assumption, it is totally unobstructed too.
In [Fino, T., J. London Math. Soc. 2011] we study Hermitian metrics (on n-dimensional complex manifolds and, more generally, on complex orbifolds) whose fundamental form (its (n-2)-power respectively) is dd-bar closed. Such metrics have been studied by many authors, e.g., [Bismut, Math. Ann. 1989], [Gauduchon, C. R. Acad. Sci. Paris Sér. A-B 1977], [Jost, Yau, Acta Math. 1993].
The second research theme deals with the study of cohomological properties of almost complex manifolds. The study of such a topic is motivated, from one hand, by the aim of generalizing cohomological decomposition in the non Kähler context, and, on the other hand, by the study of the connections between the tamed symplectic cone and the compatible symplectic cone
Furthermore [Draghici, Li, Zhang, Internat. Math. Res. Notices, 2010] prove that on any compact almost complex 4-manifold the second de Rham cohomology group decomposes as the direct sum of the subgroups formed by cohomology classes admitting invariant and anti-invariant representatives respectively.
In [Angella,T., J. Symplectic Geom. 2011] we show that, on a given complex manifold, the existence of such a cohomological decomposition is not stable under the deformation of the complex structure.
In [Angella, A.T., Zhang, Proc. AMS (2014)], the almost-Kähler case is considered. Analogue notions in the symplectic settings are studied in [Angella, T., J. Symplectic Geom. (2014)].
For further results on anti-invariant cohomology and L2-cohomology see [Hind, Medori, T., J. Geom. Anal. 2015] and [Hind, T., J. Symplectic Geom. (2019)].
Towards the aim to better understand complex non-Kähler manifolds, we investigated Bott-Chern and Aeppli cohomologies. The research in this direction has been developed, (and is still developing,) under several aspects. The relation between Bott-Chern and Aeppli cohomologies and de Rham and Dolbeault cohomologies is investigated in [Angella, T., Invent. Math (2013)]. There, we proved an inequality à la Frölicher involving the dimensions of the Bott-Chern cohomology and the Betti numbers: the equality characterizes the validity of the d-dbar-Lemma, a central property in Kähler geometry. A generalization in the generalized-complex setting, here included the symplectic case, is studied in [Angella, T., J. Noncommut. Geom. (2015)]. The results are applied and refined for compact complex surfaces in [Angella, Dloussky, T., Ann. Mat. Pura Appl. 2016]. In [Angella, Suwa, Tardini, T. Complex Manifolds (2020)] the behaviour of $partialoverlinepartial$-Lemma under modifications of compact complex manifolds is investigated. For other results on cohomological properties of compact symplectic and complex manifolds we refer to [T., Wang, Internat. J. of Math. (2018)] e [Rollenske, T., Wang, Ann. Mat. Pura Appl. (2020)], [Sferruzza, T., J. of Geom. Physics, (2022)].
As an attempt to understand the algebraic structure of Bott-Chern cohomology, we introduced a notion of geometric formality with respect to the Bott-Chern cohomology. A first attempt is done in [Angella, T., J. Geom. Phys. (2015)], and continued in [Tardini, T., Ann. Mat. Pura App. (2017)].
For a compact almost complex manifold (M,J), endowed with a J-Hermitian g, one can define the Dolbeault and Bott-Chern Laplacian, using the natural operators del and del-bar an their adjoints. It turns out that such differential operators are elliptic and consequently their kernels are finite dimensional complex vector spaces. Let h^{p,q} be the dimension of the space of Dolbeault harmonic (p,q)-forms. In Problem 20, [Hirzebruch, Annals of Math. (1954)]: Kodaira and Spencer asked the following

Question I Let (M,J) be an almost-complex manifold. Choose a Hermitian metric on (M,J) and consider the numbers h^{p,q}. Is h^{p,q} independent of the choice of the Hermitian metric?
Holt and Zhang in [Harmonic Forms on the Kodaira-Thurston Manifold, (2020)] answered negatively to this question, showing that there exist almost-complex structures on the Kodaira-Thurston manifold such that the Hodge number h^{0,1} varies with different choices of Hermitian metrics. Furthermore, the authors showed that for a compact 4-dimensional almost-Kaehler manifold h^{1,1} is independent of the metric, and more precisely h^{1,1}=b_-+1, where b_- denotes the dimension of the space of the anti-self-dual harmonic 2-forms. In [HZ] the authors asked the following
Question II Let (M,J) be a compact almost-complex 4-dimensional manifold which admit an almost-Kaehler structure. Does it have a non almost-Kaehler Hermitian metric such that
h^{1,1} is different from b_-+1 ?
In [Tardini, T., to appear in Math. Res. Lett.] we prove that:
h^{1,1}=b_-, respectively h^{1,1}=b_- + 1 if the metric is strictly locally conformally Kaehler respectively globally conformally Kaehler. For further results see [Piovani, T., Math. Zeit. (2022)]. In [Chen, Zhang, Kodaira dimension of almost complex manifolds I, II] the authors introduced the notion of Kodaira dimension of almost complex manifolds. In [Cattaneo, Nannicini, T., Internat. J. Math. (2021)] we provide several computations of Kodaira dimension of almost complex manifolds in dimension 4 and 6.

Total Publications: 79
Total citations: 648
h-index SCOPUS: 15

Total Publications: 72
Total citations: 580
h-index WOS: 14