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Hodge ideals. (English) Zbl 1442.14004

Memoirs of the American Mathematical Society 1268. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3781-7/pbk; 978-1-4704-5509-5/ebook). v, 80 p. (2019).
Let \(X\) be a nonsingular complex variety of dimension \(n>0\) and let \(D\) be a reduced divisor on \(X\). Then \(\mathcal O_X(*D) = \cup_{k\geq 0} O_X(k D)\) is the left \(\mathcal D_X\)-module of functions with poles of any orders along the divisor. From the general theory (see, e.g., [M. Saito, Publ. Res. Inst. Math. Sci. 26, No. 2, 221–333 (1990; Zbl 0727.14004); Math. Ann. 295, No. 1, 51–74 (1993; Zbl 0788.32025)] it follows that this module is endowed with a natural Hodge filtration \(F_k\mathcal O_X(*D)\) contained in the usual pole order filtration \(F_k\mathcal O_X(*D) \subseteq \mathcal O_X ((k + 1)D)\), \(k \geq 0\). In particular, a sequence of coherent sheaves of ideals, called Hodge ideals and denoted by \(\mathcal I_k(D)\), \(k\geq 0\), is determined by the equality \(F_k\mathcal O_X(*D) = \mathcal O_X ((k + 1)D)\otimes \mathcal I_k(D)\). For instance, \(\mathcal I_0(D)\) can be identified with a multiplier ideal (see Ch.9 in [R. Lazarsfeld, Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals. Berlin: Springer (2004; Zbl 1093.14500)]). In a certain sense, these ideals measure the difference between both filtrations.
The authors describe in detail the basic properties of Hodge ideals using standard tools and methods from birational geometry, such as reducing to simple normal crossings, using blow-ups and logarithmic resolutions, etc.
Thus, they compute \(\mathcal I_k(D)\) in several particular cases explicitly including the case where \(D\) is a simple normal crossing divisor or the pair \((X,D)\) is \(k\)-log-canonical (i.e., \(\mathcal I_\ell(D) = \mathcal O_X\) for all \(\ell =0,\dots,k\)). Among other things, they also prove that for an ordinary singular point of multiplicity \(m\geq 2\) the equality \(\mathcal I_k(D)_x = \mathcal O_{X,x}\) holds iff \(k\leq [m/n]-1\). Then, a series of applications, concerning singularities and classical Hodge theory in the context of the theory of projective varieties, is discussed. In the case of principally polarized abelian varieties, the authors obtain an upper bound for the multiplicity of points on theta divisors whose singularities are isolated, etc. An interested reader can find some further applications of Hodge ideals in the subsequent paper [M. Mustaţǎ and M. Popa, J. Éc. Polytech., Math. 6, 283–328 (2019; Zbl 1427.14045)].
The content of the book can be summarized as follows. It consists of nine sections, including an introduction, preliminary materials, an appendix, and a list of bibliographies. The introduction contains a brief overview of the results of the authors and related topics studied previously by many others. The preliminary section discusses some aspects of the general theory of filtered \(\mathcal D\)-modules. The main topics are then studied in the following order. The third section describes the basic properties of Saito’s Hodge filtration and Hodge modules with applications to the Hodge theory of hypersurfaces and their complements. The next three sections contain a birational definition of Hodge ideals, a description of their basic properties, including behavior under pullback and restriction to hypersurfaces, vanishing conditions along exceptional divisors, and so on. In the seventh section a number of vanishing theorems for Hodge ideals is proved, including vanishing on \(\mathbb P^n\), on toric varieties and polarized abelian varieties. Then several applications are discussed for singularities of theta divisors and singular points on ample divisors on abelian varieties. In the appendix, the authors consider several local vanishing statements for higher direct images of bundles of differential forms with logarithmic poles (see, e.g., [H. Esnault and E. Viehweg, Lect. Notes Math. 947, 241–250 (1982; Zbl 0493.14012)], [M. Saito, “Hodge ideals and microlocal \(V\)-filtration”, Preprint, arXiv:1612.08667]).
This book is a well-written treatment of a relevant contemporary topic, which many mathematicians can use as an opportunity to learn and evaluate some new ideas and results, to understand how they interact with previous ones. On the other hand, the main part of the book is almost self-contained and does not require much knowledge from the reader. That is why it can also be recommended for graduate students and beginners who are interested in the theory of \(\mathcal D\)-modules and applications.

MSC:

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14F17 Vanishing theorems in algebraic geometry
32C38 Sheaves of differential operators and their modules, \(D\)-modules
14D07 Variation of Hodge structures (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
32S25 Complex surface and hypersurface singularities
14F18 Multiplier ideals
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References:

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