Research interest


I am interested in algebraic geometry, more precisely in the following topics:
(1) enumerative invariants in algebraic geometry, and their symplectic counterparts;
(2) geometry of K3 surfaces;
(3) vector bundles on projective varieties: splitting criteria and exceptional sequences;
(4) partial ampleness for subvarieties of projective varieties.




Publications


[16]

M. Halic, Restriction properties for the Krull-Schmidt decomposition of vector bundles. Internat. J. Math. (accepted).
We compare the Krull-Schmidt decompositions of a vector bundle on a projective variety and that of its restriction to an ample subvariety. The main result is a general Horrocks-type criterion for Krull-Schmidt decompositions. Finally, we obtain a simple splitting criterion for vector bundles on partial flag varieties.


[15]

M. Halic, Semi-stable vector bundles on fibred varieties. Eur. J. Math. (accepted).
We introduce and investigate a semi-stability concept relative to morphisms π:Y→X between projective manifolds, different of Maruyama-Simpson. For torsion-free sheaves on Y and for appropriate polarizations, we relate the (semi-)stability on Y to the slope (semi-)stability along the generic fibre of π.
We apply the result to describe in great detail the moduli spaces of semi-stable vector bundles on Hirzebruch surfaces and on threefolds which are P2-bundles over P1. We establish their irreducibility and rationality, an intensely investigated issue.


[14]

M. Halic, Splitting criteria for vector bundles induced by restrictions to divisors. Michigan Math. J. (accepted).
Inhere we obtain splitting and triviality criteria for vector bundles by restricting them to partially positive divisors. In particular, we deduce an algorithmic method to check the splitting of vector bundles on arbitrary products of minuscule rational homogeneous varieties; no similar criteria were known even for a product of two Grassmannians.


[13]

M. Halic, R. Tajarod, About the cohomological dimension of certain stratified varieties. Proc. Amer. Math. Soc. 145 (2017), 5157 - 5167.
We determine an upper bound for the cohomological dimension of the complement of a closed subset in a projective variety which possesses an appropriate stratification. We apply the result to several particular cases, including the Bialynicki-Birula stratification; in this latter case, the bound is optimal.


[12]

M. Halic, Splitting criteria for vector bundles on minuscule homogeneous varieties. Transform. Groups 22 (2017), 753 - 765.
A vector bundle on a minuscule homogeneous variety splits into a direct sum of line bundles if and only if its restriction to the union of the two-dimensional Schubert subvarieties splits. The result is illustrated with examples.


[11]

M. Halic, R. Tajarod, On the q-ampleness of the tensor product of two line bundles. manuscripta math. 151 (2016), 177 - 182.
The tensor product of two line bundles, one being q-ample and the other with sufficiently low-dimensional stable base locus, is still q-ample.


[10]

M. Halic, Erratum to: Modular properties of nodal curves on K3 surfaces. Math. Z. 280 (2015), 1203 - 1211.
I correct an error in [8] and I also obtain a lower bound for the multiple Seshadri constants of general K3 surfaces.


[9]

M. Halic, R.Tajarod, A cohomological splitting criterion for locally free sheaves on arithmetically Cohen-Macaulay surfaces. Math. Proc. Cambridge Philos. Soc. 155 (2013), 517 - 527.


[8]

M. Halic, Modular properties of nodal curves on K3 surfaces. Math. Z. 270 (2012), 871 - 887.


[7]

M. Halic, Cohomological properties of invariant quotients of affine spaces. J. London Math. Soc. 82 (2010), 376 - 394.


[6]

M. Halic, A remark about the rigidity of curves on K3 surfaces. Collectanea Math. 61 (2010), 323 - 336.


[5]

M. Halic, Gauge theoretical/Hamiltonian Gromov-Witten invariants of toric varieties. Math. Z. 252 (2006), 157 - 208.


[4]

M. Halic, S. Stupariu, Rings of invariants for representations of quivers. C.R. Acad. Sci. Paris 340 (2005), 135 - 140.


[3]

M. Halic, Families of toric varieties. Math. Nachr. 261-262 (2003), 60 - 84.


[2]

M. Halic, GW invariants and invariant quotients. Comment. Math. Helv. 77 (2002), 145 - 191.


[1]

M. Halic, On the geography of symplectic 6-manifolds. manuscripta math. 99 (1999), 371 - 381.


[0]

Ph.D. thesis ''Symplectique vs Projectif''.



Pre-publications


M. Halic, Subvarieties with q-ample normal bundle and q-ample subvarieties.
The goal of this article is twofold. On one hand, we study the properties of subvarieties of projective varieties which possess partially ample normal bundle; we prove that they are G2 in the ambient space. This generalizes results of Hartshorne and Bădescu-Schneider. We work with the cohomological partial ampleness introduced by Totaro.
On the other hand, we define the concept of a partially ample subvariety, which is analogous to the notion of an ample subvariety introduced by Ottem, and study their properties. We prove that partially ample subvarieties enjoy the stronger G3 property.
Both cases are illustrated with examples.


M. Halic, Vector bundles on projective varieties which split along q-ample subvarieties.
Let Y be a subvariety of a smooth projective variety X, and V a vector bundle on X. Given that the restriction of V to Y splits into a direct sum of line bundles, we ask whether V splits on X.
I answer this question in affirmative if holds: Y is a q-ample subvariety of X (for appropriate q), it admits sufficiently many embedded deformations, and is very general within its own deformation space. The result goes beyond the previously known splitting criteria for vector bundles corresponding to restrictions. It allows to treat in a unified way examples arising in totally different situations.
I discuss the particular cases of zero loci of sections in globally generated vector bundles, on one hand, and sources of multiplicative group actions (corresponding to Bialynicki-Birula decompositions), on the other hand. Finally, I elaborate on the symplectic and orthogonal Grassmannians; I prove that the splitting of any vector bundle on them can be read off from the restriction to a low dimensional `sub'-Grassmannian.

M. Halic, Strong exceptional sequences of vector bundles over a certain class of Fano varieties.
Motivated by a problem posed by King, we give an algorithmic method to construct strong exceptional sequences of vector bundles on projective varieties which are obtained as geometric invariant quotients of affine spaces by linear actions of reductive groups. Several explicit examples are worked out.

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